Weak-disorder limit at criticality for directed polymers on hierarchical graphs

TL;DR

This paper proves a distributional limit theorem for directed polymer partition functions on hierarchical graphs at criticality, revealing new behavior in the marginally relevant disorder case with joint scaling of layers and temperature.

Contribution

It establishes the first distributional convergence result for the critical marginally relevant case of directed polymers on hierarchical graphs, using a novel Stein's method approach.

Findings

Distributional convergence of partition functions at criticality.

Limit theorem applies to models with edge and vertex disorder.

Analysis introduces a perturbative Stein's method at a critical scale.

Abstract

We prove a distributional limit theorem conjectured in [Journal of Statistical Physics 174, No. 6, 1372-1403 (2019)] for partition functions defining models of directed polymers on diamond hierarchical graphs with disorder variables placed at the graphical edges. The limiting regime involves a joint scaling in which the number of hierarchical layers, $n\in \mathbb{N}$, of the graphs grows as the inverse temperature, $\beta\equiv \beta(n)$, vanishes with a fine-tuned dependence on $n$. The conjecture pertains to the marginally relevant disorder case of the model wherein the branching parameter $b \in \{2,3,\ldots\}$ and the segmenting parameter $s \in \{2,3,\ldots\}$ determining the hierarchical graphs are equal, which coincides with the diamond fractal embedding the graphs having Hausdorff dimension two. Unlike the analogous weak-disorder scaling limit for random polymer models on hierarchical graphs in the disorder relevant $b<s$ case (or for the (1+1)-dimensional polymer on the rectangular lattice), the distributional convergence of the partition function when $b=s$ cannot be approached through a term-by-term convergence to a Wiener chaos expansion, which does not exist for the continuum model emerging in the limit. The analysis proceeds by controlling the distributional convergence of the partition functions in terms of the Wasserstein distance through a perturbative generalization of Stein's method at a critical step. In addition, we prove that a similar limit theorem holds for the analogous model with disorder variables placed at the vertices of the graphs.

Equations

$$\displaystyle\mathbf{M}^{\omega}_{\beta,n}(p)\,=\,\frac{1}{|\Gamma^{b,s}_{n}|}\frac{e^{\beta H_{n}^{\omega}(p)}}{\mathbb{E}\big{[}e^{\beta H_{n}^{\omega}(p)}\big{]}}\quad\quad\text{ for path energy }\quad\quad H_{n}^{\omega}(p)\,:=\,\sum_{a\in p}\omega_{a}\,,$$

$$\displaystyle W_{n}^{\omega}(\beta)\,:=\,\mathbf{M}^{\omega}_{\beta,n}\big{(}\Gamma^{b,s}_{n}\big{)}\,=\,\frac{1}{|\Gamma_{n}^{b,s}|}\sum_{p\in\Gamma_{n}^{b,s}}\prod_{a\in p}\frac{e^{\beta\omega_{a}}}{\mathbb{E}[e^{\beta\omega_{a}}]}$$

$$W_{n+1}^{\omega }(\beta )\stackrel{d}{=}\frac{1}{b}\sum _{i=1}^b\prod _{j=1}^s{W}_n^{(i,j)}(\beta ),$$

$$M_{b,s}(x):=$$

$$=$$

$$\displaystyle\beta_{n,r}^{b,s}\,=\,\sqrt{r}\Big{(}\frac{b}{s}\Big{)}^{n/2}\,+\,\mathit{o}\bigg{(}\Big{(}\frac{b}{s}\Big{)}^{n/2}\bigg{)}\,.$$

$${\mathbf{W}}_{\frac{s}{b}r}\stackrel{d}{=}\frac{1}{b}\sum _{i=1}^b\prod _{j=1}^s{\mathbf{W}}_r^{(i,j)},$$

$$\displaystyle\beta_{n,r}^{(b)}\,:=\,\frac{\kappa_{b}}{\sqrt{n}}\,-\,\frac{\kappa_{b}^{2}\tau}{2n}\,+\,\frac{\kappa_{b}\eta_{b}\log n}{2n^{\frac{3}{2}}}\,+\,\frac{\kappa_{b}r+\kappa_{b}^{3}(\frac{5}{4}\tau^{2}-\frac{7}{12}\tau^{\prime}-\frac{1}{2})}{2n^{\frac{3}{2}}}\,+\,\mathit{o}\Big{(}\frac{1}{n^{\frac{3}{2}}}\Big{)}\,,$$

$${\kappa }_b:=\sqrt{\frac{2}{b-1}}\text{and}{\eta }_b:=\frac{b+1}{3(b-1)}.$$

$$\displaystyle\varrho_{n}\big{(}\beta_{n,r}^{(b)}\big{)}\,=\,$$

$$\displaystyle\varrho_{0}\big{(}\beta_{n,r}^{(b)}\big{)}\,:=\,$$

$$\displaystyle W_{n}^{\omega}\big{(}\hat{\beta}/\sqrt{n}\big{)}\,\stackrel{{\scriptstyle d}}{{\approx}}\,1\,+\,\frac{1}{\sqrt{n}}\cdot\mathcal{N}\bigg{(}0,\frac{1}{1/\hat{\beta}^{2}-1/\kappa_{b}^{2}}\bigg{)}$$

$$\displaystyle W_{n}^{\omega}\big{(}\hat{\beta}/\sqrt{n}\big{)}\,\stackrel{{\scriptstyle d}}{{\approx}}1\,+\,\frac{1}{\sqrt{\log n}}\cdot\mathcal{N}\Big{(}0,\frac{6}{b+1}\Big{)}$$

$$W_n^{\omega }(\hat{\beta }/\sqrt{n})$$

$$R_{b}(r)\,=\,-\frac{\kappa_{b}^{2}}{r}\,+\,\frac{\kappa_{b}^{2}\eta_{b}\log(-r)}{r^{2}}\,+\,\mathit{O}\bigg{(}\frac{\log^{2}(-r)}{|r|^{3}}\bigg{)}\,.$$

$$R^{\prime}_{b}(r)\,=\,\lim_{n\rightarrow\infty}\frac{\kappa_{b}^{2}}{n^{2}}\prod_{k=1}^{n}\big{(}1+R_{b}(r-k)\big{)}^{b-1}\,.$$

$$\displaystyle x^{n,r}=\kappa_{b}^{2}\bigg{(}\frac{1}{n}+\frac{\eta_{b}\log n}{n^{2}}+\frac{r}{n^{2}}\bigg{)}\,+\,\mathit{o}\Big{(}\frac{1}{n^{2}}\Big{)}\,,$$

$$\displaystyle\mathbb{E}\Big{[}\big{(}W_{n}^{\omega}\big{(}\beta_{n,r}^{(b)}\big{)}-1\big{)}^{m}\Big{]}\hskip 14.22636pt\stackrel{{\scriptstyle n\rightarrow\infty}}{{\longrightarrow}}\hskip 14.22636ptR^{(m)}_{b}(r)\,.$$

$$R_{b}^{(m)}(r+1)\,=\,P_{m}\big{(}R_{b}^{(2)}(r),R_{b}^{(3)}(r),\ldots,R_{b}^{(m)}(r)\big{)}\,.$$

$$W_{n}^{\omega}\big{(}\beta_{n,r}^{(b)}\big{)}\hskip 28.45274pt\Longrightarrow\hskip 28.45274ptL_{r}^{(b)}$$

$${\mathbf{W}}_{r+1}\stackrel{d}{=}\frac{1}{b}\sum _{1\le i\le b}\prod _{1\le j\le b}{\mathbf{W}}_r^{(i,j)}.$$

$${\hat{W}}_n^{\omega }(\beta ):=\frac{1}{|{\Gamma }_n^{b,s}|}\sum _{p\in {\Gamma }_n^{b,s}}\prod _{a\in p}\frac{e^{\beta {\omega }_a}}{\mathbb{E}[e^{\beta {\omega }_a}]},$$

$$\displaystyle\widehat{W}_{n+1}^{\omega}(\beta)\,\stackrel{{\scriptstyle d}}{{=}}\,\frac{1}{b}\sum_{i=1}^{b}\Bigg{(}\prod_{j=1}^{s}\widehat{W}_{n}^{(i,j)}(\beta)\Bigg{)}\Bigg{(}\prod_{\ell=1}^{s-1}\frac{e^{\beta\omega_{i,\ell}}}{\mathbb{E}\big{[}e^{\beta\omega_{i,\ell}}\big{]}}\Bigg{)}\,,$$

$$\displaystyle\widehat{\beta}_{n,r}^{(b)}\,=\,\frac{\widehat{\kappa}_{b}}{n}\,+\,\frac{\widehat{\kappa}_{b}\eta_{b}\log n}{n^{2}}\,+\,\frac{\widehat{\kappa}_{b}r-\widehat{\kappa}_{b}^{2}\frac{\tau}{2}}{n^{2}}\,+\,\mathit{o}\Big{(}\frac{1}{n^{2}}\Big{)}\,,$$

$$\displaystyle\widehat{W}_{n}^{\omega}\big{(}\hat{\beta}/n\big{)}\,\stackrel{{\scriptstyle d}}{{\approx}}\,1\,+\,\frac{1}{n}\cdot\mathcal{N}\big{(}0,\upsilon_{b}(\hat{\beta})\big{)}$$

$$\displaystyle\widehat{W}_{n}^{\omega}\big{(}\hat{\beta}/n\big{)}\,\stackrel{{\scriptstyle d}}{{\approx}}1\,+\,\frac{1}{\sqrt{\log n}}\cdot\mathcal{N}\Big{(}0,\frac{6}{b+1}\Big{)}$$

$${\hat{W}}_n^{\omega }(\hat{\beta }/n)$$

$${\partial }_t{\mathcal{Z}}_{\hat{\beta }}=\frac{1}{2}{\partial }_x^2{\mathcal{Z}}_{\hat{\beta }}+\hat{\beta }W{\mathcal{Z}}_{\hat{\beta }},{\mathcal{Z}}_{\hat{\beta }}(t,x^{\prime };t,x)={\delta }_0(x^{\prime }-x).$$

$$M_{\hat{\beta }}(dp)=e^{\hat{\beta }\hat{W}(p)-\frac{{\hat{\beta }}^2}{2}\mathbb{E}[\hat{W}(p)]}\mathbf{P}(dp)\text{for}p\in C([0,1]),$$

$$\displaystyle\mathbb{E}\big{[}M_{\hat{\beta}}(dp)\big{]}\,=\,\mathbf{P}(dp)\hskip 28.45274pt\text{and}\hskip 28.45274pt\mathbb{E}\big{[}M_{\hat{\beta}}(dp)M_{\hat{\beta}}(dq)\big{]}\,=\,e^{\beta^{2}T(p,q)}\mathbf{P}(dp)\mathbf{P}(dq)\,.$$

$$\displaystyle Z_{L,\beta_{L}}\quad\stackbin[L\rightarrow\infty]{\mathcal{L}}{\Longrightarrow}\quad\mathcal{Z}_{\hat{\beta}}\,:=\,\begin{cases}\textup{exp}\big{\{}\sigma_{\hat{\beta}}\chi-\frac{1}{2}\sigma_{\hat{\beta}}^{2}\big{\}}&\hat{\beta}<1\,,\\ 0&\hat{\beta}\geq 1\,,\end{cases}$$

$${\mathcal{Z}}_{Lt,{\beta }_{L,r}}(dx):=\frac{1}{L}\sum _{y\in \frac{1}{\sqrt{L}}{\mathbb{Z}}^2}{Z}_{Lt,{\beta }_{L,r}}(y\sqrt{L}){\delta }_y(x),$$

$$\displaystyle\mathbb{E}\Bigg{[}\bigg{(}\int_{{\mathbb{R}}^{2}}\phi(x)\mathcal{Z}_{t,r}(dx)\bigg{)}^{2}\Bigg{]}\,=\,\int_{{\mathbb{R}}^{2}\times{\mathbb{R}}^{2}}\phi(z)\phi(z^{\prime})K_{t,r+\alpha}(z-z^{\prime})dzdz^{\prime}\,,$$

Full text

1 Introduction

In probabilistic frameworks, a disordered system usually refers to a relatively simple and familiar random object whose “pure” probabilistic law is distorted through its coupling to a random “environment” formed by an array of random variables (local impurities) or a random field. If the size of the model depends on a parameter $L\in\mathbb{N}$, a central question for these disordered systems is whether typical realizations of the random environment create either a qualitative or only a quantitative change in the law of the random object as $L\nearrow\infty$. For a given coupling strength $\beta\in[0,\infty)$ of the system to the environment, these large-scale behaviors are respectively referred to as strongly disordered or weakly disordered. A disordered system is further classified as disorder relevant if it exhibits strong disorder for any fixed $\beta$ as the system size grows or as disorder irrelevant otherwise. Finally, models at the border between the disorder relevant and disorder irrelevant regimes are referred to as marginally relevant or marginally irrelevant, and these boundary models manifest anomalous finer scaling behavior as the coupling strength vanishes.

One of the most closely studied disorder models is the directed polymer in a random environment, which usually refers to a $d$-dimensional simple symmetric random walk (SSRW) whose trajectories are reweighed within a Gibbsian formalism that depends on an inverse temperature parameter, $\beta$, and an array of centered i.i.d. random variables labeled by the time-space lattice $\{1,\ldots,L\}\times{\mathbb{Z}}^{d}$ for a polymer length $L\in\mathbb{N}$. The parameter $\beta$ effectively controls the strength of the polymer’s coupling to the environment, and $\beta=0$ corresponds to a pure SSRW. Established results in this field imply that the $(d+1)$-dimensional polymer model is disorder relevant when $d=1$, marginally relevant when $d=2$, and disorder irrelevant in all higher dimensions; see Comets’s recent book [14].

In this article, we prove a distributional limit theorem for partition functions defined from a hierarchical model for directed polymers in a random environment for which the disorder is marginally relevant. Our limiting regime, which involves a joint scaling wherein the number of hierarchical layers of the model grows while the disorder strength decays to zero, is similar to the critical weak-disorder scaling regime for $(2+1)$-dimensional polymers proposed by Caravenna, Sun, and Zygouras in [7, 9]. While [9] proves the existence of a subsequential distributional limit of the partition functions within this critical scaling regime and fully characterizes the correlation structure of any such limit, the uniqueness of the subsequential distributional limit currently remains open. Although the hierarchical symmetry of the model considered in this article makes a detailed limit analysis within the critical weak-disorder regime less difficult than for the rectangular lattice polymer model with marginally relevant disorder, the hierarchical setting provides some insights that are likely general for weak-disorder scaling limits at criticality for marginally relevant systems.

The continuum polymer model corresponding to the scaling limit of this article is studied in [12, 13]. We will return to a broader discussion of related work in Section 4 after defining our hierarchical model and presenting a first version of our main result.

2 The setup and a statement of the main result

This section begins by defining a family of random measures on directed paths crossing diamond hierarchical graphs and concludes with the statement of a limit theorem for the total masses of the measures (Theorem 2.7), which was conjectured in [10]. The models in this section have bond-disorder, i.e., disorder variables placed at the edges of the graphs, while the models discussed in the next section have disorder at the vertices.

2.1 Construction of the diamond hierarchical graphs

Hierarchical diamond graphs $D_{n}^{b,s}$, $n\in\mathbb{N}_{0}$ are recursively defined through a construction determined by a branching number $b\in\{2,3,\ldots\}$ and a segmenting number ${s}\in\{2,3,\ldots\}$. The zeroth graph, $D_{0}^{b,s}$, is simply two root vertices, $A$ and $B$, with an edge between them. The first-generation graph, $D_{1}^{b,s}$, is formed by $b$ parallel branches connecting $A$ and $B$, wherein each branch has $s$ edges running in sequence. For $n\geq 2$ the graph $D_{n}^{b,s}$ is defined recursively from $D_{n-1}^{b,s}$ by embedding a copy of $D_{1}^{b,s}$ in place of each edge on $D_{n-1}^{b,s}$. The set of edges, $E_{n}^{b,s}$, on $D_{n}^{b,s}$ thus contains $(bs)^{n}$ elements.

The first three recursively-defined diamond graphs with ${b}=3$ and ${s}=3$.

A directed path on $D_{n}^{b,s}$ is a function $p:\{1,\ldots,s^{n}\}\rightarrow E^{b,s}_{n}$ for which $p(1)$ is incident to $A$, $p(s^{n})$ is incident to $B$, and successive edges $p(k)$, $p(k+1)$ share a common vertex for $1\leq k<s^{n}$. In other terms, the path moves monotonically upwards from $A$ up to $B$, as seen in the figure. We denote the set of directed paths on $D_{n}^{b,s}$ by $\Gamma_{n}^{b,s}$.

2.2 Random Gibbsian measure on directed paths

Next we define a random Gibbs measure on the space $\Gamma^{b,s}_{n}$ of directed paths. Let $\omega_{h}$ be an i.i.d. family of random variables labeled by $h\in E_{n}^{b,s}$ and having mean zero, variance one, and finite exponential moments, \mathbb{E}\big{[}\exp\{\beta\omega_{h}\}\big{]} for $\beta\geq 0$. Given an inverse temperature value $\beta\in[0,\infty)$, we define a random path measure on directed paths such that the weight assigned to $p\in\Gamma^{b,s}_{n}$ is given by

[TABLE]

where $a\in p$ means that the edge $a\in E^{b,s}_{n}$ lies along the path $p$. At infinite temperature ($\beta=0$), $\mathbf{M}^{\omega}_{\beta,n}$ is a uniform probability measure on $\Gamma_{n}^{b,s}$. We denote the total mass of $\mathbf{M}^{\omega}_{\beta,n}$ by

[TABLE]

in terms of the disorder variables $\omega_{a}$. The recursive construction of the diamond graphs implies the following distributional recursive relation for the partition functions $W_{n}^{\omega}(\beta)$:

[TABLE]

where the $W_{n}^{(i,j)}(\beta)$’s are independent copies of the random variable $W_{n}^{\omega}(\beta)$. The variances \varrho_{n}(\beta):=\textup{Var}\big{(}W_{n}^{\omega}(\beta)\big{)} are recursively related as \varrho_{n+1}(\beta)=M_{b,s}\big{(}\varrho_{n}(\beta)\big{)} with $M_{b,s}:[0,\infty)\rightarrow[0,\infty)$ defined as

[TABLE]

Thus the fixed point is linearly attractive when $b>s$, linearly repelling when $b<s$, and marginally repelling when $b=s$.

2.3 High-temperature scaling limits for the Gibbs measure

Our focus is on high-temperature (i.e., weak-disorder) scaling limits in which the hierarchical level parameter, $n$, grows as the inverse temperature $\beta=\beta(n)$ decays under an appropriate tuning in $n$ such that the random path measures $\mathbf{M}^{\omega}_{\beta,n}$ converge in distribution to a limiting random measure on paths. This article concerns only the total mass of the measures while [12] extends this limit analysis to the full measures and discusses some delicate properties of the limiting path measures. High-temperature scaling limits are only of interest in the cases $b<s$ and $b=s$ for which $x=0$ is a repelling fixed point of the variance map $M_{b,s}$. The article [1] contains a limit theorem for $W_{n}^{\omega}(\beta)$ in the case $b<s$, where for a fixed parameter value $r\in{\mathbb{R}}_{+}$ the inverse temperature $\beta\equiv\beta_{n,r}^{b,s}$ has the large $n$ asymptotic form

[TABLE]

The sequences of random variables $\{W_{n}^{\omega}(\beta_{n,r}^{b,s})\}_{n\in\mathbb{N}}$ converge in distribution as $n\rightarrow\infty$ to a family of limit laws $\mathbf{W}_{r}$ supported on $(0,\infty)$ that satisfy the distributional recursion relation

[TABLE]

where $\mathbf{W}_{r}^{(i,j)}$ are i.i.d. copies of $\mathbf{W}_{r}$. The variance, $R_{b,s}(r)$, of $\mathbf{W}_{r}$ satisfies M_{b,s}\big{(}R_{b,s}(r)\big{)}\,=\,R_{b,s}(\frac{s}{b}r). Of course, the exponential form of the inverse temperature scaling (2.4) corresponds to the linear repelling (2.2) of the map $M_{b,s}$ from $x=0$ that occurs in the $b<s$ case.

The main result of the current article is a proof of an analogous limit theorem for $W_{n}^{\omega}(\beta)$ in the $b=s$ case. An inverse temperature scaling—see below in (2.5)—was proposed in [10] although the results therein were confined to proving convergence of the positive integer moments.222The scaling (2.5) includes a correction pointed out by an anonymous referee that ensures consistency with the variance asymptotics (2.7) below; see Appendix A for an outline of the computation determining (2.5) from (2.7). The variance asymptotics is what plays a direct role in all subsequent analysis. Although the convergence of the positive integer moments implies the existence of subsequential distributional limits, it does not imply convergence in law because the higher limiting moments increase super-factorially; see (III) of Theorem 2.4 below. For fixed $b\in\{2,3,4,\ldots\}$ and $r\in{\mathbb{R}}$, let the sequence $(\beta_{n,r}^{(b)})_{n\in\mathbb{N}}$ have the large $n$ asymptotics

[TABLE]

where $\tau:=\mathbb{E}[\omega_{a}^{3}]$ and $\tau^{\prime}:=\mathbb{E}[\omega_{a}^{4}]-3$ are respectively the third and fourth cumulants of the disorder variables, $\omega_{a}$, and the constants $\kappa_{b},\eta_{b}>0$ are defined as

[TABLE]

If we let $M_{b,b}^{n}$ denote the $n$-fold composition of $M_{b,b}$, the variance, \varrho_{n}\big{(}\beta_{n,r}^{(b)}\big{)}, of W_{n}^{\omega}\big{(}\beta_{n,r}^{(b)}\big{)} can be written explicitly as

[TABLE]

The basic observations above combined with Lemma 2.3 below imply that \varrho_{n}\big{(}\beta_{n,r}^{(b)}\big{)} converges as $n\rightarrow\infty$ to a limit $R_{b}(r)$ for any $r\in{\mathbb{R}}$.

Remark 2.1**.**

Let us set the skewness, $\tau$, of the disorder variables to zero for simplicity. Theorem 7.1 of [1] states that if $\beta_{n,r}^{(b)}$ is replaced by a coarser scaling of the form $\hat{\beta}/\sqrt{n}$ for a parameter $\hat{\beta}\in{\mathbb{R}}_{+}$, then W_{n}^{\omega}\big{(}\hat{\beta}/\sqrt{n}\big{)} has the distributional behaviors listed below depending on $\hat{\beta}$ as $n\rightarrow\infty$.

[TABLE]

In the above, we use the notation $\stackrel{{\scriptstyle d}}{{\approx}}$ heuristically to mean that the random variables are “close” in distribution. Thus $\kappa_{b}$ is a critical point for the parameter $\hat{\beta}$ in the moment behavior of W_{n}^{\omega}\big{(}\hat{\beta}/\sqrt{n}\big{)} when $n\gg 1$, and $\beta_{n,r}^{(b)}$ falls within a critical window around $\kappa_{b}$. The variance blowup after $\kappa_{b}$ coincides with the transition to strong disorder as can be seen in the limit model emerging under the scaling (2.5) as $n\rightarrow\infty$; see Remark 2.9.

Remark 2.2**.**

The critical inverse temperature scaling for (2+1)-dimensional directed polymers considered in [9] has the form \beta_{L,r}=\frac{\sqrt{\pi}}{(\log L)^{1/2}}-\frac{\pi\tau}{2\log L}+\frac{\sqrt{\pi}r+\pi^{3/2}(\frac{5}{4}\tau^{2}-\frac{7}{12}\tau^{\prime}-\frac{1}{2})}{2(\log L)^{3/2}}+\mathit{o}\big{(}\frac{1}{(\log L)^{3/2}}\big{)} for $L\gg 1$, where $L$ is the polymer length, $r\in{\mathbb{R}}$ is a parameter, and $\tau,\tau^{\prime}$ are the third and fourth cumulants of the disorder variables; see [9, Remark 1.1]. In terms of the length $L=b^{n}$ of the diamond graph polymers, the asymptotic form (2.5) is fairly similar except for the inclusion of the term $\frac{\log\log L}{(\log L)^{3/2}}$.

2.4 Previous results on the centered moments

The lemma and theorem below are results from [10].

Lemma 2.3** (Variance function).**

For any $b\in\{2,3,\ldots\}$, there exists a unique continuously differentiable increasing function $R_{b}:{\mathbb{R}}\rightarrow{\mathbb{R}}_{+}$ satisfying the properties (I)-(III) below.

1. (I)

Composition of $R_{b}(r)$ with the map $M_{b,b}$ translates the parameter $r$:  M_{b,b}\big{(}R_{b}(r)\big{)}\,=\,R_{b}(r+1). 2. (II)

As $r\rightarrow\infty$, $R_{b}(r)$ diverges to $\infty$. As $r\rightarrow-\infty$, $R_{b}(r)$ has the vanishing asymptotics

[TABLE] 3. (III)

The derivative $R_{b}^{\prime}(r)$ admits the limiting form

[TABLE]

Moreover, if for some $r\in{\mathbb{R}}$ the sequence of positive real numbers $(x^{n,r})_{n\in\mathbb{N}}$ has the large $n$ asymptotics

[TABLE]

then $M^{n}_{b,b}(x^{n,r})$ converges as $n\rightarrow\infty$ to $R_{b}(r)$.

Appendix B contains an elementary but instructive calculation showing the consistency between properties (I) and (II) above. The higher centered moments of W_{n}^{\omega}\big{(}\beta_{n,r}^{(b)}\big{)} converge to limits $R_{b}^{(m)}(r)$ characterized as follows.

Theorem 2.4** (Limiting higher moments).**

Fix $b\in\{2,3,\ldots\}$ and let $s=b$. For each $m\in\{2,3,\ldots\}$ there is a continuous, increasing function $R^{(m)}_{b}:{\mathbb{R}}\rightarrow[0,\infty)$ such that for any $r\in{\mathbb{R}}$

[TABLE]

The limit functions $R^{(m)}_{b}$ satisfy properties (I)-(III) below.

1. (I)

There are multivariate polynomials $P_{m}:{\mathbb{R}}^{m-1}\rightarrow{\mathbb{R}}$ with nonnegative coefficients such that for all $r\in{\mathbb{R}}$

[TABLE] 2. (II)

$R^{(m)}_{b}(r)$* diverges to $\infty$ as $r\rightarrow\infty$ and vanishes as $r\rightarrow-\infty$ with the asymptotics $R^{(m)}_{b}(r)\sim\kappa_{b}^{m}\frac{m!}{2^{m/2}(m/2)!}|r|^{-m/2}$ for $m$ even and R^{(m)}_{b}(r)=\mathit{O}\big{(}|r|^{-(m+1)/2}\big{)} for $m$ odd.* 3. (III)

There is a $c>0$ such that $\frac{\log\log(R_{b}^{(m)}(r))}{m}>c$ holds for any fixed $r\in{\mathbb{R}}$ and large enough $m\in\mathbb{N}$.

Remark 2.5**.**

The function $R_{b}(r)$ in the statement of Lemma 2.3 is equal to $R^{(2)}_{b}(r)$ in the statement of Theorem 2.4.

Remark 2.6**.**

The quantity $\kappa_{b}^{m}\frac{m!}{2^{m/2}(m/2)!}|r|^{-m/2}$ in (II) for $m$ even agrees with the $m^{th}$ moment of a centered normal random variable with variance $\kappa^{2}_{b}/|r|$.

2.5 A first version of the main result

As mentioned above, Theorem 2.4 does not imply that W_{n}^{\omega}\big{(}\beta_{n,r}^{(b)}\big{)} converges in law as $n\rightarrow\infty$ since $R^{(m)}_{b}(r)$ grows super-factorially with $m\in\mathbb{N}$ by (III) of Theorem 2.4. Thus the following theorem was left as a conjecture in [10].

Theorem 2.7**.**

Fix $b\in\{2,3,\ldots\}$ and $r\in{\mathbb{R}}$, and let the sequence $(\beta_{n,r}^{(b)})_{n\in\mathbb{N}}$ have the form (2.5). When $s=b$ there is convergence in distribution as $n\rightarrow\infty$

[TABLE]

to a family of limit laws \big{\{}L_{r}^{(b)}\big{\}}_{r\in{\mathbb{R}}} uniquely determined by (I)-(IV) below.

1. (I)

$L_{r}^{(b)}$* has mean $1$ and variance $R_{b}(r)$.* 2. (II)

For $m\in\{3,4,\ldots\}$, the $m^{th}$ centered moment of $L_{r}^{(b)}$ is equal to $R^{(m)}_{b}(r)$. 3. (III)

Let $\mathbf{W}_{r}$ be a random variable with distribution $L_{r}^{(b)}$. The centered variables $\sqrt{-r}(\mathbf{W}_{r}-1)$ converge in law as $r\rightarrow-\infty$ to a centered normal with variance $\kappa_{b}^{2}$. 4. (IV)

If $\mathbf{W}^{(i,j)}_{r}$ are independent variables with distribution $L_{r}^{(b)}$, then there is equality in distribution

[TABLE]

Remark 2.8**.**

The convergence in distribution of $\sqrt{-r}(\mathbf{W}_{r}-1)$ to $\mathcal{N}(0,\kappa^{2}_{b})$ as $r\rightarrow-\infty$ follows from the asymptotics for the centered moments $R^{(m)}_{b}(r)$ in (II) of Theorem 2.4.

Remark 2.9**.**

The family of limit laws in Theorem 2.7 exhibits a transition from weak disorder to strong disorder as $r$ goes from $-\infty$ to $+\infty$ in the sense that the random variables $\mathbf{W}_{r}$ converge in probability to one as $r\rightarrow-\infty$ and to zero as $r\rightarrow\infty$, where the latter is proved in [13, Section 5] using a conditional Gaussian multiplicative chaos structure that we will describe at the end of Section 4.

3 A similar limit theorem for the site-disorder model

Next we will state an analogous result to Theorem 2.7 corresponding to when the environmental disorder is built into the partition function through the vertices of the diamond graphs rather than the edges.

For $n\in\mathbb{N}_{0}$ and $b,s\in\{2,3,\ldots\}$, let $V^{b,s}_{n}$ denote the set of vertices on the $n^{th}$ diamond graph $D_{n}^{b,s}$ with the roots $A$ and $B$ excluded. Thus $V^{b,s}_{0}=\emptyset$, and for $n\geq 1$ the number of non root vertices is given by \big{|}V^{b,s}_{n}\big{|}=b(s-1)\frac{(bs)^{n}-1}{bs-1}. The hierarchical construction of the sequence of diamond graphs in Section 2.1 implies that $V^{b,s}_{n-1}$ is canonically identifiable with a subset of $V^{b,s}_{n}$ for each $n\in\mathbb{N}$, and we refer to $V^{b,s}_{n}\backslash V^{b,s}_{n-1}$ as the set of generation-$n$ vertices.

As before, let $\{\omega_{a}\}_{a\in V_{n}^{b,s}}$ be an i.i.d. family of centered random variables with variance one and finite exponential moments. We define the partition function $\widehat{W}_{n}^{\omega}(\beta)$ in analogy to $W_{n}^{\omega}(\beta)$ in (2.1) except with the product of random variables $e^{\beta\omega_{a}}/\mathbb{E}[e^{\beta\omega_{a}}]$ running over all vertices $a\in V_{n}^{b,s}$ along the path $p\in\Gamma_{n}^{b,s}$:

[TABLE]

where the notation $a\boldsymbol{\in}p$ is used for a vertex $a\in V_{n}^{b,s}$ and a path $p:\{1,\ldots,s^{n}\}\rightarrow E_{n}^{b,s}$ to indicate that one of the edges $p(k)\in E_{n}^{b,s}$ for $k\in\{2,\dots,s^{n}-1\}$ is incident to $a$. When $n=0$ the partition function $\widehat{W}_{n}^{\omega}(\beta)$ is simply equal to $1$ since $V_{0}^{b,s}=\emptyset$, and the hierarchical symmetry of the model implies the following distributional equality, which is similar to (2.2):

[TABLE]

where $\widehat{W}_{n}^{(i,j)}(\beta)$ are i.i.d. copies of $\widehat{W}_{n}^{\omega}(\beta)$ and $\omega_{i,\ell}$ are i.i.d. copies of the disorder variable. The terms $e^{\beta\omega_{i,\ell}}/\mathbb{E}[e^{\beta\omega_{i,\ell}}]$ correspond to the generation-$1$ vertices of the diamond graph $D_{n+1}^{b,s}$.

The following theorem is the counterpart to Theorem 2.7 for the site-disorder model, and its proof is in Section 14.

Theorem 3.1**.**

Fix $b\in\{2,3,\ldots\}$ and $r\in{\mathbb{R}}$, and assume $s=b$. Define $\widehat{\kappa}_{b}:=\frac{\pi\sqrt{b}}{\sqrt{2}(b-1)}$, and let $\tau$ and $\eta_{b}$ be defined as in (2.5). If the sequence \big{\{}\widehat{\beta}_{n,r}^{(b)}\big{\}}_{n\in\mathbb{N}} has the asymptotic form

[TABLE]

then \widehat{W}_{n}^{\omega}\big{(}\widehat{\beta}_{n,r}^{(b)}\big{)} converges in distribution as $n\rightarrow\infty$ to the limit law $\mathbf{W}_{r}$ of Theorem 2.7.

Remark 3.2**.**

Define \upsilon_{b}:\big{[}0,\widehat{\kappa}_{b}\big{)}\rightarrow[0,\infty) by \upsilon_{b}(\hat{\beta}):=\hat{\beta}\frac{\sqrt{2}}{\sqrt{b}}\tan\big{(}\frac{\pi}{2}\frac{\hat{\beta}}{\widehat{\kappa}_{b}}\big{)}. In the case of $s=b$, [1, Thm. 2.5] states that the partition function $\widehat{W}_{n}^{\omega}(\hat{\beta}/n)$ has the large $n$ distributional behaviors listed below depending on the parameter $\hat{\beta}\geq 0$.

[TABLE]

We use $\stackrel{{\scriptstyle d}}{{\approx}}$ in the same heuristic sense as in Remark 2.1. Thus $\widehat{\kappa}_{b}$ is a critical point for the large $n$ behavior of \widehat{W}_{n}^{\omega}\big{(}\hat{\beta}/n\big{)} that is analogous to $\kappa_{b}$ for W_{n}^{\omega}\big{(}\hat{\beta}/\sqrt{n}\big{)} as described in Remark 2.1.

Remark 3.3**.**

Our proof of Theorem 3.1 proceeds by showing that \widehat{W}_{n}^{\omega}\big{(}\widehat{\beta}_{n,r}^{(b)}\big{)} is close in $L^{2}$ norm to a similarly-defined partition function in which the disorder variables $e^{\beta\omega_{a}}/\mathbb{E}[e^{\beta\omega_{a}}]$ are only attached to vertices of generation greater than $\lfloor\log n\rfloor$. This effectively reduces the generation-$n$ site-disorder model to a generation-$\lfloor\log n\rfloor$ bond-disorder model. The results developed to prove Theorem 2.7 can then be applied to prove Theorem 3.1.

4 Further discussion

As mentioned in Section 1, the $(d+1)$-dimensional polymer model is disorder relevant when $d=1$ and marginally relevant when $d=2$. In principle, disorder relevance opens up the possibility that there exists a continuum disorder model that emerges in a joint limit in which the polymer length, $L$, grows as the inverse temperature $\beta\equiv\beta(L)$ vanishes with an appropriate dependence on $L$.333The general relationship between disorder relevance and continuum limits is argued for in [8]. A rigorous mathematical result in this direction was developed by Alberts, Khanin, and Quastel in the article [2], which proved that the partition function for (1+1)-dimensional polymers converges in law to a nontrivial distributional limit, $\mathcal{Z}_{\hat{\beta}}$, as $L\nearrow\infty$ and the inverse temperature has the asymptotic form \beta=\big{(}\hat{\beta}+\mathit{o}(1)\big{)}L^{-1/4} for a fixed parameter value $\hat{\beta}\in{\mathbb{R}}_{+}$. This scaling limit is referred to as the intermediate disorder regime since it magnifies a parameter region between the weak ($\beta=0$) and the strong ($\beta>0$) domains of disorder behavior for the $(1+1)$-dimensional polymer, and it amounts to a continuum/weak-disorder limiting regime in which the polymers are diffusively rescaled towards Brownian motion trajectories while the environmental disorder variables are renormalized towards a white noise field $W\equiv W(t,x)$ on $[0,1]\times{\mathbb{R}}$. The authors construct the limiting partition functions $\mathcal{Z}_{\hat{\beta}}$ in terms of Wiener chaos expansions of the field $W(t,x)$ involving the one-dimensional heat kernel \varrho(t^{\prime},x^{\prime};t,x)=\frac{1}{\sqrt{2\pi(t-t^{\prime})}}\textup{exp}\big{\{}-\frac{(x-x^{\prime})^{2}}{2(t-t^{\prime})}\big{\}}.

A model of continuum directed polymers corresponding to the limiting partition function laws $\mathcal{Z}_{\hat{\beta}}$ in [2] was discussed more explicitely in [3], where $\mathcal{Z}_{\hat{\beta}}$ is equal in distribution to the total mass of a random measure on $C([0,1])$, i.e., the space of Brownian trajectories. Moreover, the authors use the point-to-point form, $\mathcal{Z}_{\hat{\beta}}\equiv\mathcal{Z}_{\hat{\beta}}(t^{\prime},x^{\prime};t,x)$, of these limiting partition function laws to construct a solution to the one-dimensional stochastic heat equation (SHE):

[TABLE]

In the case where $\mathcal{Z}_{\hat{\beta}}\equiv\mathcal{Z}_{\hat{\beta}}(0,0;1,*)$ corresponds to the limit of point-to-line partition functions for polymers starting at the origin, $\mathcal{Z}_{\hat{\beta}}$ is equal in law to the total mass of a random measure $M_{\hat{\beta}}$ on $C([0,1])$ that can be formally expressed as

[TABLE]

where $\mathbf{P}$ is the Wiener measure on $C([0,1])$ for a standard Brownian motion and $\widehat{W}(p):=\int_{0}^{1}W(t,p_{t})dt$ defines a Gaussian field444The field $\widehat{W}(p)$ yields a Gaussian random variable when integrated against a test function \psi\in L^{2}\big{(}C([0,1]),\mathbf{P}\big{)}. over $C([0,1])$ with correlation kernel given by the intersection time between paths: T(p,q)=\mathbb{E}\big{[}\widehat{W}(p)\widehat{W}(q)\big{]}=\int_{0}^{1}\delta(p_{t}-q_{t})dt. Random measures formally expressed in terms of exponentials of Gaussian fields as in (4.1) are the focus of the theory of Gaussian multiplicative chaos (GMC), and $M_{\hat{\beta}}$ is a subcritical GMC for any $\hat{\beta}\in{\mathbb{R}}_{+}$ that can be understood through the general approach to GMC theory in [26]. The random measures $M_{\hat{\beta}}$ are a.s. mutually singular to $\mathbf{P}$ and satisfy

[TABLE]

In particular, $\mathbb{E}[M_{\hat{\beta}}\times M_{\hat{\beta}}]$ is absolutely continuous with respect to $\mathbf{P}\times\mathbf{P}$, which is a necessary feature of subcritical GMCs.555See Lemma 34 of [26].

Weak-disorder limits analogous to [2] for the marginally relevant $(2+1)$-dimensional polymer involve fundamental new mathematical difficulties and are not as well understood as the weak-disorder regime for the $(1+1)$-polymer despite significant progress in a series of articles [5, 6, 7, 8, 9] by Caravenna, Sun, and Zygouras. In [6] the authors proved that the partition function $Z_{L,\beta}$ for (2+1)-dimensional polymers has the following distributional limit behavior as $L\nearrow\infty$ when the inverse temperature tends to zero as \beta\equiv\beta_{L}=\frac{\sqrt{\pi}}{(\log L)^{1/2}}\big{(}\hat{\beta}+\mathit{o}(1)\big{)} for fixed $\hat{\beta}\in{\mathbb{R}}_{+}$:

[TABLE]

where $\chi$ is a standard normal random variable and \sigma_{\hat{\beta}}^{2}:=\log\big{(}\frac{1}{1-\hat{\beta}^{2}}\big{)}. In other terms, for $\hat{\beta}<1$ the limit law, $\mathcal{Z}_{\hat{\beta}}$, is a mean-one lognormal that converges in probability to zero (while having exploding variance) as $\hat{\beta}\nearrow 1$. Thus a phase transition from weak disorder to strong disorder occurs at $\hat{\beta}=1$ within this weak-coupling limit regime.

A further study of the (2+1)-dimensional directed polymer around the critical point $\hat{\beta}=1$ within the weak-disorder limit is undertaken in [9] by choosing the more refined inverse temperature scaling $\beta\equiv\beta_{L,r}$ in Remark 2.2, which depends on a fixed parameter value $r\in{\mathbb{R}}$. This scaling satisfies \beta_{L,r}=\frac{\sqrt{\pi}}{(\log L)^{1/2}}\big{(}1+\mathit{o}(1)\big{)} for $L\gg 1$, i.e., falls within the critical window of the phase transition (4.3) and is determined by the requirement that the variance of \textup{exp}\{\beta_{L,r}\omega\}/\mathbb{E}\big{[}\textup{exp}\{\beta_{L,r}\omega\}\big{]}, where $\omega$ is a disorder variable, has the large $L$ asymptotic form \frac{\pi}{\log L}+\frac{\pi r}{\log^{2}L}+\mathit{o}\big{(}\frac{1}{\log^{2}L}\big{)}.666The parameter $r\in\mathbb{R}$ is related to the parameter $\vartheta\in{\mathbb{R}}$ used in [7, 9] through $r=\vartheta-\alpha$ for $\alpha$ defined below (4.5). For a time parameter $t\geq 0$, the authors define the following random measures $\mathscr{Z}_{Lt,\beta_{L,r}}$ on ${\mathbb{R}}^{2}$:

[TABLE]

where $Z_{L,\beta}(x)$ is the partition function for length $L$ polymers starting from position $x\in{\mathbb{Z}}^{2}$. Using a tightness argument involving bounds for the third moments of the variables $Z_{Lt,\beta_{L,r}}(\phi):=\int_{{\mathbb{R}}^{2}}\phi(x)\mathscr{Z}_{Lt,\beta_{L,r}}(dx)$ for $\phi\in C_{c}({\mathbb{R}}^{2})$, the authors prove the existence of subsequential limits as $L\rightarrow\infty$ such that $\mathscr{Z}_{Lt,\beta_{L,r}}$ converges in law to a random measure $\mathcal{Z}_{t,r}$ on ${\mathbb{R}}^{2}$ satisfying

[TABLE]

where $\alpha:=\gamma+\log 16-\pi$ for the Euler-Mascheroni constant $\gamma$, and $K_{t,r}(z-z^{\prime})$ is a correlation kernel with logarithmic blowup around its diagonal from Bertini and Cancrini’s article [4] on the two-dimensional SHE. The above is related to a recent breakthough on the moments of the two-dimensional SHE at criticality by Gu, Quastel, and Tsai [20]. When $t=1$ the form (4.5) is consistent with the existence of a $(2+1)$-dimensional continuum random polymer measure $M_{r}^{\phi}(dp)$ on $C([0,1],{\mathbb{R}}^{2})$, with total mass equal in distribution to the random variable $\int_{{\mathbb{R}}^{2}}\phi(x)\mathcal{Z}_{1,r}(dx)$, that is analogous to the (1+1)-dimensional case in [3] when the starting point of the polymer has an appropriate probability density $\phi:{\mathbb{R}}^{2}\rightarrow[0,\infty)$ (i.e., diffuse initial position). If $\mathbf{P}^{\phi}$ denotes Wiener measure on $C([0,1],{\mathbb{R}}^{2})$ for trajectories starting with initial position density $\phi$, then two independently chosen trajectories will a.s. not intersect. In other words, the product Wiener measure $\mathbf{P}^{\phi}\times\mathbf{P}^{\phi}$ assigns probability zero to the set of pairs of trajectories that intersect. If a continuum disordered polymer measure $M_{r}^{\phi}$ exists, $\mathbb{E}[M_{r}^{\phi}\times M_{r}^{\phi}]$ would not be absolutely continuous with respect to $\mathbf{P}^{\phi}\times\mathbf{P}^{\phi}$, unlike the continuum $(1+1)$-dimensional polymer case (4.2).

Next we outline the rough analogy between models for directed polymers in a random environment on diamond hierarchical graphs and on rectangular lattices. Hierarchical graphs (“lattices”) are a frequent setting for statistical mechanical toy models because they may retain key characteristics of interest from their non-hierarchical analogs while providing a decomposability in terms of renormalization transformations; see for instance [17, 18, 21, 22, 23, 25, 28] for recent mathematical work. By the nature of their recursive construction, hierarchical models embed copies of themselves after a change in the controlling parameters for the embedded copies. The articles [15, 16] were the first to study models of directed polymers in a random environment on diamond hierarchical graphs.777This assertion about the history of directed polymers on the diamond lattice is from [14, Page 73]. In [23], Lacoin and Moreno analyzed the phase diagram of polymers on diamond graphs when the disorder variables are placed on the vertices, showing that

• •

strong disorder holds for any $\beta>0$ when $b\leq s$, and

• •

when $b>s$ there is a critical inverse temperature $\beta_{c}>0$ for which weak disorder holds when $\beta\leq\beta_{c}$ and strong disorder holds for $\beta$ above $\beta_{c}$.

In terms of their disorder relevance, the cases $b<s$, $b=s$, and $b>s$ are analogous respectively to the $d=1$, $d=2$, and $d\geq 3$ cases of (d+1)-dimensional polymers on the rectangular lattice. In the disorder relevant $b<s$ case, [1] proves a limit theorem for the partition functions in an intermediate disorder regime analogous to [2], and [11] defines a continuum polymer model similar to [3], although using GMC for the construction rather than Wiener chaos.

When the model is altered by placing disorder variables on the edges of the graphs rather than the vertices (as in Section 2), the analysis in [23] goes through essentially unchanged when $b<s$ or $b>s$, but for the marginal case of $b=s$ there is a basic combinatorial difference: for two directed polymers $p$ and $q$ chosen independently and uniformly at random,

• •

the expected number of vertices shared by $p$ and $q$ has order $\log L$ for $L\gg 1$, where $L$ is the length888In terms of the parameter $s$, the polymer length has the form $L=s^{n}=b^{n}$. of the polymers, and

• •

the expected number of edges shared by $p$ and $q$ is exactly $1$, independent of $L$. A closer look shows that when $L\gg 1$ the polymers will share no edges at all with a probability $1-\mathit{O}(1/\log L)$, and that the expected number of common edges will be of order $\log L$ in the complementary event.

Thus, when $b=s$, the diamond graph polymer model with edge disorder is similar to the polymer measures underlying the mollified partition functions in (4.4) in the sense that two independent two-dimensional SSRW trajectories of length $L$ with initial spatial probability densities spread out on the order of $\sqrt{L}$ have a probability of intersecting that vanishes with order $1/\log L$ and, when conditioned on the event that the paths intersect, an expected number of intersections on the order of $\log L$.

We will briefly summarize the continuum polymer model defined in [12] and its conditional Gaussian multiplicative chaos structure [13]. The limiting partition function law, $\mathbf{W}_{r}$, derived in later sections is equal in distribution to the total mass of a random measure $\mathbf{M}_{r}$ on the space $\Gamma$ of directed paths crossing a compact diamond fractal, $D$, having Hausdorff dimension two. Each directed path $p\in\Gamma$ is an isometric embedding of the unit interval $[0,1]$ into the fractal, and there is a natural “uniform” probability measure $\mu$ on $\Gamma$ (serving as the analog of Wiener measure for the continuum (1+1)-dimensional polymer) for which $\mathbb{E}[\mathbf{M}_{r}]=\mu$. For directed paths $p,q\in\Gamma$, the set of intersection times is $\mathcal{I}_{p,q}:=\{t\in[0,1]\,|\,p(t)=q(t)\}$, and two paths chosen uniformly at random, i.e., according to the product measure $\mu\times\mu$, have a finite (trivial) number of intersections with probability one. In contrast, the random product measures $\mathbf{M}_{r}\times\mathbf{M}_{r}$ a.s. assign positive weight to the set of pairs $(p,q)\in\Gamma\times\Gamma$ for which $\mathcal{I}_{p,q}$ is uncountable, albeit of Hausdorff dimension zero. The size of typical $\mathcal{I}_{p,q}$ can be characterized through the exponent $\mathfrak{h}=1$ case of the generalized Hausdorff measure $\mathcal{H}^{\textup{log}}_{\mathfrak{h}}$ on $[0,1]$ of the form

[TABLE]

where $S\subset[0,1]$, and the infimum is over all coverings of $S$ by intervals $I$ of length $|I|$ less than $\delta>0$; see the monograph [24] for a discussion of the general theory of Hausdorff measures.

The qualitative difference (trivial to nontrivial) between the typical behavior of the intersection-times set $I_{p,q}$ under the pure measure $\mu\times\mu$ and realizations of the disordered product measure $\mathbf{M}_{r}\times\mathbf{M}_{r}$ is a strong localization property that is not present in the subcritical continuum models [3, 11]. To compare with the (1+1)-dimensional continuum polymer measures $M_{\hat{\beta}}$ discussed above, the set of intersection times $I_{p,q}$ is appropriately measured by $T(p,q)=\int_{0}^{1}\delta_{0}(p_{t}-q_{t})dt$—which is closely related to the dimension-$1/2$ Hausdorff measure of $I_{p,q}$—for both the product Wiener measure $\mathbf{P}\times\mathbf{P}$ and realizations of $M_{\hat{\beta}}\times M_{\hat{\beta}}$. Secondly, in contrast with (4.2), the expectation of $\mathbf{M}_{r}\times\mathbf{M}_{r}$ has Lebesgue decomposition with respect to $\mu\times\mu$ given by

[TABLE]

where the measure $\varpi_{r}$ assigns full weight to the set of pairs $(p,q)\in\Gamma\times\Gamma$ such that $\mathcal{H}^{\textup{log}}_{\mathfrak{h}}(I_{p,q})=\infty$ for all $\mathfrak{h}<1$ and $\mathcal{H}^{\textup{log}}_{\mathfrak{h}}(I_{p,q})=0$ for all $\mathfrak{h}>1$, in other terms, for which $I_{p,q}$ has log-Hausdorff exponent one. The fact that \mathbb{E}\big{[}\mathbf{M}_{r}\times\mathbf{M}_{r}\big{]} is not absolutely continuous with respect to $\mathbb{E}[\mathbf{M}_{r}]\times\mathbb{E}[\mathbf{M}_{r}]=\mu\times\mu$ implies that $\mathbf{M}_{r}$ is not a subcritical GMC.

The random measure $\mathbf{M}_{r}$ is also not a “critical” GMC since the expectation \mathbb{E}\big{[}\mathbf{M}_{r}]=\mu is a probability measure and thus $\sigma$-finite. The family of random measure laws $(\mathbf{M}_{r})_{r\in{\mathbb{R}}}$, however, has a conditional interrelational GMC structure wherein for any $a\in{\mathbb{R}}_{+}$ the law of the random measure $\mathbf{M}_{r+a}$ can be constructed from $\mathbf{M}_{r}$ as

[TABLE]

where $\widehat{W}_{\mathbf{M}_{r}}(p)$ is a field over $(\Gamma,\mathbf{M}_{r})$ that is Gaussian when conditioned on $\mathbf{M}_{r}$ and has a correlation kernel T(p,q)=\mathbb{E}\big{[}\widehat{W}_{\mathbf{M}_{r}}(p)\widehat{W}_{\mathbf{M}_{r}}(q)\,|\,\mathbf{M}_{r}\big{]} roughly equivalent to the generalized Hausdorff measure with exponent $\mathfrak{h}=1$, $\mathcal{H}^{\textup{log}}_{1}(\mathcal{I}_{p,q})$, of the set of intersection times. Because the random measures $\mathbf{M}_{r}$ converge in law to the pure measure $\mu$ as $r\searrow-\infty$, the above formally implies that an infinite field strength is required to generate $\mathbf{M}_{r}$ as a GMC on $\mu$.

5 Notation and organization

Notation: In the remainder of the article, we refer exclusively to the case when the branching parameter and the segmenting parameter of the diamond graphs are equal ($b=s$). The dependence of all previously defined expressions on the parameter $b\in\{2,3,\ldots\}$ will be suppressed as in the following list of notational identifications:

[TABLE]

$\mathbb{N}$ denotes the positive integers and $\mathbb{N}_{0}:=\mathbb{N}\cup\{0\}$. In heuristic discussions, we write $X\stackrel{{\scriptstyle d}}{{\approx}}Y$ for random variables $X$ and $Y$ that are “close” in distribution.

Article organization:

• •

Section 6 builds up to the statement of Theorem 6.23 (bond-disorder #2), which is a slightly strengthened version of Theorem 2.7 (bond-disorder #1) that is couched in the language used in the proofs. Theorem 7.3 (bond-disorder #3) is a third version of this type of distributional convergence result that leverages more stringent moment conditions for greater control of the rate of convergence.

• •

Taken together, Sections 8 & 9 complete the proof of Theorem 6.23 (bond-disorder #2) after stating the key technical results in Proposition 9.1 and Lemmas 9.7-9.9 that support the proof.

• •

Sections 10 & 11 contain the proofs of Proposition 9.1 & Lemmas 9.7-9.9 with some of the relatively routine elements delayed to Section 12.

• •

Theorem 7.3 (bond-disorder #3) is proved in Section 13.

• •

Theorem 3.1 (site-disorder) is proved in Section 14.

• •

Proofs of propositions that are technical variations of results from [10] are placed in Section 15.

• •

Appendix A derives the inverse temperature scaling (2.5) from the variance scaling (2.7), Appendix B carries through an instructive consistency check between (I) and (II) of Lemma 2.3, and Appendix C provides some background on the zero bias approach [19] to Stein’s method.

6 Reformulation in terms of arrays and Wasserstein distance

This section defines the notation and terminology needed for the statement of Theorem 6.23, which is a more flexible version of Theorem 2.7. The language defined here is used throughout the remainder of the article.

6.1 Edge-labeled array notation

The recursive construction of the diamond hierarchical graphs outlined in Section 2.1 implies a canonical one-to-one correspondence between the set of edges, $E_{k}$, of the $k^{th}$-generation diamond graph $D_{k}$ and the $2k$-fold product set (\{1,\ldots,b\}\times\{1,\ldots,b\}\big{)}^{k}; see the diagram below illustrating this correspondence in the first- and second-generation graphs when $b=2$. The hierarchical structure of the graphs also implies that for $l,k\in\mathbb{N}_{0}$ with $l<k$ each element $\mathbf{a}\in E_{l}$ is canonically identifiable with a $b^{2(k-l)}$-element subset of $E_{k}$.

Notation 6.1** (Arrays).**

Let $x_{a}$ be real numbers labeled by $E_{k}$ for some $k\in\mathbb{N}_{0}$. **

• •

The notation $\{x_{a}\}_{a\in E_{k}}$ denotes an element of ${\mathbb{R}}^{b^{2k}}$, which we refer to as an* array.*

• •

If $\mathbf{a}\in E_{l}$ for some $l\in\mathbb{N}$ with $l\leq k$, then $\{x_{a}\}_{a\in\mathbf{a}\cap E_{k}}$ denotes an element in ${\mathbb{R}}^{b^{2(k-l)}}$, where we have abused notation by identifying $\mathbf{a}$ with its canonically corresponding subset of $E_{k}$.**

Next we define an operation on edge-labeled arrays that can be used (see Proposition 6.5) to express the partition function (2.1).

Definition 6.2** (Array maps).**

For $k\in\mathbb{N}_{0}$ and $a\in E_{k}$, define $a{\times}(i,j)$ for $i,j\in\{1,\ldots,b\}$ as the element in $E_{k+1}$ corresponding to the $j^{th}$ segment along the $i^{th}$ branch of the embedded copy of $D_{1}$ in $D_{n+1}$ identified with $a$.999This is to be understood in the context of the recursive construction of $D_{n+1}$ from $D_{n}$ in Section 2.1.

• •

We define $\mathcal{Q}$ as the map that sends an array of real numbers $\{x_{a}\}_{a\in E_{k}}$ to the contracted array

[TABLE]

• •

We define $\mathcal{L}$ as the linearization of $\mathcal{Q}$ around the zero array:

[TABLE]

• •

We define $\mathcal{E}:=\mathcal{Q}-\mathcal{L}$, i.e., the “error” of the linearization.

• •

For $N\in\mathbb{N}_{0}$, $\mathcal{Q}^{N}$ and $\mathcal{L}^{N}$ refer to the $N$-fold composition of the maps $\mathcal{Q}$ and $\mathcal{L}$, respectively.

Remark 6.3**.**

Note the ambiguity of the notations $\mathcal{Q}$, $\mathcal{L}$, $\mathcal{E}$ since we use them to denote maps from ${\mathbb{R}}^{E_{k}}$ to ${\mathbb{R}}^{E_{k-1}}$ for any $k\in\mathbb{N}$.

Remark 6.4**.**

For $a\in E_{k}$, our notational conventions imply that

[TABLE]

The following proposition relates the array map $\mathcal{Q}$ to the partition function $W^{\omega}_{n}(\beta)$. The proof is placed in Section 12.1.

Proposition 6.5**.**

The partition function $W^{\omega}_{n}(\beta)$ in (2.1) can be written in terms of the map $\mathcal{Q}$ as

[TABLE]

Remark 6.6**.**

Let $\{x_{a}\}_{a\in E_{k}}$ be an array of i.i.d. centered random variables with variance $\sigma^{2}$.

1. (i)

$\mathcal{Q}\{x_{a}\}_{a\in E_{k}}$ and $\mathcal{L}\{x_{a}\}_{a\in E_{k}}$ are i.i.d. arrays of centered random variables with variance $M(\sigma^{2})$ and $\sigma^{2}$, respectively. In particular, the operation $\mathcal{L}$ preserves the variance of the array variables. 2. (ii)

For $\{y_{a}\}_{a\in E_{k-1}}:=\mathcal{L}\{x_{a}\}_{a\in E_{k}}$ and $\{z_{a}\}_{a\in E_{k-1}}:=\mathcal{E}\{x_{a}\}_{a\in E_{k}}$, the random variables $y_{a}$ and $z_{a}$ are uncorrelated. Thus the variables in the array $\mathcal{E}\{x_{a}\}_{a\in E_{k}}$ have variance $M(\sigma^{2})-\sigma^{2}$. 3. (iii)

Moreover, the random variable $\mathcal{Q}^{k}\{x_{a}\}_{a\in E_{k}}$ can be written as the following sum of uncorrelated terms: $\mathcal{Q}^{k}\{x_{a}\}_{a\in E_{k}}=\mathcal{L}^{k}\{x_{a}\}_{a\in E_{k}}\,+\,\sum_{l=1}^{k}\mathcal{L}^{l-1}\mathcal{E}\mathcal{Q}^{k-l}\{x_{a}\}_{a\in E_{k}}$.

The lemma below generalizes (iii) in Remark 6.6 and identifies the main source of uncorrelated terms found in this article. The proof follows easily from the multilinear polynomial forms of the maps $\mathcal{Q}$, $\mathcal{E}$, $\mathcal{L}$.

Lemma 6.7**.**

Let $\{x_{a}\}_{a\in E_{k}}$ be an array of independent centered random variables with finite second moments. If $A_{l},B_{l}\in\{\mathcal{Q},\mathcal{E},\mathcal{L}\}$ for $l\in\{1,\ldots,k\}$, then the random variables $A_{1}\cdots A_{k}\{x_{a}\}_{a\in E_{k}}$ and $B_{1}\cdots B_{k}\{x_{a}\}_{a\in E_{k}}$ are uncorrelated when at least one of the following sets is nonempty:

[TABLE]

Proof.

Suppose that $\ell\in S_{A}$. The multilinear polynomial $A_{1}\cdots A_{k}\{x_{a}\}_{a\in E_{k}}$ is a linear combination of monomials $\prod_{a\in U}x_{a}$ for which the set $U\subset E_{k}$ must contain a pair $a_{1},a_{2}\in U$ satisfying the following property: there exist $f_{1},f_{2}\in E_{\ell}$ and $e\in E_{\ell-1}$ such that $a_{1}\in f_{1}$, $a_{2}\in f_{2}$, $f_{1}\neq f_{2}$, and $f_{1},f_{2}\in e$. On the other hand, the multilinear polynomial $B_{1}\cdots B_{k}\{x_{a}\}_{a\in E_{k}}$ does not contain any monomials of this type, so $A_{1}\cdots A_{k}\{x_{a}\}_{a\in E_{k}}$ and $B_{1}\cdots B_{k}\{x_{a}\}_{a\in E_{k}}$ are uncorrelated. ∎

Remark 6.8**.**

Note that if $\{x_{h}\}_{h\in E_{n}}$ is an array of i.i.d. centered random variables with variance $\sigma^{2}$, then $\mathcal{L}^{n}\{x_{h}\}_{h\in E_{n}}=\frac{1}{b^{n}}\sum_{h\in E_{n}}x_{h}$ has the form of a central limit-type normalized sum since $b^{n}=|E_{n}|^{1/2}$. More generally, if $n\geq k$, then $\{z_{a}\}_{a\in E_{k}}:=\mathcal{L}^{n-k}\{x_{h}\}_{h\in E_{n}}$ is an array of central limit-type normalized sums $z_{a}=\frac{1}{b^{n-k}}\sum_{h\in a\cap E_{n}}x_{h}$ since $b^{n-k}=|a\cap E_{n}|^{1/2}$.

In the following, we define terminology for the multilayer arrays determined by repeated application of $\mathcal{Q}$ when starting from a given edge-labeled array.

Definition 6.9**.**

Let $\mathcal{Q}$ be defined as in Definition 6.2 and $n\in\mathbb{N}_{0}$.

• •

A $\mathcal{Q}$-pyramidic array is a finite sequence in $k=0,1,\ldots,n$ of arrays of real numbers $\{x_{a}^{(k,n)}\}_{a\in E_{k}}$ satisfying \big{\{}x_{a}^{(k-1,n)}\big{\}}_{a\in E_{k-1}}=\mathcal{Q}\big{\{}x_{a}^{(k,n)}\big{\}}_{a\in E_{k}} for all $k\neq 0$.

• •

When $k=n$ we condense the superscript as $x_{h}^{(n,n)}\equiv x_{h}^{(n)}$ for $h\in E_{n}$. Moreover, \big{\{}x_{a}^{(k,n)}\big{\}}_{a\in E_{k}}=Q^{k}\{x_{h}^{(n)}\}_{h\in E_{n}} is referred to as the $\mathcal{Q}$-pyramidic array generated from \big{\{}x_{h}^{(n)}\big{\}}_{h\in E_{n}}.

Remark 6.10**.**

When $k=0$ we remove the subscript from $x_{a}^{(0,n)}\equiv x^{(0,n)}$ since $|E_{0}|=1$.

Remark 6.11**.**

To distinguish the entire $\mathcal{Q}$-pyramidic array from one of its subarray layers, \big{\{}x_{a}^{(k,n)}\big{\}}_{a\in E_{k}}, we will sometimes write \big{\{}x_{a}^{(*,n)}\big{\}}_{a\in E_{*}}.

6.2 Regular sequences of $\mathcal{Q}$-pyramidic arrays of random variables

Next we narrow our focus to sequences of $\mathcal{Q}$-pyramidic arrays of random variables. The following definition characterizes the assumptions that we use in our limit theorem in the next subsection.

Definition 6.12**.**

A sequence \big{(}\{X_{a}^{(*,n)}\}_{a\in E_{*}}\big{)}_{n\in\mathbb{N}} of $\mathcal{Q}$-pyramidic arrays of random variables taking values in $[-1,\infty)$ will be said to be regular with parameter $r\in{\mathbb{R}}$ if the sequence of generating arrays \big{(}\{X_{h}^{(n)}\}_{h\in E_{n}}\big{)}_{n\in\mathbb{N}} satisfies the properties below.

1. (I)

For each $n\in\mathbb{N}$, the random variables in the array $\{X_{h}^{(n)}\}_{h\in E_{n}}$ are centered and i.i.d. 2. (II)

The variance of the random variables in the array $\{X_{h}^{(n)}\}_{h\in E_{n}}$ has the large $n$ asymptotics

[TABLE] 3. (III)

For each $m\in\{4,6,\ldots\}$, the $m^{th}$ moment of the random variables in the array $\{X_{h}^{(n)}\}_{h\in E_{n}}$ vanishes as $n\rightarrow\infty$.

Moreover, \big{(}\{X_{a}^{(*,n)}\}_{a\in E_{*}}\big{)}_{n\in\mathbb{N}} is minimally regular if (I)-(II) hold, but (III) is only assumed for $m=4$.

Remark 6.13**.**

The first example of a regular sequence \big{(}\big{\{}X_{a}^{(*,n)}\big{\}}_{a\in E_{*}}\big{)}_{n\in\mathbb{N}} of $\mathcal{Q}$-pyramidic arrays that we have in mind is when the random variables in the generating arrays \big{\{}X_{h}^{(n)}\}_{h\in E_{n}} are defined as in (6.1) with $\beta\equiv\beta_{n,r}$ having the large $n$ asymptotics (2.5) for some $r\in{\mathbb{R}}$. The variance criterion (II) in Definition 6.12 holds by (2.7) and the higher even moment criterion (III) merely follows from the fact that $\beta_{n,r}$ vanishes as $n\rightarrow\infty$.

Proposition 6.14 generalizes the result (2.9) in Theorem 2.4 about the convergence of the higher centered moments of $W^{\omega}_{n}(\beta_{n,r})$. We omit the proof, which is the same as that of part (i) of Theorem 3.3 of [10], or said differently, the proof of part (i) of Theorem 3.3 of [10] proceeds by implicitly proving Proposition 6.14.

Proposition 6.14**.**

Let \big{(}\big{\{}X_{a}^{(*,n)}\big{\}}_{a\in E_{*}}\big{)}_{n\in\mathbb{N}} be a sequence of $\mathcal{Q}$-pyramidic arrays of random variables generated from a sequence of arrays \big{(}\{X_{h}^{(n)}\}_{h\in E_{n}}\big{)}_{n\in\mathbb{N}} satisfying properties (I)-(II) in Definition 6.12 for some $r\in{\mathbb{R}}$. If the $\mathfrak{p}^{th}$ even moment of the random variables in the array \big{(}\{X_{h}^{(n)}\}_{h\in E_{n}}\big{)}_{n\in\mathbb{N}} vanishes as $n\rightarrow\infty$, then for each $m\in\{2,3,\ldots,2\mathfrak{p}\}$ the $m^{th}$ moment of the random variables X^{(0,n)}=\mathcal{Q}^{n}\big{\{}X_{h}^{(n)}\big{\}}_{h\in E_{n}} converges to $R^{(m)}(r)$ as $n\rightarrow\infty$, where $R^{(m)}:{\mathbb{R}}\rightarrow[0,\infty)$ is the function in Theorem 2.4.

The statement of the following lemma is formulated to emphasize the connection with the properties (I)-(III) in Theorem 6.16 below that we use to characterize the limit law emerging as $n\rightarrow\infty$.

Lemma 6.15**.**

The statements below hold for any regular sequence \big{(}\big{\{}X_{a}^{(*,n)}\big{\}}_{a\in E_{*}}\big{)}_{n\in\mathbb{N}} of $\mathcal{Q}$-pyramidic arrays with parameter $r\in{\mathbb{R}}$.

1. (I)

For each $n$ and $k$, the variables in the array \big{\{}X_{a}^{(k,n)}\big{\}}_{a\in E_{n}} are i.i.d. 2. (II)

For each $n$ and $k\geq 1$, the array \mathcal{Q}\big{\{}X_{a}^{(k,n)}\big{\}}_{a\in E_{k}} is equal to \big{\{}X_{a}^{(k-1,n)}\big{\}}_{a\in E_{k-1}}. 3. (III)

For each $n$ and $k$, the variables in the array $\{X_{a}^{(k,n)}\}_{a\in E_{k}}$ are centered, and the variables have finite $m^{th}$ moment that converges to $R^{(m)}(r-k)$ as $n\rightarrow\infty$ for every $k$ and $m\in\{2,3,\ldots\}$.

The above hold for minimally regular sequences except the convergence in (III) is only for $m\in\{2,3,4\}$.

Proof.

Statements (I) and (II) of Lemma 6.15 are immediate consequences of the definition of the variable arrays \big{\{}X_{a}^{(k,n)}\big{\}}_{a\in E_{k}}. To see (III), note that for $a\in E_{k}$ we have X_{a}^{(k,n)}=\mathcal{Q}^{n-k}\big{\{}X_{h}^{(n)}\big{\}}_{h\in a\cap E_{n}}. By definition, the random variables $X_{h}^{(n)}$ have variance satisfying the large $n$ asymptotics (2.7), which we can rewrite in the form

[TABLE]

Notice that (6.3) has the form (2.7) with $n$ and $r$ replaced by $n-k$ and $r-k$, respectively. It follows from Proposition 6.14 that the $m^{th}$ moment of X_{a}^{(k,n)}=\mathcal{Q}^{n-k}\big{\{}X_{h}^{(n)}\big{\}}_{h\in a\cap E_{n}} converges to $R^{(m)}(r-k)$ as $n\rightarrow\infty$ for each $m\in\{2,3,\ldots\}$.∎

6.3 A limit theorem for $\mathcal{Q}$-pyramidic arrays

Theorems 6.16 & 6.23 below are the main technical results of this article, and they are jointly proved in Section 9.3. Theorem 6.16 characterizes the limiting law for the distributional convergence statement in Theorem 6.23.

Theorem 6.16** (Limit law).**

For any $r\in{\mathbb{R}}$, there exists a unique law on sequences in $k\in\mathbb{N}_{0}$ of edge-labeled arrays of random variables, \big{\{}\mathbf{X}_{a}^{(k)}\big{\}}_{a\in E_{k}}, taking values in $[-1,\infty)$ and holding the properties (I)-(III) below.

1. (I)

For each $k\in\mathbb{N}_{0}$, the variables in the array \big{\{}\mathbf{X}_{a}^{(k)}\big{\}}_{a\in E_{k}} are i.i.d. 2. (II)

For each $k\in\mathbb{N}$, the array \big{\{}\mathbf{X}_{a}^{(k-1)}\big{\}}_{a\in E_{k-1}} is equal to \mathcal{Q}\big{\{}\mathbf{X}_{a}^{(k)}\big{\}}_{a\in E_{k}}. 3. (III)

For each $k\in\mathbb{N}_{0}$, the variables in the array \big{\{}\mathbf{X}_{a}^{(k)}\big{\}}_{a\in E_{k}} are centered and have $m^{th}$ moment equal to $R^{(m)}(r-k)$ for all $m\in\{2,3,\ldots\}$.

Notation 6.17**.**

In the $k=0$ case of the random variables $\mathbf{X}_{a}^{(k)}$ from Theorem 6.16, i.e., the peak of the infinite $\mathcal{Q}$-pyramidic array of random variables, we will drop the scripts $a$ & $(k)$ and optionally attach the parameter $r\in{\mathbb{R}}$ as a subscript: $\mathbf{X}_{a}^{(0)}\equiv\mathbf{X}\equiv\mathbf{X}_{r}$.

Remark 6.18**.**

By hierarchical symmetry, the random variables in the arrays \big{\{}\mathbf{X}_{a}^{(k)}\big{\}}_{a\in E_{k}} from Theorem 6.16 with parameter $r\in{\mathbb{R}}$ are equal in distribution to $\mathbf{X}_{r-k}$.

Remark 6.19**.**

The limit law $\mathbf{W}_{r}$ in Theorem 2.7 is equal in distribution to $1+\mathbf{X}_{r}$.

Remark 6.20**.**

Let \big{(}\big{\{}\mathbf{X}_{a}^{(k)}\big{\}}_{a\in E_{k}}\big{)}_{k\in\mathbb{N}_{0}} be a sequence of arrays of random variables satisfying the properties in the statement of Theorem 6.16. For the purpose of proving the uniqueness in Theorem 6.16, it will be useful to make the trivial observation that the sequence of $\mathcal{Q}$-pyramidic arrays \big{(}\big{\{}\mathbf{X}_{a}^{(*,n)}\big{\}}_{a\in E_{*}}\big{)}_{n\in\mathbb{N}} defined by \big{\{}\mathbf{X}_{a}^{(k,n)}\big{\}}_{a\in E_{k}}\equiv\big{\{}\mathbf{X}_{a}^{(k)}\big{\}}_{a\in E_{k}} for $0\leq k\leq n$ is regular with parameter $r$.

In the sequel we will evaluate the distance between measures on ${\mathbb{R}}$ using Wasserstein-$1$ & -$2$ metrics.

Definition 6.21** (Wasserstein distance).**

For two Borel probability measures $\mu$ and $\nu$ on ${\mathbb{R}}$, let $\mathcal{M}_{\mu,\nu}$ be the set of joint measures $J(dx,dy)$ on ${\mathbb{R}}^{2}$ with marginals $\mu$ and $\nu$. For $p\geq 1$ assume that $\mu$ and $\nu$ satisfy $\int_{{\mathbb{R}}}|x|^{p}\mu(dx)<\infty$ and $\int_{{\mathbb{R}}}|x|^{p}\nu(dx)<\infty$. We define the Wasserstein-$p$ distance between $\mu$ and $\nu$ as

[TABLE]

If $X$ and $Y$ are random variables with distributional measures $\mu$ and $\nu$, respectively, then we extend our notation through the interpretation $\rho_{p}(X,Y)\equiv\rho_{p}(\mu,\nu)$.

We prove the following proposition on the distributional continuity of $r\,\mapsto\,\mathbf{X}_{r}$ in Section 12.1.

Proposition 6.22**.**

Let $\mathbf{X}_{r}$ be defined as in Notation 6.17. The law of $\mathbf{X}_{r}$ is a locally $\frac{1}{2}$-Hölder continuous function of $r\in{\mathbb{R}}$ with respect to the Wasserstein-$2$ metric.

By Remark 6.13 the limit theorem below implies Theorem 2.7.

Theorem 6.23**.**

Let \big{(}\{X_{a}^{(*,n)}\}_{a\in E_{*}}\big{)}_{n\in\mathbb{N}} be a minimally regular sequence of $\mathcal{Q}$-pyramidic arrays of random variables with parameter $r\in{\mathbb{R}}$. For any $k\in\mathbb{N}_{0}$ and $a\in E_{k}$, the Wasserstein-2 distance between $X_{a}^{(k,n)}$ and $\mathbf{X}_{a}^{(k)}$ vanishes as $n\rightarrow\infty$, and, in particular, the i.i.d. array \big{\{}X_{a}^{(k,n)}\big{\}}_{a\in E_{k}} (viewed as taking values in ${\mathbb{R}}^{b^{2k}}$) converges in law to \big{\{}\mathbf{X}_{a}^{(k)}\big{\}}_{a\in E_{k}} for each $k\in\mathbb{N}_{0}$.

Remark 6.24**.**

The hierarchical symmetry of the model implies that it is sufficient to prove Theorem 6.23 for the case $k=0$ in which the arrays \big{\{}X_{a}^{(k,n)}\big{\}}_{a\in E_{k}} and \big{\{}\mathbf{X}_{a}^{(k)}\big{\}}_{a\in E_{k}} are single random variables $X^{(0,n)}$ and $\mathbf{X}$, respectively. The proof of Theorem 6.23 involves writing X^{(0,n)}=\mathcal{Q}^{N}\big{\{}X_{e}^{(N,n)}\big{\}}_{e\in E_{N}} and \mathbf{X}=\mathcal{Q}^{N}\big{\{}\mathbf{X}_{e}^{(N)}\big{\}}_{e\in E_{N}} for $N\in\mathbb{N}$ with $1\ll N\ll n$ and introducing arrays of random variables \big{\{}\mathbf{\widetilde{X}}_{e}^{(N)}\big{\}}_{e\in E_{N}} (Definition 9.5) for which we show that $X_{e}^{(N,n)}\stackrel{{\scriptstyle d}}{{\approx}}\mathbf{\widetilde{X}}_{e}^{(N)}$ and $\mathbf{X}_{e}^{(N)}\stackrel{{\scriptstyle d}}{{\approx}}\mathbf{\widetilde{X}}_{e}^{(N)}$ in an appropriately strong sense that is characterized in Proposition 9.1.

7 Rate of convergence under stricter moment assumptions

In this section we will state an alternative version of the limit result in Theorem 6.23 that offers more explicit rates of distributional convergence as $n\rightarrow\infty$ under stronger moment assumptions on the arrays of random variables from which the $\mathcal{Q}$-pyramidic arrays are generated. The conditions of the limit theorem easily translate into conditions for checking that a family of regular sequences of $\mathcal{Q}$-pyramidic arrays of random variables depending on an auxiliary parameter $s\in S$ is uniformly convergent with respect to the Wasserstein-$2$ metric (Corollary 7.5). The following definition characterizes our new assumptions.

Definition 7.1**.**

Fix some $\alpha\in(0,1)$. A regular sequence of $\mathcal{Q}$-pyramidic arrays of random variables \big{(}\{X_{a}^{(*,n)}\}_{a\in E_{*}}\big{)}_{n\in\mathbb{N}} with parameter $r\in{\mathbb{R}}$ is said to be $\alpha$-sharply regular if the sequence of generating arrays \big{\{}X_{h}^{(n)}\big{\}}_{h\in E_{n}} satisfies the following more restrictive forms of (II) and (III) in Definition 6.12:

1. (II*)

The variance of the random variables in the array \big{\{}X_{h}^{(n)}\big{\}}_{h\in E_{n}} has the asymptotics (6.2) with \mathit{o}\big{(}\frac{1}{n^{2}}\big{)} replaced by \mathit{O}\big{(}\frac{1}{n^{2+\alpha}}\big{)}. 2. (III*)

For each $m\in\{4,6,\ldots\}$, the $m^{th}$ moment of the random variables in the array \big{\{}X_{h}^{(n)}\big{\}}_{h\in E_{n}} is $\mathit{O}(n^{-m/2})$ as $n\rightarrow\infty$.

Remark 7.2**.**

The sequence \big{(}\{X_{a}^{(*,n)}\}_{a\in E_{*}}\big{)}_{n\in\mathbb{N}} of $\mathcal{Q}$-pyramidic arrays generated by arrays \big{\{}X_{h}^{(n)}\}_{h\in E_{n}} defined as in (6.1) where $\beta\equiv\beta_{n,r}$ has the large $n$ asymptotics (2.5) with \mathit{o}\big{(}\frac{1}{n^{3/2}}\big{)} replaced by \mathit{O}\big{(}\frac{1}{n^{3/2+\alpha}}\big{)} is $\alpha$-sharply regular. Property (III*) holds since $\beta_{n,r}$ is \mathit{O}\big{(}\frac{1}{n^{1/2}}\big{)} as $n\rightarrow\infty$ and property (II*) follows from the computation in Appendix A.

The following theorem, which we prove in Section 13, implies that if \big{(}\{X_{a}^{(*,n)}\}_{a\in E_{*}}\big{)}_{n\in\mathbb{N}} is an $\alpha$-sharply regular sequence of $\mathcal{Q}$-pyramidic arrays of random variables with parameter $r\in{\mathbb{R}}$, then the Wasserstein-$2$ distance between $X^{(0,n)}$ (i.e, the peak of the $n^{th}$ $\mathcal{Q}$-pyramidic array in the sequence) and the limit law $\mathbf{X}_{r}$ vanishes with order $n^{-\upsilon}$ as $n\rightarrow\infty$ for any choice of $\upsilon\in(0,\alpha/9)$. By hierarchical symmetry, this generalizes to the convergence of the random variables \big{\{}X^{(k,n)}_{a}\big{\}}_{a\in E_{k}} in the higher generation (i.e., $k\geq 1$) array layers. The statement of Theorem 7.3 is formulated to provide easily verifiable conditions under which a family of $\alpha$-sharply regular sequences of $\mathcal{Q}$-pyramidic arrays of random variables can be shown to be uniformly convergent in law; see Corollary 7.5.

Theorem 7.3**.**

Fix $\mathbf{v},\varkappa>0$, $\alpha\in(0,1)$, $\upsilon\in(0,\alpha/9)$, and a bounded interval $\mathcal{I}\subset{\mathbb{R}}$. Define $\mathfrak{p}:=\lceil\frac{2\alpha}{\alpha-9\upsilon}\rceil+1$. There exists a positive number $C\equiv C(\mathcal{I},\mathbf{v},\varkappa,\alpha,\upsilon)$ such that for any $r\in\mathcal{I}$, $n\in\mathbb{N}$, and i.i.d. array of centered random variables \big{\{}X_{h}^{(n)}\big{\}}_{h\in E_{n}} satisfying

1. (I)

\left|\textup{Var}\big{(}X_{h}^{(n)}\big{)}\,-\,\kappa^{2}\big{(}\frac{1}{n}+\frac{\eta\log n}{n^{2}}+\frac{r}{n^{2}}\big{)}\right|\,<\,\frac{\mathbf{v}}{n^{2+\alpha}}* and* 2. (II)

\mathbb{E}\left[\big{|}X_{h}^{(n)}\big{|}^{2\mathfrak{p}}\right]\,<\,\frac{\varkappa}{n^{\mathfrak{p}}},

the peak, $X^{(0,n)}$, of the $\mathcal{Q}$-pyramidic array, \big{\{}X_{a}^{(*,n)}\big{\}}_{a\in E_{*}}, generated by \big{\{}X_{h}^{(n)}\big{\}}_{h\in E_{n}} has distance less than $Cn^{-\upsilon}$ from $\mathbf{X}_{r}$ with respect to the Wasserstein-2 metric.

Remark 7.4**.**

Our proof of Theorem 7.3 follows essentially the same track as the proof of Theorem 6.23 except for the use of technical lemmas that fit with this particular formulation of the distributional convergence. Through a different proof method, it may be possible to extend the range of the exponent $\upsilon$ to a larger interval, e.g., $(0,\alpha/6)$.

The next corollary is a direct consequence of Theorem 7.3.

Corollary 7.5**.**

Fix $\mathbf{v},\varkappa>0$, $\alpha\in(0,1)$, $\upsilon\in(0,\alpha/9)$, and a bounded interval $\mathcal{I}\subset{\mathbb{R}}$. Let $\mathfrak{r}$ be a function from a set $S$ into $\mathcal{I}$. For some $n\in\mathbb{N}$ and all $s\in S$, let \big{\{}X_{h}^{(n)}(s)\big{\}}_{h\in E_{n}} be an i.i.d. array of random variables satisfying conditions (I)-(II) in Theorem 7.3 with parameter $r\equiv\mathfrak{r}(s)$. The inequality below holds for the $C\equiv C(\mathcal{I},\mathbf{v},\varkappa,\alpha,\upsilon)$ in Theorem 7.3.

[TABLE]

Fix $T>0$ and $r\in\mathbb{R}$. The following example applies Corollary 7.5 to uniformly approximate the random variables $\mathbf{X}_{r+t}$ for $t$ in the interval $[0,T]$ by $\mathcal{Q}^{n}$ applied to an i.i.d. array $\{X_{h}^{(n)}(r,t)\}_{h\in E_{n}}$, where the variables $X_{h}^{(n)}(r,t)$ are log-normal perturbations of the variables $\mathbf{X}_{h}^{(n)}$ from Theorem 6.16. The construction below is used in the proof of Proposition 6.22 and is closely related to the Gaussian multiplicative chaos construction in (4.7).

Example 7.6**.**

Let the array of random variables $\{\mathbf{X}_{h}^{(n)}\}_{h\in E_{n}}$ be defined as in Theorem 6.16 for some parameter value $r\in{\mathbb{R}}$ and $\{\mathbf{B}^{h}\}_{h\in E_{n}}$ be an array of independent standard Brownian motions independent of $\{\mathbf{X}_{h}^{(n)}\}_{h\in E_{n}}$. For $t\in[0,T]$ define

[TABLE]

Note that when $t=0$ the random variable $X^{\mathbf{B}}_{n,r,t}$ is equal in distribution to $\mathbf{X}_{r}$ by (II) of Theorem 6.16. The variance of $X_{h}^{(n)}(r,t)$ has the large $n$ asymptotic form

[TABLE]

where we have used (II) of Lemma 2.3. Moreover, the error term is uniformly bounded by a single multiple of $\frac{\log^{2}n}{n^{3}}$ for all $t\in[0,T]$. By writing $X_{h}^{(n)}(r,t)$ as a sum of $\mathbf{X}_{h}^{(n)}$ and \big{(}1+\mathbf{X}_{h}^{(n)}\big{)}\big{(}e^{\frac{\kappa}{n}\mathbf{B}^{h}_{t}-\frac{\kappa^{2}}{2n^{2}}t}-1\big{)}, the $\mathfrak{p}^{th}$ even moment of $X_{h}^{(n)}(r,t)$ can be shown to be \mathit{O}\big{(}\frac{1}{n^{\mathfrak{p}}}\big{)} using that

[TABLE]

The approximation above for $R^{(2\mathfrak{p})}(s)$ when $-s\gg 1$ is from (II) of Theorem 2.4. It follows that the arrays \big{\{}X_{h}^{(n)}(r,t)\big{\}}_{h\in E_{n}} satisfy the conditions (I)-(II) of Theorem 7.3 for any fixed $\alpha\in(0,1)$ and all $n\in\mathbb{N}$ and $t\in[0,T]$ for large enough $\mathbf{v},\varkappa>0$. By Corollary 7.5, the random variables $X^{\mathbf{B}}_{n,r,t}$ converge uniformly to $\mathbf{X}_{r+t}$ over $t\in[0,T]$ with respect to the Wasserstein-$2$ metric as $n\rightarrow\infty$.

8 Existence of a limiting $\mathcal{Q}$-pyramidic array of random variables

In this section we prove the existence of the infinite $\mathcal{Q}$-pyramidic array of random variables described in Theorem 6.16. The proof is based on a routine tightness argument involving nested subsequences.

Proof of Theorem 6.16 (existence).

Let \big{(}\big{\{}X_{a}^{(*,n)}\big{\}}_{a\in E_{*}}\big{)}_{n\in\mathbb{N}} be a regular sequence of $\mathcal{Q}$-pyramidic arrays of random variables with parameter $r\in{\mathbb{R}}$, e.g., of the form in Remark 6.13. For any $k\in\mathbb{N}_{0}$, and $a\in E_{k}$, the variance of $X_{a}^{(k,n)}$ converges to $R(r-k)$ as $n\rightarrow\infty$ by Lemma 6.15. In particular, for any fixed $k$ the sequence $\{X_{a}^{(k,n)}\}_{a\in E_{k}}$ of random arrays indexed by $n\in\mathbb{N}$, viewed as a random vector in $\mathbb{R}^{b^{2k}}$, is tight. We define $\xi_{n}^{(k)}\in\mathbb{N}$ inductively in $k\in\mathbb{N}_{0}$ as a nested sequence of subsequences as follows:

• •

Let $(\xi_{n}^{(0)})_{n\in\mathbb{N}}$ be a subsequence of $n=1,2,3,\ldots$ such that the single-element array \big{\{}X_{a}^{(0,\,\xi_{n}^{(0)})}\big{\}}_{a\in E_{0}} converges in law as $n\rightarrow\infty$ to a limit \big{\{}\mathbf{X}_{a}^{(0)}\big{\}}_{a\in E_{0}}.

• •

If for $k\in\mathbb{N}_{0}$ the sequence $(\xi_{n}^{(k)})_{n\in\mathbb{N}}$ has been chosen so that the array \big{\{}X_{a}^{(k,\,\xi_{n}^{(k)})}\big{\}}_{a\in E_{k}} converges in law as $n\rightarrow\infty$ to a limiting array \big{\{}\mathbf{X}_{a}^{(k)}\big{\}}_{a\in E_{k}}, then we choose $(\xi_{n}^{(k+1)})_{n\in\mathbb{N}}$ to be a subsequence of $(\xi_{n}^{(k)})_{n\in\mathbb{N}}$ such that \big{\{}X_{a}^{(k+1,\,\xi^{(k+1)}_{n})}\big{\}}_{a\in E_{k+1}} converges in law to some limit \big{\{}\mathbf{X}_{a}^{(k+1)}\big{\}}_{a\in E_{k+1}}.

With the sequence in $k\in\mathbb{N}_{0}$ of limiting array laws $\{\mathbf{X}_{a}^{(k)}\}_{a\in E_{k}}$ constructed above, we will next consider properties (I)-(III). When it comes to property (II), we will first verify the equality in a distributional sense—see (8.1)—because the arrays $\{\mathbf{X}_{a}^{(k)}\}_{a\in E_{k}}$ constructed above may be defined on different probability spaces for different $k\in\mathbb{N}_{0}$.

Property (I) follows immediately from the construction since all of the arrays, \big{\{}X_{a}^{(k,n)}\big{\}}_{a\in E_{k}}, used in the construction are i.i.d. For property (II) notice that for any $k\in\mathbb{N}$

[TABLE]

where the second equality follows from part (II) of Lemma 6.15, and the third holds by the continuity of the map $\mathcal{Q}$. It follows that for each $k\in\mathbb{N}$ the $\mathcal{Q}$-pyramidic array generated from \big{\{}\mathbf{X}_{a}^{(k-1)}\big{\}}_{a\in E_{k-1}} is equal in distribution to the top $k-1$ layers of the $\mathcal{Q}$-pyramidic array generated by \big{\{}\mathbf{X}_{a}^{(k)}\big{\}}_{a\in E_{k}}. By the Kolmogorov extension theorem, the sequence in $k\in\mathbb{N}_{0}$ of arrays of random variables \big{\{}\mathbf{X}_{a}^{(k)}\big{\}}_{a\in E_{k}} can be defined on a single probability space such that \big{\{}\mathbf{X}_{a}^{(k)}\big{\}}_{a\in E_{k}} is a.s. equal to \mathcal{Q}\big{\{}\mathbf{X}_{a}^{(k-1)}\big{\}}_{a\in E_{k-1}}. For property (III), Lemma 6.15 implies that the $m^{th}$ moment of $X_{a}^{(k,n)}$ converges to the limit $R^{(m)}(r-k)$ for any $a\in E_{k}$ and $m\in\{2,3,\ldots\}$. Since this holds for all $m$, we have that \mathbb{E}\big{[}(\mathbf{X}_{a}^{(k)})^{m}\big{]}=R^{(m)}(r-k) for all $m$ by uniform integrability.

The limiting random variables $\{\mathbf{X}_{a}^{(k)}\}_{a\in E_{k}}$ take values in $[-1,\infty)$ since the random variables \big{\{}1+X_{h}^{(n)}\big{\}}_{h\in E_{n}} are nonnegative by their definition (6.1), and the form of the map $\mathcal{Q}$ implies that the arrays \big{\{}1+X_{a}^{(k,n)}\big{\}}_{a\in E_{k}} for \big{\{}X_{a}^{(k,n)}\big{\}}_{a\in E_{k}}:=\mathcal{Q}^{n-k}\big{\{}X_{h}^{(n)}\big{\}}_{h\in E_{n}} must also be nonnegative. ∎

9 Uniqueness of the limiting $\mathcal{Q}$-pyramidic array and universality

The goal of this section is to prove Theorem 6.23 and, simultaneously, the uniqueness part of Theorem 6.16 after stating the key propositions that enter into the proof. Section 9.1 contains the statement of Proposition 9.1, which is central to the organization of our analysis. In Section 9.2, we heuristically motivate the definitions of the arrays of random variables that have a role in the proof of Theorem 6.23, which is in Section 9.3.

9.1 $\mathbf{L^{2}}$-bound for a contractive dynamics on arrays of random variables

The following proposition provides a condition template by which we can show that the random variables \mathcal{Q}^{N}\big{\{}U_{e}^{(N)}\big{\}}_{e\in E_{N}} and \mathcal{Q}^{N}\big{\{}V_{e}^{(N)}\big{\}}_{e\in E_{N}} are close together under the $L^{2}$ metric on random variables provided that \big{\{}\big{(}U_{e}^{(N)},V_{e}^{(N)}\big{)}\big{\}}_{e\in E_{N}} is an i.i.d.  array of (${\mathbb{R}}^{2}$-valued) random variables and the variables $U_{e}^{(N)}$ and $V_{e}^{(N)}$ are close together in $L^{2}$. In loose terms, we are bounding the sensitivity of the “dynamics” on arrays generated by the map $\mathcal{Q}$ to the initial conditions.

Proposition 9.1**.**

Fix some $s\in{\mathbb{R}}$, and let $N\in\mathbb{N}$. There exist $\delta>0$ and $C>0$ depending only on $s\in{\mathbb{R}}$ such that the statements (i)-(ii) below hold for any i.i.d. array \big{\{}\big{(}U_{e}^{(N)},V_{e}^{(N)}\big{)}\big{\}}_{e\in E_{N}} of centered ${\mathbb{R}}^{2}$-valued random variables for which $U_{e}^{(N)}$ has the variance bound

[TABLE]

1. (i)

If \mathbb{E}\big{[}\big{(}V_{e}^{(N)}-U_{e}^{(N)}\big{)}^{2}\big{]}<\delta/N^{4}, then

[TABLE] 2. (ii)

If \mathbb{E}\big{[}\big{(}V_{e}^{(N)}-U_{e}^{(N)}\big{)}^{2}\big{]}<\delta/N^{2} and the variables $U_{e}^{(N)}$ and $V_{e}^{(N)}-U_{e}^{(N)}$ are uncorrelated, then

[TABLE]

Remark 9.2**.**

In particular, if \big{\{}\big{(}U_{e}^{(N)},V_{e}^{(N)}\big{)}\big{\}}_{e\in E_{N}} is a sequence in $N\in\mathbb{N}$ of arrays of random variables satisfying the conditions of Proposition 9.1 and \mathbb{E}\big{[}\big{(}V_{e}^{(N)}-U_{e}^{(N)}\big{)}^{2}\big{]}=\mathit{o}\big{(}1/N^{4}), then the $L^{2}$ distance between \mathcal{Q}^{N}\big{\{}V_{e}^{(N)}\big{\}}_{e\in E_{N}} and \mathcal{Q}^{N}\big{\{}U_{e}^{(N)}\big{\}}_{e\in E_{N}} vanishes with large $N$.

Remark 9.3**.**

By the asymptotics for $R(r)$ as $r\rightarrow-\infty$ in (II) of Lemma 2.3, the right side of (9.1) is equal to R(s-N)+\mathit{o}\big{(}\frac{1}{N^{2}}\big{)}. The statement of Proposition 9.1 is equivalent if $R(-N)+\frac{\kappa^{2}s}{N^{2}}$ is replaced by $R(s-N)$.

9.2 Defining intermediary distributional approximations

After the heuristic discussion below, we will state Definition 9.5, which defines the arrays of random variables appearing in the proof of Theorem 6.23. Lemmas 9.7-9.9 in the next subsection state bounds for the $L^{2}$ distance/Wasserstein-$2$ distance between the random variables in these arrays, providing opportunities to apply Proposition 9.1.

Let $\{X^{(*,n)}\}_{a\in E_{*}}$ be a minimally regular sequence in $n\in\mathbb{N}$ of $\mathcal{Q}$-pyramidic arrays of random variables. Proposition 9.1 combined with Remark 6.24 suggests a path for proving Theorem 6.23 by showing that for $1\ll N\ll n$ and $e\in E_{N}$ the $L^{2}$ distance between the random variables $X^{(N,n)}_{e}$ and $\mathbf{X}_{e}^{(N)}$ is small for some coupling of the variables. To help orient the reader towards the framework of the analysis in coming sections, we will motivate the definitions of three distributional approximations for the random variable $X^{(N,n)}_{e}$ that have roles in the proof of Theorem 6.23. The analysis will be founded on the introduction of intermediary generational scales $\mathbf{n}(N),\mathbf{\widehat{n}}(N)\in\mathbb{N}$ between $N$ and $n$ that allow us to identify two sources of central limit-type renormalized sums—see (I) and (II) below—within an approximation for $X^{(N,n)}_{e}$. It suffices for us to take

[TABLE]

for a large enough choice of $\mathfrak{m}>0$.101010For the purpose of proving Theorem 6.23, $\mathfrak{m}\log N$ can also be replaced by $N^{\epsilon}$ for any choice of $0<\epsilon<1/2$ in the definitions of $\mathbf{\widehat{n}}(N)$ and $\mathbf{n}(N)$, however, this is not optimal for Theorem 7.3. In particular, when $1\ll N\ll n$

[TABLE]

For notational neatness, we will suppress the dependence of these generational parameters on $N$: $\mathbf{\widehat{n}}(N)\equiv\mathbf{\widehat{n}}$ and $\mathbf{n}(N)\equiv\mathbf{n}$.

Remark 9.4**.**

To enable the reader to distinguish at a glance between arrays having the four distinct generational parameters $N<\mathbf{\widehat{n}}<\mathbf{n}\ll n$, we will maintain a rigid indexing convention in which the arrays with generation numbers $N$, $\mathbf{\widehat{n}}$, $\mathbf{n}$, $n$ are respectively dummy indexed by the letters $e$, $f$, $g$, $h$:

[TABLE]

Recall from (ii) of Notation 6.1 that given an array $\{x_{a}\}_{a\in E_{k}}$ and some $\mathbf{a}\in E_{\ell}$ with $0\leq\ell\leq k$, then $\{x_{a}\}_{a\in\mathbf{a}\cap E_{k}}$ refers to the subarray labeled by all $a\in E_{k}$ canonically embedded in $\mathbf{a}$. From Definition 6.9 we can write X^{(N,n)}_{e}=\mathcal{Q}^{n-N}\big{\{}X_{h}^{(n)}\big{\}}_{h\in e\cap E_{n}}. For any $\mathbf{n}$ defined as above with $n\geq\mathbf{n}$, this equality can be rewritten using the identity $\mathcal{Q}=\mathcal{L}+\mathcal{E}$ as

[TABLE]

The braced expressions above are central limit-type normalized sums (recall Remark 6.8), and thus admit Gaussian approximations when $\mathbf{n}-\mathbf{\widehat{n}}\gg 1$ and $\mathbf{\widehat{n}}-N\gg 1$:

1. (I)

For $e\in E_{N}$ the variables in the array \big{\{}Y_{f}^{N,n}\big{\}}_{f\in e\cap E_{\mathbf{\widehat{n}}}}:=\mathcal{L}^{\mathbf{n}-\mathbf{\widehat{n}}}\mathcal{Q}^{n-\mathbf{n}}\big{\{}X_{h}^{(n)}\big{\}}_{h\in e\cap E_{n}} are approximately distributed as

[TABLE]

because the variables in the array \mathcal{Q}^{n-\mathbf{n}}\big{\{}X_{h}^{(n)}\big{\}}_{h\in e\cap E_{n}} have variance approximately equal to $R(r-\mathbf{n})$ when $n\gg 1$ by Lemma 6.15. 2. (II)

For \displaystyle Z_{f}^{N,n}:=\sum_{k=1}^{\mathbf{n}-\mathbf{\widehat{n}}}\mathcal{L}^{k-1}\mathcal{E}\mathcal{L}^{\mathbf{n}-\mathbf{\widehat{n}}-k}\mathcal{Q}^{n-\mathbf{n}}\big{\{}X_{h}^{(n)}\big{\}}_{h\in f\cap E_{n}}, the variable \displaystyle\macc@depth\char 1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a 111{Z}_{e}^{N,n}:=\mathcal{L}^{\mathbf{\widehat{n}}-N}\big{\{}Z_{f}^{N,n}\big{\}}_{f\in e\cap E_{\mathbf{\widehat{n}}}} has approximate distribution

[TABLE]

The variance $\varsigma_{N}^{2}$ is the asymptotic variance of $Z_{e}^{N,n}$ as $n\rightarrow\infty$ as will be shown in Lemma 11.3.

The above line of heuristic reasoning suggests that variables in the array \big{\{}X^{(N,n)}_{e}\}_{e\in E_{N}} are close in distribution to variables in the array \big{\{}\mathbf{\widetilde{X}}^{(N)}_{e}\}_{e\in E_{N}} defined in (iii) of Definition 9.5 below. The random variables $\widehat{X}^{N,n}_{e}$ and $\mathbf{\widehat{X}}^{N,n}_{e}$ in (i) & (ii) of Definition 9.5 serve as distributional intermediaries between $X^{(N,n)}_{e}$ and $\mathbf{\widetilde{X}}^{(N)}_{e}$; see the Wasserstein-$2$ bounds for their differences in Lemmas 9.7-9.9. Note that $\widehat{X}^{N,n}_{e}$ in (i) is merely a different way of writing (9.2).

Definition 9.5**.**

Let $\mathbf{\widehat{n}},\mathbf{n}\in\mathbb{N}$ be defined as in (9.2) for a given value of $\mathfrak{m}>0$, and let the i.i.d. arrays of random variables \big{\{}Y_{f}^{N,n}\big{\}}_{f\in E_{\mathbf{\widehat{n}}}}, \big{\{}\macc@depth\char 1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a 111{Z}_{e}^{N,n}\big{\}}_{e\in E_{N}}, \big{\{}\mathbf{Y}_{f}^{(N)}\big{\}}_{f\in E_{\mathbf{\widehat{n}}}} and \big{\{}\mathbf{Z}_{e}^{(N)}\big{\}}_{e\in E_{N}} be defined as in (I) and (II) above.

1. (i)

We define variables in the array \big{\{}\widehat{X}^{N,n}_{e}\big{\}}_{e\in E_{N}} as

[TABLE] 2. (ii)

For \big{\{}Y_{f}^{N,n}\big{\}}_{f\in E_{\mathbf{\widehat{n}}}} and \big{\{}\mathbf{Z}_{e}^{(N)}\big{\}}_{e\in E_{N}} independent, we define the i.i.d. array \big{\{}\mathbf{\widehat{X}}^{N,n}_{e}\big{\}}_{e\in E_{N}} to have variables with distribution

[TABLE] 3. (iii)

For \big{\{}\mathbf{Y}_{f}^{(N)}\big{\}}_{f\in E_{\mathbf{\widehat{n}}}} and \big{\{}\mathbf{Z}_{e}^{(N)}\big{\}}_{e\in E_{N}} independent, we define the i.i.d. array \big{\{}\mathbf{\widetilde{X}}_{e}^{(N)}\big{\}}_{e\in E_{N}} to have variables with distribution

[TABLE]

Remark 9.6**.**

The superscripts of the variables $\widehat{X}^{N,n}_{e}$, $\mathbf{\widehat{X}}^{N,n}_{e}$, $\mathbf{\widetilde{X}}_{e}^{(N)}$, $Y_{f}^{N,n}$, $\mathbf{Y}_{f}^{(N)}$, $Z_{f}^{N,n}$, $\macc@depth\char 1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a 111{Z}_{e}^{N,n}$, and $\mathbf{Z}_{e}^{(N)}$ refer to their dependence on the underlying generational parameters $N,n\in\mathbb{N}$ with $\mathbf{n}\leq n$, whereas the superscript of $X^{(N,n)}_{e}$ (with the parenthesis and two indices) denotes more specifically that the random variable $X^{(N,n)}_{e}$ is an element of the $N^{th}$ layer of a $\mathcal{Q}$-pyramidic array generated from a generation-$n$ array, $\{X^{(n)}_{h}\}_{h\in E_{n}}$.

9.3 Proof of Theorem 6.23

We will prove Theorem 6.23 and the uniqueness part of Theorem 6.16 after stating the crucial Lemmas 9.7-9.9, whose proofs in Section 11 form the core of our technical analysis.

For $N,n\in\mathbb{N}$ with $n\geq\mathbf{n}$ and $e\in E_{N}$, let the random variables $\widehat{X}^{N,n}_{e}$, $\mathbf{\widehat{X}}_{e}^{N,n}$, $\mathbf{\widetilde{X}}_{e}^{(N)}$ be defined as in Section 9.2 for a minimally regular sequence, \big{(}\{X_{a}^{(*,n)}\}_{a\in E_{*}}\big{)}_{n\in\mathbb{N}}, of $\mathcal{Q}$-pyramidic arrays with parameter $r\in{\mathbb{R}}$ and a choice of the parameter $\mathfrak{m}>0$ in the equations (9.2) defining $\mathbf{n}$ and $\mathbf{\widehat{n}}$. The lemmas below imply that the pairs \big{(}X^{(N,n)}_{e},\widehat{X}^{N,n}_{e}\big{)}, \big{(}\widehat{X}^{N,n}_{e},\mathbf{\widehat{X}}_{e}^{N,n}\big{)}, and \big{(}\mathbf{\widehat{X}}_{e}^{N,n},\mathbf{\widetilde{X}}^{(N)}_{e}\big{)} satisfy the conditions (i) or (ii) of Proposition 9.1 when $\mathfrak{m}\geq\frac{5}{\log b}$ after appropriate couplings of the variables for the latter two pairs. The constants $\mathbf{c}>0$ in the statements of the next three lemmas depend on $\mathfrak{m}>0$ and the sequence \big{(}\{X_{a}^{(*,n)}\}_{a\in E_{*}}\big{)}_{n\in\mathbb{N}}.

Lemma 9.7, which is proved in Section 11.1, bounds the error in $L^{2}$ resulting from the partial linear approximation in (9.2).

Lemma 9.7**.**

The random variables $X^{(N,n)}_{e}-\widehat{X}^{N,n}_{e}$ and $\widehat{X}^{N,n}_{e}$ are uncorrelated. There is a positive number $\mathbf{c}$ such that for any $N\in\mathbb{N}$ the inequality below holds for all large enough $n\in\mathbb{N}$.

[TABLE]

Lemma 9.8 provides a bound for the error, when measured in terms of the Wasserstein-2 distance, of the Gaussian approximation heuristically motivated in (I) of Section 9.2. The proof is in Section 11.3 and uses a perturbative generalization of Stein’s method that is discussed in Section 11.2.

Lemma 9.8**.**

There exists a positive number $\mathbf{c}$ such that for any $N\in\mathbb{N}$ the inequality below holds for all large enough $n\in\mathbb{N}$.

[TABLE]

Lemma 9.9 bounds the Wasserstein-2 distance error resulting from the Gaussian approximation heuristically motivated in (II) of Section 9.2. The proof is in Section 11.4 and uses a bound (Lemma 11.6) that follows from the zero bias approach to Stein’s method, which is discussed in Appendix C.

Lemma 9.9**.**

There exists a positive number $\mathbf{c}$ such that for any $N\in\mathbb{N}$ the inequality below holds for all large enough $n\in\mathbb{N}$.

[TABLE]

Remark 9.10**.**

By definition of $\rho_{2}$, Lemmas 9.8 & 9.9 imply that there are couplings \big{(}\widehat{X}_{e}^{N,n},\mathbf{\widehat{X}}_{e}^{N,n}\big{)} and \big{(}\mathbf{\widehat{X}}_{e}^{N,n},\mathbf{\widetilde{X}}_{e}^{(N)}\big{)} such that \mathbb{E}\big{[}\big{(}\widehat{X}_{e}^{N,n}-\mathbf{\widehat{X}}^{N,n}_{e}\big{)}^{2}\big{]} and \mathbb{E}\big{[}\big{(}\mathbf{\widehat{X}}_{e}^{N,n}-\mathbf{\widetilde{X}}^{(N)}_{e}\big{)}^{2}\big{]} are $<\mathbf{c}N^{-\frac{\mathfrak{m}}{3}\log b-\frac{1}{3}}$ for large $n$.

Remark 9.11**.**

When applying Proposition 9.1 in the proof of Theorem 6.23, we only need that the bounds $\mathbf{a}_{N}:=\mathbf{c}N^{-3/2}\log(N+1)$, $\mathbf{b}_{N}:=\mathbf{c}N^{-1/3-\frac{\mathfrak{m}}{3}\log b}\log^{-1/6}(N+1)$, and $\mathbf{c}_{N}:=\mathbf{c}N^{-1/2-\frac{\mathfrak{m}}{3}\log b}$ in Propositions 9.7-9.9 are respectively $\mathit{o}(N^{-1})$, $\mathit{o}(N^{-2})$, and $\mathit{o}(N^{-2})$ for which it is sufficient to assume that $\mathfrak{m}\geq 5/\log b$ for $\mathbf{b}_{N}$ and $\mathbf{c}_{N}$.

The following easy corollary of Lemmas 9.7 - 9.9 verifies the condition (9.1) in the statement of Proposition 9.1 for the pairs of random variables discussed above, and its proof is in Section 12.2.

Corollary 9.12**.**

Define $\mathfrak{m}:=5/\log b$. For any $s\in(r,\infty)$ the inequality \mathbb{E}\big{[}\big{(}U_{e}^{(N)}\big{)}^{2}\big{]}<R(-N)+\frac{\kappa^{2}s}{N^{2}} holds for $U_{e}^{(N)}$ equal to $\widehat{X}^{N,n}_{e}$, $\mathbf{\widehat{X}}^{N,n}_{e}$, and $\mathbf{\widetilde{X}}^{(N)}_{e}$ for large enough $N$ and $n\geq\mathbf{n}$.

Remark 9.13**.**

The relevant sense of a given statement holding “for large enough $N$ and $n$” will always be that there exists a constant $\lambda>0$ and an increasing function $\Lambda:\mathbb{N}\rightarrow(0,\infty)$ such that the statement is true whenever $N>\lambda$ and $n>\Lambda(N)$.

Let us temporarily assume Proposition 9.1, Lemmas 9.7 - 9.9, and Corollary 9.12 to complete the remainder of the proof of Theorem 6.23. As in Corollary 9.12, we will define $\mathfrak{m}:=5/\log b$ for the reason explained in Remark 9.11.

Proof of Theorem 6.23 and Theorem 6.16 (uniqueness part).

Let \big{(}\{X_{a}^{(*,n)}\}_{a\in E_{*}}\big{)}_{n\in\mathbb{N}} be a minimally regular sequence of $\mathcal{Q}$-pyramidic arrays of random variables with parameter $r\in{\mathbb{R}}$. By Remark 6.24 it suffices for us to focus on distributional convergence in the case $k=0$ in which the array \big{\{}X^{(k,n)}_{a}\big{\}}_{a\in E_{k}} consists of a single random variable, $X^{(0,n)}$. We have divided the analysis below into parts (a)-(d).

(a) Setting up: For $n\geq\mathbf{n}$ let the arrays of random variables \big{\{}\widehat{X}_{e}^{N,n}\big{\}}_{e\in E_{N}}, \big{\{}\mathbf{\widehat{X}}_{e}^{N,n}\big{\}}_{e\in E_{N}}, and \big{\{}\mathbf{\widetilde{X}}_{e}^{(N)}\big{\}}_{e\in E_{N}} be defined as in Definition 9.5. We will show that the Wasserstein-$2$ distance between $X^{(0,n)}$ and \mathcal{Q}^{N}\big{\{}\mathbf{\widetilde{X}}_{e}^{(N)}\big{\}}_{e\in E_{N}} converges to zero as $N$ and $n$ grow. Writing X^{(0,n)}=\mathcal{Q}^{N}\big{\{}X_{e}^{(N,n)}\big{\}}_{e\in E_{N}} and applying the triangle inequality yields

[TABLE]

The random variables \mathcal{Q}^{N}\big{\{}\widehat{X}_{e}^{N,n}\big{\}}_{e\in E_{N}} and \mathcal{Q}^{N}\big{\{}X_{e}^{(N,n)}\big{\}}_{e\in E_{N}} are already defined in the same probability space, and we will not require any special coupling between them. Notice that the expressions on the right side above have the form of those expressions bounded in Proposition 9.1.

(b) Verifying the conditions of Proposition 9.1: By Lemma 9.7 the variables $X_{e}^{(N,n)}-\widehat{X}_{e}^{N,n}$ and $\widehat{X}_{e}^{N,n}$ are uncorrelated, and there is a positive sequence $\{\mathbf{a}_{N}\}_{N\in\mathbb{N}}$ with $\mathbf{a}_{N}=\mathit{o}(N^{-1})$ such that

[TABLE]

for any fixed $N$ and large enough $n$. By Lemmas 9.8 & 9.9 and Remark 9.11, there is a positive sequence $\{\mathbf{b}_{N}\}_{N\in\mathbb{N}}$ with $\mathbf{b}_{N}=\mathit{o}(N^{-2})$ and i.i.d. couplings \big{\{}\big{(}\widehat{X}_{e}^{N,n},\mathbf{\widehat{X}}_{e}^{N,n}\big{)}\big{\}}_{e\in E_{N}} and \big{\{}\big{(}\mathbf{\widehat{X}}_{e}^{N,n},\mathbf{\widetilde{X}}_{e}^{(N)}\big{)}\big{\}}_{e\in E_{N}} such that

[TABLE]

for any fixed $N$ and large enough $n\geq\mathbf{n}$. Corollary 9.12 implies that the arrays \big{\{}\widehat{X}_{e}^{N,n}\big{\}}_{e\in E_{N}}, \big{\{}\mathbf{\widehat{X}}_{e}^{N,n}\big{\}}_{e\in E_{N}}, \big{\{}\mathbf{\widetilde{X}}_{e}^{(N)}\big{\}}_{e\in E_{N}} satisfy condition (9.1) of Proposition 9.1 for any $s\in(r,\infty)$ and large $N$ and $n$. Moreover, the above considerations imply that for large enough $N$ and $n$ we have the following:

• •

the array \big{\{}\big{(}\widehat{X}_{e}^{N,n},X_{e}^{(N,n)}\big{)}\big{\}}_{e\in E_{N}} satisfies the conditions for part (ii) of Proposition 9.1 with \big{(}\widehat{X}_{e}^{N,n},X_{e}^{(N,n)}\big{)}=\big{(}U_{e}^{(N)},V_{e}^{(N)}\big{)},

• •

the arrays \big{\{}\big{(}\widehat{X}_{e}^{N,n},\mathbf{\widehat{X}}_{e}^{N,n}\big{)}\big{\}}_{e\in E_{N}} satisfy the conditions for part (i) of Proposition 9.1, and

• •

the arrays \big{\{}\big{(}\mathbf{\widehat{X}}_{e}^{N,n},\mathbf{\widetilde{X}}_{e}^{(N)}\big{)}\big{\}}_{e\in E_{N}} satisfy the conditions for part (i) of Proposition 9.1.

(c) Returning to (9.7): Therefore with three applications of Proposition 9.1 to the right side of (9.7) there is a $C>0$ such that for large enough $N$ and $n\geq\mathbf{n}$ we have the first inequality below.

[TABLE]

The second inequality holds by Lemmas 9.7 - 9.9. As $N\rightarrow\infty$ the above goes to zero by the asymptotic properties of $\mathbf{a}_{N}$ and $\mathbf{b}_{N}$.

(d) Connecting with the random array constructed in Section 8: We have established that the Wasserstein-$2$ distance between $X^{(0,n)}$ and \mathcal{Q}^{N}\big{\{}\mathbf{\widetilde{X}}_{e}^{(N)}\big{\}}_{e\in E_{N}} vanishes as $n$ and $N$ grow. Let \big{\{}\mathbf{X}^{(k)}_{a}\big{\}}_{a\in E_{k}} be the sequence in $k\in\mathbb{N}_{0}$ of arrays of random variables for parameter $r\in{\mathbb{R}}$ constructed in Section 8 through subsequential distributional limits of \big{\{}X_{a}^{(k,n)}\big{\}}_{a\in E_{k}} as $n\rightarrow\infty$. As mentioned in Remark 6.20, the arrays \big{\{}\mathbf{X}^{(k)}_{a}\big{\}}_{a\in E_{k}} form a parameter-$r$ regular sequence of $Q$-pyramidic arrays of random variables with no $n\in\mathbb{N}$ dependence. Thus we can apply the distributional convergence result that we have just proved to the special case \big{\{}X_{a}^{(k,n)}\big{\}}_{a\in E_{k}}:=\big{\{}\mathbf{X}_{a}^{(k)}\big{\}}_{a\in E_{k}} to get that the Wassertstein-$2$ distance between $\mathbf{X}$ and \mathcal{Q}^{N}\big{\{}\mathbf{\widetilde{X}}_{e}^{(N)}\big{\}}_{e\in E_{N}} converges to zero as $N\rightarrow\infty$. Therefore, \rho_{2}\big{(}X^{(0,n)},\mathbf{X}\big{)} vanishes with large $n$ and the law of $\mathbf{X}$ must be unique. ∎

10 Proof of Proposition 9.1

Proof.

By Remark 9.3, the condition (9.1) is equivalent to assuming that the variance of $U_{e}^{(N)}$ is smaller than $R(s-N)$. For $0\leq k\leq N$, define the i.i.d. arrays of random variables

[TABLE]

and $W_{a}^{(k,N)}\,:=\,V_{a}^{(k,N)}\,-\,U_{a}^{(k,N)}$. The variables $U_{a}^{(k,N)}$, $V_{a}^{(k,N)}$, $W_{a}^{(k,N)}$ have mean zero, and $U_{a}^{(k,N)}$ has variance

[TABLE]

for \big{(}\sigma^{(N)}\big{)}^{2}:=\mathbb{E}\big{[}\big{(}U_{e}^{(N)}\big{)}^{2}\big{]}, where the second equality above holds by Remark 6.6 (note that $U_{a}^{(k,N)}$ has the same law as a generation $N-k$ partition function). The inequality uses our assumption that the variance of $U_{e}^{(N)}$ is smaller than $R(s-N)$, and the last equality is property (I) of Lemma 2.3.

We have the following recursive relation for the variables $W_{a}^{(k,N)}$

[TABLE]

Since the arrays are i.i.d. and centered, the recursive formula above shows, by induction, that if $W_{e}^{(N)}:=V_{e}^{(N)}-U_{e}^{(N)}$ is uncorrelated with $U_{e}^{(N)}$ for $e\in E_{N}$ then $W_{a}^{(k,N)}$ is uncorrelated with $U_{a}^{(k,N)}$ for all $0\leq k<N$ and $a\in E_{k}$. In particular if $U_{e}^{(N)}$ and $V_{e}^{(N)}-U_{e}^{(N)}$ are uncorrelated, then \mathcal{Q}^{N}\big{\{}U_{e}^{(N)}\big{\}}_{e\in E_{N}} and \mathcal{Q}^{N}\big{\{}V_{e}^{(N)}\big{\}}_{e\in E_{N}}-\mathcal{Q}^{N}\big{\{}U_{e}^{(N)}\big{\}}_{e\in E_{N}} are uncorrelated.

Define the multivariate polynomial

[TABLE]

The form of the polynomial $P$ implies that there exists a $\mathbf{c}>0$ such that

[TABLE]

for all $(x,y,z)$ with $0\leq x,y\leq 1$ and $|z|\leq\sqrt{xy}$. To see the above inequality, notice that a single term $x^{|A|-u}y^{|B|-u}z^{2u}$ has absolute value bounded by $x^{|A|}y^{|B|}$, and we have $x^{|A|}y^{|B|}\leq x^{2}$ when $|A|\geq 2$ and $x^{|A|}y^{|B|}\leq xy^{2}$ when $|B|\geq 2$ since $|A|\geq 1$.

Let \big{(}\varrho_{k}^{(N)}\big{)}^{2} denote the second moment of $W_{a}^{(k,N)}$, and define u_{k}^{(N)}:=\mathbb{E}\big{[}U_{a}^{(k,N)}W_{a}^{(k,N)}\big{]}. Taking the second moment of (10.2) yields

[TABLE]

where the last inequality follows from (10.1).

In the following analysis, we will temporarily assume that $s<-1$ and that $s$ is sufficiently far in the negative direction so that $R(r)>\frac{\kappa^{2}}{-r}$ for all $r\in(-\infty,s]$, which is possible by the asymptotics for $R(r)$ as $r\rightarrow-\infty$ in (II) of Lemma 2.3. These assumptions ensure that the terms $R(s-\ell)-\frac{\kappa^{2}}{\ell-s}$ in the sums over $\ell\in\mathbb{N}$ below are positive and that the denominator $\ell-s$ is bounded away from zero. Recall that \big{(}\varrho_{N}^{(N)}\big{)}^{2}:=\mathbb{E}\Big{[}\big{(}V_{e}^{(N)}-U_{e}^{(N)}\big{)}^{2}\Big{]} for $e\in E_{N}$. Suppose that \big{(}\varrho_{N}^{(N)}\big{)}^{2}<\delta/N^{2+\epsilon}, where

[TABLE]

Note that $\delta>0$ because property (II) in Lemma 2.3 implies that the series \sum_{\ell=1}^{\infty}\big{(}R(s-\ell)\,-\,\frac{\kappa^{2}}{\ell-s}\big{)} and \sum_{\ell=1}^{\infty}\big{(}R(s-\ell)\big{)}^{2} are summable and because the asymptotics $R(-r)\sim\frac{\kappa^{2}}{r}$ for $r\gg 1$ implies that the infimum above is finite.

Let $\mathbf{k}^{(N)}$ be the smallest $k\in\mathbb{N}_{0}$ such that \big{(}\varrho_{k}^{(N)}\big{)}^{2}\leq\big{(}R(s-k)\big{)}^{2}. Note that the inequality \big{(}\varrho_{N}^{(N)}\big{)}^{2}\leq\big{(}R(s-N)\big{)}^{2} holds by the assumption \big{(}\rho_{N}^{(N)}\big{)}^{2}<\delta/N^{2+\epsilon} and the definition of $\delta$, and thus we must have $\mathbf{k}^{(N)}\leq N$. For $k\in\mathbb{N}_{0}$ with k+1\in\big{[}\mathbf{k}^{(N)},N\big{]}, we have the inequality

[TABLE]

Notice that \big{(}\varrho_{k}^{(N)}\big{)}^{2} is smaller than \big{(}R(s-k)\big{)}^{2} because \big{(}\varrho^{(N)}_{N}\big{)}^{2}<\delta/N^{2+\epsilon}. Hence, $k\geq\mathbf{k}^{(N)}$ and by induction on $k$ we can deduce that $\mathbf{k}^{(N)}=0$. Therefore we can apply the above inequality with $k=0$ to get

[TABLE]

where $C:=R(s)/\delta^{\frac{1}{2}}$. Since \big{(}\varrho^{(N)}_{N}\big{)}^{2}:=\mathbb{E}\big{[}\big{(}V_{e}^{(N)}-U_{e}^{(N)}\big{)}^{2}\big{]}, the proof is complete in the case when $s\in(-\infty,-1)$ is sufficiently far in the negative direction, i.e., for all $s\in(-\infty,\theta]$ for some $\theta<-1$.

For the general case of $s\in{\mathbb{R}}$, pick $n\in\mathbb{N}$ large enough so that $s-n\leq\theta$. Our previous result for $s\in(-\infty,\theta]$ implies that there exist $\delta^{\prime},C^{\prime}>0$ such for any $N>n$,

[TABLE]

The above uses that \big{(}\varrho^{(N)}_{n}\big{)}^{2}:=\mathbb{E}\big{[}\big{(}U_{a}^{(n,N)}-V_{a}^{(n,N)}\big{)}^{2}\big{]}, where $U_{a}^{(n,N)}$, $V_{a}^{(n,N)}$ are distributed as generation $N-n$ partition functions and that we can write the argument of $R$ in the inequality \mathbb{E}\big{[}\big{(}U_{e}^{(N)}\big{)}^{2}\big{]}<R(s-N) in the form $s-N:=s^{\prime}-(N-n)$ for $s^{\prime}:=s-n$ with $s^{\prime}\leq\theta$. Through iterating (10.4) $n$ times, we can get an inequality of the form \big{(}\varrho_{0}^{(N)}\big{)}^{2}\,\leq\,Q_{s}\big{(}\big{(}\varrho_{n}^{(N)}\big{)}^{2}\big{)} for a degree $2^{n}$ polynomial $Q_{s}$ with nonnegative coefficients that depend on $s\in{\mathbb{R}}$ and no constant term. Since $Q_{s}$ is differentiable and $Q_{s}(0)=0$, there are $\lambda,\mathbf{c}>0$ such that $Q_{s}(x)\leq\mathbf{c}x$ for all $x\in[0,\lambda]$. Combining (10.5) with this inequality for $Q_{s}$ yields that for any $N>n$,

[TABLE]

This implies that the desired inequalities hold in the general case of $s\in{\mathbb{R}}$. ∎

11 The three approximation lemmas

In this section, we will prove Lemmas 9.7-9.9. Recall from Sections 9.2 & 9.3 that Lemma 9.7 involves bounding the error of the partial linearization (9.2) and Lemmas 9.8 & 9.9 are Gaussian approximations of the terms (I) and (II) in (9.4) driven by central limit-type normalized sums that occur at different generational scales.

As before, let \big{(}\big{\{}X_{a}^{(*,n)}\big{\}}_{a\in E_{*}}\big{)}_{n\in\mathbb{N}} denote a minimally regular sequence of $\mathcal{Q}$-pyramidic arrays with parameter $r\in{\mathbb{R}}$. For $0\leq k\leq n$, $a\in E_{k}$, and $h\in E_{n}$, we will frequently use the notation

[TABLE]

Note that $\sigma_{k,n}^{2}\rightarrow R(r-k)$ as $n\rightarrow\infty$ by (III) of Lemma 6.15 with $m=2$.

11.1 Proof of Lemma 9.7

Proof of Lemma 9.7.

The variables $X^{(N,n)}_{e}-\widehat{X}^{N,n}_{e}$ and $\widehat{X}^{N,n}_{e}$ are uncorrelated by Lemma 6.7 and have mean zero, so the square of the $L^{2}$ distance between $X^{(N,n)}_{e}$ and $\widehat{X}^{N,n}_{e}$ can be written as

[TABLE]

The proof is complete since $\xi_{N}(n)\rightarrow 0$ as $n\rightarrow\infty$. ∎

11.2 A generalization of Stein’s auxiliary functions

Before moving to the proof of Lemma 9.8 we will discuss a generalized version of the auxiliary functions used in Stein’s method [27], which is a general strategy for proving the central limit theorem under the Wasserstein-$1$ metric. For random variables $X$ and $Y$ with $\mathbb{E}[|X|],\mathbb{E}[|Y|]<\infty$, the Wasserstein-$1$ distance has the dual form

[TABLE]

where $\textup{Lip}_{1}$ is the collection of all Lipshitz functions on ${\mathbb{R}}$ with Lipshitz constant $\leq 1$. Given $H\in\textup{Lip}_{1}$ define the auxiliary function $f:{\mathbb{R}}\rightarrow{\mathbb{R}}$

[TABLE]

The function $f$ solves the differential equation

[TABLE]

and has the following convenient uniform bounds on its first two derivatives:

[TABLE]

Thus if $X$ is a random variable with finite variance and $\mathcal{X}\sim\mathcal{N}(0,1)$ then

[TABLE]

A useful feature of the auxiliary function, $f$, is that the Wasserstein-$1$ distance between the distributions of $X$ and $\mathcal{X}$ can be reduced to a quantity only involving $X$.

We will require a perturbative generalization of Stein’s method that bounds the Wasserstein-$1$ distance between random variables of the form $X:=Y+Z$ and $\mathcal{X}:=Y+\mathcal{Z}$ for variables $Y$, $Z$, $\mathcal{Z}$ satisfying that $Z$ is centered with $\textup{Var}(Z)=1$ and $\mathcal{Z}\sim\mathcal{N}(0,1)$ is independent of $Y$. In other words, we would like to show how to bound the error of replacing the random variable $Z$ with a standard normal $\mathcal{Z}$ independent of $Y$. In this case we will define an auxiliary function $F:{\mathbb{R}}^{2}\rightarrow{\mathbb{R}}$ for a given $H\in\textup{Lip}_{1}$ that satisfies the following partial differential equation analogous to (11.6):

[TABLE]

The following proposition, whose proof is in Section 12.3, provides bounds for the first- and second-order partial derivatives of $F$ in analogy to (11.7).

Proposition 11.1**.**

Define $F:{\mathbb{R}}^{2}\rightarrow{\mathbb{R}}$ for $H\in\textup{Lip}_{1}$ through the formula

[TABLE]

For all $(y,z)\in{\mathbb{R}}^{2}$,

[TABLE]

The trivial corollary below generalizes Proposition 11.1 to arbitrary variance $\sigma^{2}>0$.

Corollary 11.2**.**

Define $F_{\sigma}:{\mathbb{R}}^{2}\rightarrow{\mathbb{R}}$ for $H\in\textup{Lip}_{1}$ through the formula

[TABLE]

The function $F_{\sigma}(y,z)$ solves the partial differential equation

[TABLE]

and for all $(y,z)\in{\mathbb{R}}^{2}$,

[TABLE]

Proof.

Define $\widehat{F}_{\sigma}(y,z)\,:=\,\frac{1}{\sigma}F_{\sigma}(\sigma y,\sigma z)$ and $\widehat{H}_{\sigma}(z):=\frac{1}{\sigma}H(\sigma z)$. Notice that we can write $\widehat{F}_{\sigma}$ as

[TABLE]

Since $\widehat{H}_{\sigma}(z)\in\textup{Lip}_{1}$, it follows that the first- and second-order derivatives of $\widehat{F}_{\sigma}$ have the bounds in Proposition 11.1. From the equation $F_{\sigma}(y,z)=\sigma\widehat{F}_{\sigma}(\frac{y}{\sigma},\frac{z}{\sigma})$ we see that the derivatives of $F_{\sigma}$ have the desired bounds. ∎

11.3 Proof of Lemma 9.8

For $N,n\in\mathbb{N}$ with $n\geq\mathbf{n}$, we will maintain the usual convention that $e\in E_{N}$, $f\in E_{\mathbf{\widehat{n}}}$, and $g\in E_{\mathbf{n}}$. Recall that $Y_{f}^{N,n}$ is defined in (9.5), $Z^{N,n}_{f}$ is defined above (9.6), and $\widehat{X}^{N,n}_{e}$, $\mathbf{\widehat{X}}^{N,n}_{e}$ are defined in Definition 9.5. We will need the following lemma, which collects some statements about the second and fourth moments of these random variables. The proof is in Section 12.3.

Lemma 11.3**.**

Let the random variables $Y_{f}^{N,n}$, $Z^{N,n}_{f}$, $\widehat{X}^{N,n}_{e}$, $\mathbf{\widehat{X}}^{N,n}_{e}$ be defined in terms of a minimally regular sequence of $\mathcal{Q}$-pyramidic arrays of random variables \big{(}\big{\{}X^{(*,n)}_{a}\big{\}}_{a\in E_{*}}\big{)}_{n\in\mathbb{N}} with parameter $r\in{\mathbb{R}}$.

1. (i)

The variance of Y_{f}^{N,n}:=\mathcal{L}^{\mathbf{n}-\mathbf{\widehat{n}}}\big{\{}X_{g}^{(\mathbf{n},n)}\big{\}}_{g\in f\cap E_{\mathbf{n}}} is $\sigma_{\mathbf{n},n}^{2}$, and $\displaystyle\lim_{n\rightarrow\infty}\sigma_{\mathbf{n},n}^{2}=R(r-\mathbf{n})$. Moreover, $\sigma_{\mathbf{n},n}^{2}$ is bounded from above and below by constant multiples of $\frac{1}{N}$ for all $n,N\in\mathbb{N}$ with $n\geq\mathbf{n}$. 2. (ii)

The variance of Z_{f}^{N,n}:=\sum_{k=1}^{\mathbf{n}-\mathbf{\widehat{n}}}\mathcal{L}^{k-1}\mathcal{E}\mathcal{L}^{\mathbf{n}-\mathbf{\widehat{n}}-k}\big{\{}X_{g}^{(\mathbf{n},n)}\big{\}}_{g\in f\cap E_{\mathbf{n}}} has the large $n$ convergence

[TABLE]

Moreover, $\varsigma_{N,n}^{2}$ is bounded by a constant multiple of $\frac{\log(N+1)}{N^{2}}$ for all $n,N\in\mathbb{N}$ with $n\geq\mathbf{n}$. 3. (iii)

There is a $C>0$ such that the fourth moments of the random variables $Y_{f}^{N,n}$ and $Z_{f}^{N,n}$ are respectively bounded by $\frac{C}{N^{2}}$ and $C\frac{\log^{2}(N+1)}{N^{4}}$ for all $N,n\in\mathbb{N}$ with $n\geq 2\mathbf{n}$. 4. (iv)

There is a $C>0$ such that the fourth moments of the random variables $X^{(\mathbf{n},n)}_{g}$, $\widehat{X}^{N,n}_{e}$, $\mathbf{\widehat{X}}^{N,n}_{e}$ are bounded by $\frac{C}{N^{2}}$ for all $N,n\in\mathbb{N}$ with $n\geq 2\mathbf{n}$.

Remark 11.4**.**

For (ii) of Lemma 11.3, note that $\varsigma_{N}^{2}$ is bounded from below by a constant multiple $c>0$ of $\frac{\log(N+1)}{N^{2}}$ for all $N\in\mathbb{N}$ as a consequence of (II) of Lemma 2.3 and since $\mathbf{n}\sim N$ for $N\gg 1$ and $\mathbf{n}-\mathbf{\widehat{n}}\propto\log N$.

The lemma below, whose proof is in Section 12.3, follows easily from Holder’s inequality and the definition of Wasserstein-$p$ distance.

Lemma 11.5**.**

For $m\in\mathbb{N}$, let $X$ and $Y$ be random variables with finite $(m+1)^{th}$ absolute moments. We have the following bound on the Wasserstein-$2$ distance between $X$ and $Y$ using the Wasserstein-$1$ distance:

[TABLE]

Proof of Lemma 9.8.

This proof is divided into parts (a)-(g).

(a) Notation: For $e\in E_{N}$ we can write $\widehat{X}^{N,n}_{e}$ and $\mathbf{\widehat{X}}^{N,n}_{e}$ in the forms

[TABLE]

where the random variables $\macc@depth\char 1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a 111{X}_{e}^{N,n}$, $\macc@depth\char 1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a 111{Y}_{e}^{N,n}$, $\macc@depth\char 1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a 111{Z}_{e}^{N,n}$ are defined as

[TABLE]

and recall that $\mathbf{Z}^{(N)}_{e}$ is the normal random variable (independent of $\macc@depth\char 1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a 111{X}_{e}^{N,n}$ and $\macc@depth\char 1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a 111{Y}_{e}^{N,n}$) defined in (9.6).

(b) Stein’s method: Next we will use Stein’s method to bound the Wasserstein-$1$ distance between $\widehat{X}^{N,n}_{e}$ and $\mathbf{\widehat{X}}^{N,n}_{e}$. By definition of Wasserstein-$1$ distance,

[TABLE]

For a given $H:{\mathbb{R}}\rightarrow{\mathbb{R}}$ with Lipschitz constant less than $1$, define $F:{\mathbb{R}}^{2}\rightarrow{\mathbb{R}}$ as in Corollary 11.2 with $\sigma:=\varsigma_{N}$. Then $F$ is a solution to the partial differential equation

[TABLE]

where the expectation is w.r.t.  \mathbf{Z}^{(N)}_{e}\sim\mathcal{N}\big{(}0,\varsigma_{N}^{2}\big{)}. By Corollary 11.2, the first-order partial derivatives of $F$ are bounded by $\sqrt{\pi/2}$ and the second-order partial derivatives are bounded by $2/\varsigma_{N}$.

By (11.10) and (11.12), to bound the expression in the supremum of (11.11), we must bound the absolute value of

[TABLE]

As in the usual implementation of Stein’s method, we would like to tease out cancellations between (I) and (II) by writing the random variable $\macc@depth\char 1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a 111{Z}^{N,n}_{e}$ in (II) as a sum of a “large” term, $\macc@depth\char 1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a 111{Z}^{N,n}_{e}-\frac{1}{b^{\mathbf{\widehat{n}}-N}}Z^{N,n}_{f}$, and a “small” term, $\frac{1}{b^{\mathbf{\widehat{n}}-N}}Z^{N,n}_{f}$, and then Taylor expanding (II). The complicating feature here is that $\macc@depth\char 1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a 111{X}_{e}^{N,n}+\macc@depth\char 1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a 111{Y}_{e}^{N,n}$ is not independent of $Z^{N,n}_{f}$.

(c) Identifying the dependent factors: Next we seek to separate out the dependence of the random variables $\macc@depth\char 1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a 111{X}_{e}^{N,n}$ and $\macc@depth\char 1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a 111{Y}_{e}^{N,n}$ on the random variable $Z^{N,n}_{f}$ for a given $f\in e\cap E_{\mathbf{\widehat{n}}}$. More precisely, we can define a term $B_{f}^{N,n}$ such that the statements (i)-(iii) below hold for the ${\mathbb{R}}^{2}$-valued random variable \Delta_{f}^{N,n}:=\frac{1}{b^{\mathbf{\widehat{n}}-N}}\big{(}Y_{f}^{N,n}+Y_{f}^{N,n}B_{f}^{N,n},\,Z^{N,n}_{f}\big{)}.

1. (i)

The random variables $Z^{N,n}_{f}$, $Y_{f}^{N,n}$, $B_{f}^{N,n}$ have mean zero. 2. (ii)

The random vector \big{(}\macc@depth\char 1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a 111{V}^{N,n}_{e}-\Delta_{f}^{N,n},B_{f}^{N,n}\big{)} is independent of \big{(}Y_{f}^{N,n},Z^{N,n}_{f}\big{)}. 3. (iii)

The random variables $Y_{f}^{N,n}$ and $Z^{N,n}_{f}$ are uncorrelated. Thus with (ii) the random variables $Y_{f}^{N,n}$, $Z^{N,n}_{f}$, $B_{f}^{N,n}$ are pairwise uncorrelated.

For $f\in e\cap E_{\mathbf{\widehat{n}}}$ the definition of $B_{f}^{N,n}$ is as follows:

[TABLE]

where the function $\mathcal{F}$, which maps arrays $\{y_{a}\}_{a\in E_{\mathbf{\widehat{n}}-N}}$ into ${\mathbb{R}}$, is defined below.111111Recall that for $e\in E_{N}$ the indexing set $e\cap E_{\mathbf{\widehat{n}}}$ is canonically identifiable with $E_{\mathbf{\widehat{n}}-N}$. The variable $\macc@depth\char 1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a 111{Y}_{e}^{N,n}$ is a multilinear function, $\mathcal{F}$, of the array \big{\{}Y^{N,n}_{f}\big{\}}_{f\in e\cap E_{\mathbf{\widehat{n}}}}, where

[TABLE]

Moreover, the partial derivative of $\mathcal{F}$ with respect to $y_{\alpha}$ has the form

[TABLE]

where $E_{k}^{\updownarrow\alpha}$ is the $(b-1)$-element subset of $E_{k}$ consisting of elements $\mathbf{\widehat{a}}$ with the following three $\mathbf{(1)}$-$\mathbf{(3)}$ restrictions: $\mathbf{(1)}$ $\alpha\not\subset\mathbf{\widehat{a}}$, $\mathbf{(2)}$ there is a path in $\Gamma_{k}$ that passes over both $\alpha$ and $\mathbf{\widehat{a}}$, and $\mathbf{(3)}$ there is an element in $E_{k-1}$ that contains both $\alpha$ and $\mathbf{\widehat{a}}$.121212The elements $\mathbf{\widehat{a}}\in E_{k}^{\updownarrow\alpha}$ correspond to the $\mathbf{a}\times(i,j)\in E_{k}$ in the above expression for \mathcal{F}\big{\{}y_{a}\big{\}}_{a\in E_{\mathbf{\widehat{n}}-N}}.

Next we justify statements (i)-(iii). For statement (i), note that the variables $X_{f}^{N,n}$, $Y_{f}^{N,n}$, $B_{f}^{N,n}$ are multilinear polynomials of the array \big{\{}X^{(\mathbf{n},n)}_{g}\big{\}}_{g\in e\cap E_{\mathbf{n}}} that have no constant term, and consequently these variables have mean zero. Statement (iii) follows from Lemma 6.7 because the random variables have the forms Y_{f}^{N,n}:=\mathcal{L}^{\mathbf{n}-\mathbf{\widehat{n}}}\big{\{}X^{(\mathbf{n},n)}_{g}\big{\}}_{g\in f\cap E_{\mathbf{n}}} and Z^{N,n}_{f}:=\sum_{k=1}^{\mathbf{n}-\mathbf{\widehat{n}}}\mathcal{L}^{k-1}\mathcal{E}\mathcal{L}^{\mathbf{n}-\mathbf{\widehat{n}}-k}\big{\{}X_{g}^{(\mathbf{n},n)}\big{\}}_{g\in f\cap E_{\mathbf{n}}}. Note, in particular, that $Y_{f}^{N,n}$ and $Z_{f}^{N,n}$ are functions of the random variables $X^{(\mathbf{n},n)}_{g}$ with $g\in f\cap E_{\mathbf{n}}$. The form (11.14) of the multilinear polynomial $\frac{\partial\mathcal{F}}{\partial y_{f}}\{y_{\hat{f}}\}_{\hat{f}\in e\cap E_{\mathbf{\widehat{n}}}}$ implies that $B_{f}^{N,n}$ only depends on variables in the array \big{\{}X^{(\mathbf{n},n)}_{g}\big{\}}_{g\in e\cap E_{\mathbf{n}}} with $g\notin f\cap E_{\mathbf{n}}$. Hence $B_{f}^{N,n}$ is independent of \big{(}Y_{f}^{N,n},Z^{N,n}_{f}\big{)}. By using that \macc@depth\char 1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a 111{Y}_{e}^{N,n}=\mathcal{F}\big{\{}Y^{N,n}_{f}\big{\}}_{f\in e\cap E_{\mathbf{\widehat{n}}}}, the difference between the ${\mathbb{R}}^{2}$-valued random variables $\macc@depth\char 1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a 111{V}^{N,n}_{e}$ and $\Delta_{f}^{N,n}$ can be written as

[TABLE]

The multilinearity of $\mathcal{F}$ implies that \mathcal{G}\big{\{}y_{\widehat{f}}\big{\}}_{\widehat{f}\in e\cap E_{\mathbf{\widehat{n}}}}:=\mathcal{F}\big{\{}y_{\widehat{f}}\big{\}}_{\widehat{f}\in e\cap E_{\mathbf{\widehat{n}}}}-y_{f}\frac{\partial\mathcal{F}}{\partial y_{f}}\big{\{}y_{\widehat{f}}\big{\}}_{\widehat{f}\in e\cap E_{\mathbf{\widehat{n}}}} does not depend on the variable $y_{f}$. The right side of the display above is a function of the variables \big{(}Y_{\widehat{f}}^{N,n},\,Z^{N,n}_{\widehat{f}}\big{)} with $\widehat{f}\in e\cap E_{\mathbf{\widehat{n}}}$ and $\widehat{f}\neq f$, and thus $\macc@depth\char 1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a 111{V}^{N,n}_{e}-\Delta_{f}^{N,n}$ is independent of \big{(}Y_{f}^{N,n},\,Z^{N,n}_{f}\big{)}. In fact, these observations imply that $B_{f}^{N,n}$ and $\macc@depth\char 1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a 111{V}^{N,n}_{e}-\Delta_{f}^{N,n}$ are are jointly independent of \big{(}Y_{f}^{N,n},Z^{N,n}_{f}\big{)}, i.e., (ii).

With (11.14) and the triangle inequality, we can bound the $L^{2}$ norm of $B^{N,n}_{f}$ for $f\in e\cap E_{\mathbf{\widehat{n}}}$ by

[TABLE]

The inequality holds for some $C>0$ and all $n\geq\mathbf{n}$ as a consequence of part (i) of Lemma 11.3.

(d) Stein analysis: Now we are ready to begin an analysis of the expression (11.3). By Taylor’s theorem to second-order, the expression inside the expectation in (II) has the form

[TABLE]

where $\mathbf{D}_{2}$ is the 2-tensor of second-order derivatives and $\mathbf{r}_{f}$ is some value between [math] and $1$ depending on $\macc@depth\char 1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a 111{V}^{N,n}_{e}$ and $\Delta_{f}^{N,n}$. The expectation of the first expression on the right side of (11.3) is zero by observations (i)-(iii) in part (c) above. By definition of $\Delta_{f}^{N,n}$, the second term on the right side of (11.3) can be written as

[TABLE]

Again by observations (i)-(iii) in part (c), the expectation of the first expression on the right side of (11.3) is zero.

As a consequence of the above remarks, taking the expectation of (11.3) leaves us with

[TABLE]

where we have used that $Z^{N,n}_{f}$ is independent of $\macc@depth\char 1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a 111{V}^{N,n}_{e}-\Delta_{f}^{N,n}$ to factor the first expectation on the right. The right-most expectation on the top line of (11.3) is equal to

[TABLE]

For \varsigma_{N,n}:=\mathbb{E}\big{[}\big{(}Z^{N,n}_{f}\big{)}^{2}\big{]}^{1/2}, combining (11.3) and (11.19) with (11.3) yields the equality

[TABLE]

In the above we have used that the expressions (III) and (IV) do not depend on the choice of $f\in e\cap E_{\mathbf{\widehat{n}}}$ and that there are $b^{2(\mathbf{\widehat{n}}-N)}$ elements in $e\cap E_{\mathbf{\widehat{n}}}$. The first term on the right side of (11.3) vanishes as $n\rightarrow\infty$ because $\partial_{2}F$ is bounded by $\sqrt{\pi/2}$ and $\varsigma_{N,n}\rightarrow\varsigma_{N}$ by part (ii) of Lemma 11.3. We will bound the last two terms on the right side of (11.3) in (e) and (f) below.

(e) Second term on the right side of (11.3): For any $(x,z)\in{\mathbb{R}}^{2}$, the norm of the 2-tensor $\mathbf{D}_{2}F(x,z)$ is bounded by $4/\varsigma_{N}$ since its components are smaller than $2/\varsigma_{N}$ as a consequence of Corollary 11.2. Thus we have the second inequality below.

[TABLE]

By (11.3), Lemma 11.3, and Remark 11.4, the above is bounded for all $n,N\in\mathbb{N}$ with $n\geq 2\mathbf{n}$ by

[TABLE]

As $N\rightarrow\infty$ the above is asymptotically proportional to $\frac{\log^{-\frac{1}{2}}(N+1)}{N^{\mathfrak{m}\log b}}$ since $\mathbf{\widehat{n}}-N\sim\mathfrak{m}\log N$.

(f) Third term on the right side of (11.3): To bound the third term on the right side of (11.3), we can use that the vector $(\nabla\partial_{2}F)(x,z)$ has norm less $\sqrt{2}$ times $2/\varsigma_{N}$, i.e., the bound for the second-order partial derivatives of $F$, and apply the Cauchy-Schwarz inequality to get

[TABLE]

By (11.3), Lemma 11.3, and Remark 11.4, the above is bounded for all $n,N\in\mathbb{N}$ with $n\geq 2\mathbf{n}$ by

[TABLE]

As $N\rightarrow\infty$ the above is asymptotically proportional to $\frac{1}{N^{\mathfrak{m}\log b+\frac{1}{2}}}$.

(g) Extension to the Wasserstein-$\mathbf{2}$ distance: Our results in parts (b)-(f) can be summarized by stating that there is a $\mathfrak{c}>0$ such that for all large $n,N\in\mathbb{N}$ with $n\geq 2\mathbf{n}$

[TABLE]

where \xi_{N}^{\prime}(n):=\sqrt{\frac{\pi}{2}}\big{|}\frac{\varsigma_{N,n}^{2}}{\varsigma_{N}}-\varsigma_{N}\big{|}. As mentioned below (11.3), $\xi_{N}^{\prime}(n)$ vanishes as $n\rightarrow\infty$ for any fixed $N$. By applying Lemma 11.5 with $m=3$, we have that

[TABLE]

The limit superior of the above as $n\rightarrow\infty$ is bounded by a constant multiple of $\frac{\log^{-\frac{1}{6}}(N+1)}{N^{\mathfrak{m}\frac{\log b}{3}+\frac{1}{3}}}$ by (11.21) and part (iv) of Lemma 11.3. ∎

11.4 Proof of Lemma 9.9

The following lemma is a central limit theorem in which the distance between a normalized sum of i.i.d. random variables and a centered normal random variable of the same variance is measured in terms of the Wasserstein-1 distance. We include a proof using the zero bias transformation of Goldstein and Reinert [19] in Appendix C.

Lemma 11.6**.**

Let $X_{1},\ldots,X_{n}$ be i.i.d. centered random variables with variance $\sigma^{2}$ and finite third absolute moment. Then for $\macc@depth\char 1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a 111{X}_{n}:=\frac{X_{1}+\cdots+X_{n}}{\sqrt{n}}$ and $\mathcal{X}\sim\mathcal{N}(0,\sigma^{2})$

[TABLE]

The next corollary applies Lemma 11.5 to the above result. The proof is at the end of Appendix C.

Corollary 11.7**.**

Let us take the conditions of Lemma 11.6 and assume in addition that the fourth moment of the random variables is finite. Then for any $n\in\mathbb{N}$

[TABLE]

Proof of Lemma 9.9.

For $e\in E_{N}$ the variables $\mathbf{\widehat{X}}_{e}^{N,n}$ and $\mathbf{\widetilde{X}}_{e}^{(N)}$ have the form

[TABLE]

for $Y_{f}:=Y_{f}^{N,n}$ and $Y_{f}:=\mathbf{Y}_{f}^{(N)}$, respectively, where \big{\{}Y_{f}^{N,n}\big{\}}_{f\in e\cap E_{\mathbf{\widehat{n}}}} and \big{\{}\mathbf{Y}_{f}^{(N)}\big{\}}_{f\in e\cap E_{\mathbf{\widehat{n}}}} are defined as in (9.5) and independent of $\mathbf{Z}_{e}^{(N)}$. In the analysis below, we bound the Wasserstein-$2$ distance between $\mathbf{\widehat{X}}_{e}^{N,n}$ and $\mathbf{\widetilde{X}}_{e}^{(N)}$ after choosing i.i.d. couplings \big{(}Y_{f}^{N,n},\mathbf{Y}_{f}^{(N)}\big{)} for $f\in e\cap E_{\mathbf{\widehat{n}}}$.

(a) Using i.i.d. couplings to bound the Wasserstein-$2$ distance: For each $f\in e\cap E_{\widehat{n}}$, let $(Y_{f}^{N,n},\mathbf{Y}_{f}^{(N)})$ be a coupling of the variables $Y^{N,n}_{f}$ and $\mathbf{Y}_{f}^{(N)}$ such that

[TABLE]

With this coupling, we can bound the Wasserstein-$2$ distance between $\mathbf{\widehat{X}}_{e}^{N,n}$ and $\mathbf{\widetilde{X}}_{e}^{(N)}$ as follows:

[TABLE]

where for $\mathbf{e}\in e\cap E_{N+k-1}$ the arrays within the expectations above are defined as

[TABLE]

(b) Bounding the inner summand on the second line of (11.4): Recall from (i) of Lemma 11.3 and (9.5), respectively, that the variables $Y^{N,n}_{f}$ and $\mathbf{Y}^{(N)}_{f}$ have variances \textup{Var}\big{(}Y^{N,n}_{f}\big{)}=\sigma_{\mathbf{n},n}^{2} and \textup{Var}\big{(}\mathbf{Y}^{N,n}_{f}\big{)}=R(r-\mathbf{n}). Consequently, elements in the above arrays have variances \textup{Var}\big{(}\widetilde{Y}^{N,n}_{\mathbf{f}}\big{)}=\sigma_{\mathbf{n},n}^{2} and \textup{Var}\big{(}\mathbf{\widetilde{Y}}^{(N)}_{\mathbf{f}}\big{)}\,=\,R(r-\mathbf{n}) since $\mathcal{L}$ preserves the variance of the array variables. For any $1\leq k\leq\mathbf{\widehat{n}}-N$, we can write the summand in (11.4) in the form

[TABLE]

(c) Going back to (11.4): The first term on the right side of (11.4) is equal to \big{(}\rho_{2}\big{(}Y^{N,n}_{f},\mathbf{Y}_{f}^{(N)}\big{)}\big{)}^{2} for any representative $f\in e\cap E_{\mathbf{\widehat{n}}}$ by definition of how the couplings in (11.22) are defined and since \big{|}e\cap E_{\mathbf{\widehat{n}}}|=b^{2(\mathbf{\widehat{n}}-N)}. Similarly, as a consequence of (11.24), the second term on the right side of (11.4) is bounded from above by \mathbf{c}\frac{\mathbf{\widehat{n}}-N}{N}\big{(}\rho_{2}\big{(}Y^{N,n}_{f},\mathbf{Y}_{f}^{(N)}\big{)}\big{)}^{2}. Thus for all $n,N\in\mathbb{N}$ with $n\geq\mathbf{n}$

[TABLE]

where the second inequality holds for some $\mathbf{C}>0$ since $\mathbf{\widehat{n}}:=N+\lfloor\mathfrak{m}\log N\rfloor$. Thus we have shown that \rho_{2}\big{(}\mathbf{\widehat{X}}_{e}^{N,n},\mathbf{\widetilde{X}}_{e}^{(N)}\big{)} is bounded by a constant multiple of \rho_{2}\big{(}Y^{N,n}_{f},\mathbf{Y}_{f}^{(N)}\big{)}.

(d) Bounding the right side of (11.25): Next we focus on bounding \rho_{2}\big{(}Y^{N,n}_{f},\mathbf{Y}_{f}^{(N)}\big{)}. Since $Y^{N,n}_{f}$ has variance $\sigma_{\mathbf{n},n}^{2}$ and $\mathbf{Y}_{f}^{(N)}$ has variance $R(r-\mathbf{n})$, it will be convenient to use the triangle inequality to get

[TABLE]

Using that \textup{Var}\big{(}\mathbf{Y}_{f}^{(N)}\big{)}=R(r-\mathbf{n}), the first term on the right side of (11.26) can simply be bounded by

[TABLE]

By definition, $Y^{N,n}_{f}$ is a sum of the i.i.d. random variables $\frac{1}{b^{\mathbf{n}-\mathbf{\widehat{n}}}}X^{(\mathbf{n},n)}_{g}$ over $g\in f\cap E_{\mathbf{n}}$, which contains $b^{2(\mathbf{n}-\mathbf{\widehat{n}})}$ elements. Hence, by Corollary 11.7 we have the inequality below for the first term on the right side of (11.26).

[TABLE]

The second inequality holds for some $C>0$ since \mathbb{E}\big{[}\big{|}X^{(\mathbf{n},n)}_{g}\big{|}^{4}\big{]} is bounded from above by a constant multiple of $\frac{1}{N^{2}}$ for all $n,N\in\mathbb{N}$ with $n\geq 2\mathbf{n}$ by (iv) of Lemma 11.3. The last term in (11.28) is asymptotically proportional to $N^{-\mathfrak{m}\frac{\log b}{3}-\frac{1}{2}}$ as $N\rightarrow\infty$ since $\mathbf{n}-\mathbf{\widehat{n}}\approx\mathfrak{m}\log N$.

(e) Conclusion: The inequalities (11.25)-(11.28) show that there is a $\mathfrak{c}>0$ such that for all $N,n\in\mathbb{N}$ with $n\geq 2\mathbf{n}$

[TABLE]

where \varsigma_{N}^{\prime\prime}(n):=\mathfrak{c}\big{|}\sigma_{\mathbf{n},n}-\sqrt{R(r-\mathbf{n})}\big{|}. The term $\varsigma_{N}^{\prime\prime}(n)$ vanishes as $n\rightarrow\infty$ since $\sigma_{\mathbf{n},n}^{2}\rightarrow R(r-\mathbf{n})$ by (III) of Lemma 6.15 with $m=2$, and hence the proof is complete. ∎

12 Miscellaneous proofs from Sections 6, 9, & 11

12.1 Proofs from Section 6

Proof of Proposition 6.5.

We will prove the identity (6.1) using induction starting from $n=0$. When $n=0$, the set $E_{n}$ contains a single element $h$, and the identity follows immediately from the definitions:

[TABLE]

Suppose that the identity (6.1) holds for some $n\in\mathbb{N}_{0}$. The hierarchical nesting that defines the sequence $\{D_{n}\}_{n\in\mathbb{N}_{0}}$ of diamond graphs implies that there is a one-to-one correspondence between the set of generation-$(n+1)$ paths, $p\in\Gamma_{n+1}$, crossing $D_{n+1}$ and the set of $(b+1)$-tuples $(i,p_{1},\ldots,p_{b})$ with $i\in\{1,\ldots,b\}$ and $p_{j}\in\Gamma_{n}$. Within this identification, $i\in\{1,\ldots,b\}$ labels the branch of $D_{n+1}$ that $p\in\Gamma_{n+1}$ traces over and $p_{j}\in\Gamma_{n}$ for $j\in\{1,\ldots,b\}$ is the trajectory of $p$ through the $j^{th}$ copy of $D_{n}$ along the branch. In particular, it follows that $|\Gamma_{n+1}|=b|\Gamma_{n}|^{b}$. Using this bijection, we can rewrite the partition function $W^{\omega}_{n+1}(\beta)$ as

[TABLE]

Hence the identity (6.1) holds for all $n\in\mathbb{N}_{0}$ by induction.∎

Proof of Proposition 6.22.

We will prove that the law of $\mathbf{X}_{r}$ is a locally $\frac{1}{2}$-Hölder continuous function of $r\in{\mathbb{R}}$ with respect to the Wasserstein-$2$ metric by showing that for all $r$ and $t\geq 0$

[TABLE]

where the function $R:{\mathbb{R}}\rightarrow(0,\infty)$ has a continuous—and thus locally bounded—derivative by Lemma 2.3. For any $n\in\mathbb{N}$ we can construct $\mathbf{X}_{r}$ as $\mathbf{X}_{r}=\mathcal{Q}^{n}\{\mathbf{X}_{h}^{(n)}\}_{h\in E_{n}}$, where the array of random variables $\{\mathbf{X}_{h}^{(n)}\}_{h\in E_{n}}$ is defined as in Theorem 6.16 for parameter $r\in{\mathbb{R}}$. Let $\{\mathbf{B}^{h}_{t}\}_{h\in E_{n}}$ be an array of independent normal random variables with mean [math] and variance $t>0$ that is independent of $\{\mathbf{X}_{h}^{(n)}\}_{h\in E_{n}}$. Define X_{n,r,t}^{\mathbf{B}}:=\mathcal{Q}^{n}\big{\{}X_{h}^{(n)}(r,t)\big{\}}_{h\in E_{n}} for X_{h}^{(n)}(r,t):=\big{(}1+\mathbf{X}_{h}^{(n)}\big{)}\textup{exp}\big{\{}\frac{\kappa}{n}\mathbf{B}^{h}_{t}-\frac{\kappa^{2}}{2n^{2}}t\big{\}}-1, i.e., as in Example 7.6. By the triangle inequality, we can bound the Wasserstein-$2$ distance between $\mathbf{X}_{r}$ and $\mathbf{X}_{r+t}$ by

[TABLE]

The second term on the right side of (12.2) converges to zero as $n\rightarrow\infty$ by the discussion in Example 7.6. The random variables $X^{\mathbf{B}}_{n,r,t}-\mathbf{X}_{r}$ and $\mathbf{X}_{r}$ are uncorrelated since $\mathbf{X}_{r}=\mathcal{Q}^{n}\{\mathbf{X}_{h}^{(n)}\}_{h\in E_{n}}$ is the conditional expectation of $X^{\mathbf{B}}_{n,r,t}$ given $\{\mathbf{X}_{h}^{(n)}\}_{h\in E_{n}}$. Thus, since $\mathbf{X}_{r}$ and $X^{\mathbf{B}}_{n,r,t}$ have mean zero,

[TABLE]

To see that \textup{Var}\big{(}X^{\mathbf{B}}_{n,r,t}\big{)} converges to $R(r+t)$ as $n\rightarrow\infty$, notice that

[TABLE]

where the first and third equalities hold by part (i) of Remark 6.6 and Lemma 2.3, respectively. The second equality above follows from (7.2). Therefore we have established the inequality (12.1). ∎

12.2 Proofs from Section 9

Proof of Corollary 9.12.

The random variables $X^{(N,n)}_{e}-\widehat{X}^{N,n}_{e}$ and $\widehat{X}^{N,n}_{e}$ are uncorrelated as a consequence of Lemma 6.7, and thus

[TABLE]

where the convergence holds by (III) of Lemma 6.15 with $m=2$. The equality holds for $N\gg 1$ by the asymptotics for $R(r)$ as $r\rightarrow-\infty$ in (II) of Lemma 2.3. If $s>r$, then the right side above is smaller than $R(-N)+\frac{\kappa^{2}s}{N^{2}}$ for $N\gg 1$. Thus we have verified the desired condition in the case $U_{e}^{(N)}:=\widehat{X}^{N,n}_{e}$ for any $s\in(r,\infty)$ and large enough $N,n\in\mathbb{N}$.

Next we extend our result to the case $U_{e}^{(N)}:=\mathbf{\widehat{X}}^{N,n}_{e}$. By Lemma 9.8 and Remark 9.11, there are couplings between the random variables $\widehat{X}^{N,n}_{e}$ and $\mathbf{\widehat{X}}^{N,n}_{e}$ such that the limit superior as $n\rightarrow\infty$ of \mathbb{E}\big{[}\big{(}\widehat{X}^{N,n}_{e}-\mathbf{\widehat{X}}^{N,n}_{e}\big{)}^{2}\big{]} is \mathit{o}\big{(}\frac{1}{N^{4}}\big{)} for $N\gg 1$. By foiling and applying Cauchy-Schwarz, we get

[TABLE]

Since \limsup_{n\rightarrow\infty}\mathbb{E}\big{[}(\widehat{X}^{N,n}_{e})^{2}\big{]}\leq R(r-N) by (12.4) and $R(r-N)$ is \mathit{O}\big{(}\frac{1}{N}\big{)} for $N\gg 1$ as a consequence of (II) of Lemma 2.3, the limit superior of the middle term above as $n\rightarrow\infty$ is \mathit{o}\big{(}\frac{1}{N^{5/2}}\big{)} with large $N$. Thus \limsup_{n\rightarrow\infty}\mathbb{E}\big{[}(\mathbf{\widehat{X}}^{N,n}_{e})^{2}\big{]} is bounded by \limsup_{n\rightarrow\infty}\mathbb{E}\big{[}(\widehat{X}^{N,n}_{e})^{2}\big{]}+\mathit{o}\big{(}\frac{1}{N^{5/2}}\big{)}, which is smaller than $R(-N)+\frac{\kappa^{2}s}{N^{2}}$ when $N\gg 1$ for any choice of $s\in(r,\infty)$. Hence we have extended our result to the case $U_{e}^{(N)}:=\mathbf{\widehat{X}}^{N,n}_{e}$, and the same reasoning applies to $U_{e}^{(N)}:=\mathbf{\widetilde{X}}^{(N)}_{e}$. ∎

12.3 Proofs from Section 11

Proof of Proposition 11.1.

The bounds $\sup_{y,z\in{\mathbb{R}}}|\partial_{z}F(y,z)|\leq 1$ and $\sup_{y,z\in{\mathbb{R}}}|\partial_{z}^{2}F(y,z)|\leq 2$ are equivalent to (11.7), so we can focus on the partial derivatives $\partial_{y}$, $\partial_{y}^{2}$, and $\partial_{y}\partial_{z}$. Define $\displaystyle\phi_{-}(t):=\int_{-\infty}^{t}\frac{1}{\sqrt{2\pi}}e^{-\frac{r^{2}}{2}}dr$ and $\phi_{+}(t):=1-\phi_{-}(t)$. We can rewrite $H$ in terms of $H^{\prime}$ as

[TABLE]

Moreover, we can rewrite $F$ in the form

[TABLE]

where $G:{\mathbb{R}}^{2}\rightarrow{\mathbb{R}}$ is the kernel

[TABLE]

The results will follow by bounding \sup_{z\in{\mathbb{R}}}\int_{{\mathbb{R}}}\big{|}(\mathbf{d}G)(z,r)\big{|}dr for the derivatives \mathbf{d}\in\big{\{}\partial_{r},\partial_{r}^{2},\partial_{z}\partial_{r}\big{\}}.

The first partial derivative with respect to $r$ has the form

[TABLE]

For any $z\in{\mathbb{R}}$, the equality \int_{{\mathbb{R}}}\big{|}\partial_{r}G(z,r)\big{|}dr\,=\,2\sqrt{2\pi}e^{\frac{z^{2}}{2}}\phi_{-}(z)\phi_{+}(z) holds, and the right side attains its maximum value, $\sqrt{\pi/2}$, when $z=0$.

The second-order partial derivatives involving $r$ have the forms $\partial_{r}^{2}G(z,r)=-\delta(z-r)+A_{1}(z,r)$ and $\partial_{z}\partial_{r}G(z,r)=-\delta(z-r)+A_{2}(z,r)$, where

[TABLE]

Notice that $1+\sqrt{2\pi}ze^{\frac{z^{2}}{2}}\phi_{-}(z)$ and $1-\sqrt{2\pi}ze^{\frac{z^{2}}{2}}\phi_{+}(z)$ are nonnegative for all $z\in{\mathbb{R}}$, and thus we simply have

[TABLE]

Therefore \sup_{z\in{\mathbb{R}}}\int_{{\mathbb{R}}}\big{|}(\mathbf{d}G)(z,r)\big{|}dr\leq 2 for $\mathbf{d}=\partial_{z}\partial_{r}$ and $\mathbf{d}=\partial_{r}^{2}$. ∎

The following proposition gives uniform bounds for the second and fourth moments of random variables from a minimally regular sequence of $\mathcal{Q}$-pyramidic arrays. We prove Proposition 12.1 in Section 15.1 using techniques and an inequality from [10].

Proposition 12.1**.**

Let \big{(}\big{\{}X^{(*,n)}_{a}\big{\}}_{a\in E_{*}}\big{)}_{n\in\mathbb{N}} be a minimally regular sequence of $\mathcal{Q}$-pyramidic arrays of random variables.

1. (i)

The variances of the random variables $X^{(k,n)}_{a}$ are bounded from above and below by positive multiples of $\frac{1}{k+1}$ for all $n\in\mathbb{N}$ and $k\in\{0,\ldots,n\}$. 2. (ii)

The fourth moments of the random variables $X^{(k,n)}_{a}$ are bounded from above by a multiple of $\frac{1}{(k+1)^{2}}$ for all $n\in\mathbb{N}$ and k\in\big{\{}0,\ldots,\lfloor n/2\rfloor\big{\}}.

We will prove the next lemma in Section 15.2. In a basic sense, the proof uses the same idea as the proof of Lemma 6.7 although the analysis is made more complex by the fourth moment.

Lemma 12.2**.**

For $n\in\mathbb{N}$, let $\{x_{a}\}_{a\in E_{n}}$ be an array of i.i.d. centered random variables with finite fourth moment. Define $Y_{\ell}:=\mathcal{L}^{\ell-1}\mathcal{E}\mathcal{L}^{n-\ell}\{x_{a}\}_{a\in E_{n}}$ for $\ell\in\{1,\ldots,n\}$. There is a $C>0$ not depending on the distribution of the variables $x_{a}$ such that the following inequality holds for all $n\in\mathbb{N}$:

[TABLE]

Proof of Lemma 11.3.

Part (i): For $f\in E_{\mathbf{\widehat{n}}}$ the variance of Y_{f}^{N,n}\,=\,\mathcal{L}^{\mathbf{n}-\mathbf{\widehat{n}}}\big{\{}X_{g}^{(\mathbf{n},n)}\big{\}}_{g\in f\cap E_{\mathbf{n}}} is \sigma_{\mathbf{n},n}^{2}:=\textup{Var}\big{(}X_{g}^{(\mathbf{n},n)}\big{)} since the operation $\mathcal{L}$ preserves the variance of the random variables in the array by Remark 6.6. The convergence of $\sigma_{\mathbf{n},n}^{2}$ to $R(r-\mathbf{n})$ as $n\rightarrow\infty$ holds by (III) of Lemma 6.15 with $m=2$. Finally, $\sigma_{\mathbf{n},n}^{2}$ is bounded from above and below by constant multiples of $\frac{1}{N}$ for all $N,n\in\mathbb{N}$ with $n\geq\mathbf{n}$ by Proposition 12.1 since $N\sim\mathbf{n}:=N+\lfloor 2\mathfrak{m}\log N\rfloor$.

Part (ii): Since terms in the sum Z^{N,n}_{f}=\sum_{k=\mathbf{\widehat{n}}+1}^{\mathbf{n}}\mathcal{L}^{k-\mathbf{\widehat{n}}-1}\mathcal{E}\mathcal{L}^{\mathbf{n}-k}\big{\{}X_{g}^{(\mathbf{n},n)}\big{\}}_{g\in f\cap E_{\mathbf{n}}} are uncorrelated by Lemma 6.7, we have the second equality below.

[TABLE]

By definition of $\varsigma_{N}^{2}$, the above expression has the form $\varsigma_{N}^{2}+\xi_{N}(n)$.

Next we argue that $\varsigma_{N,n}^{2}$ is bounded from above by a constant multiple of $\frac{\log N}{N^{2}}$. By (12.6), we have that \varsigma_{N,n}^{2}:=(\mathbf{n}-\mathbf{\widehat{n}})S\big{(}\sigma_{\mathbf{n},n}^{2}\big{)}, where the polynomial $S(x):=M(x)-x$ has no constant or linear terms. Since the lowest-order nonzero term in the polynomial $S(x)$ is quadratic, part (i) of Proposition 12.1 implies that S\big{(}\sigma_{\mathbf{n},n}^{2}\big{)} is bounded by a constant multiple of $\frac{1}{N^{2}}$ for all $n,N\in\mathbb{N}$ with $n\geq\mathbf{n}$. The result then follows because $\mathbf{n}-\mathbf{\widehat{n}}\sim\mathfrak{m}\log N$ for $N\gg 1$.

Part (iii): For $g\in E_{\mathbf{n}}$, define \sigma^{(4)}_{\mathbf{n},n}\,:=\,\mathbb{E}\Big{[}\big{(}X_{g}^{(\mathbf{n},n)}\big{)}^{4}\Big{]}. Also, for $m\in\{2,4\}$ and $a\in E_{k}$ with $k\in\{0,\ldots,\mathbf{n}\}$, we define

[TABLE]

Note that \widetilde{\sigma}^{(2)}_{k,\mathbf{n},n}=\textup{Var}\big{(}X_{g}^{(\mathbf{n},n)}\big{)}=:\sigma^{2}_{\mathbf{n},n}, and Jensen’s inequality implies that

[TABLE]

The second inequality above holds for some $C>0$ and all $n,N\in\mathbb{N}$ with $n\geq 2\mathbf{n}$ by (ii) of Proposition 12.1 and since $\mathbf{n}\sim N$ for $N\gg 1$. Applying (12.7) with $k=\mathbf{\widehat{n}}$ yields our desired bound for \mathbb{E}\big{[}\big{(}Y_{f}^{N,n}\big{)}^{4}\big{]}\,=\,\widetilde{\sigma}^{(4)}_{\mathbf{\widehat{n}},\mathbf{n},n}.

Let $f\in E_{\mathbf{\widehat{n}}}$. By Lemma 12.2 the fourth moment of $Z_{f}^{N,n}$ has the bound

[TABLE]

For $\mathbf{\widehat{n}}<k\leq\mathbf{n}$, define \big{\{}\check{X}_{a}^{N,n}\big{\}}_{a\in f\cap E_{k}}:=\mathcal{L}^{\mathbf{n}-k}\big{\{}X_{g}^{(\mathbf{n},n)}\big{\}}_{g\in f\cap E_{\mathbf{n}}}. A single term from the sum in (12.8) has the bound

[TABLE]

The first inequality above is Jensen’s, and the second inequality holds for all $n,N$ with $n\geq\mathbf{n}$ by (12.7). Thus by (12.8), (12.9), and the form of the polynomial $T(x,y)$, the fourth moment of $Z_{f}^{N,n}$ is bounded from above by a multiple of $\frac{(\mathbf{n}-\mathbf{\widehat{n}})^{2}}{N^{2}}\sim\mathfrak{m}^{2}\frac{\log^{2}(N+1)}{N^{2}}$.

Part (iv): Since $\mathbf{n}\sim N$ for $N\gg 1$, an application of (ii) of Proposition 12.1 with $k=\mathbf{n}$ yields that the fourth moment of $X_{g}^{(\mathbf{n},n)}$ is bounded by a constant multiple of $\frac{C}{N^{2}}$ for all $n,N\in\mathbb{N}$ with $n\geq 2\mathbf{n}$. The fourth moment bounds for $\widehat{X}_{e}^{N,n}$ and $\mathbf{\widehat{X}}_{e}^{N,n}$ can be proven using the techniques in the proof of (iii).131313Also, see the proof of part (iv) of Lemma 11.3 in Section 13.3, which is an analogous result for general even moments under $\alpha$-sharp regularity-type assumptions. ∎

Proof of Lemma 11.5.

Let $(X,Y)$ be a coupling such that the $L^{1}$-distance between the variables $X$ and $Y$ is equal to $\rho_{1}(X,Y)$. Since $\rho_{2}(X,Y)$ is an infimum of the $L^{2}$ distance over couplings,

[TABLE]

∎

13 Sharp regularity and rate of convergence

Next we focus on proving Theorem 7.3. To do this, we will use analogous technical results to those in Lemmas 9.7-9.9—see (i)-(iii) of Lemma 13.1 below—that assume sharp regularity-type conditions and provide bounds in terms of functions of the “microscopic” parameter $n\in\mathbb{N}$ rather than the “mesoscopic” parameter $N\in\mathbb{N}$. With Lemma 13.1 in hand, the proof of Theorem 7.3 carries through with only minor modifications of the proof of Theorem 6.23. We prove Lemma 13.1 in Section 13.2, and in Section 13.3 we prove an analog of Lemma 11.3.

13.1 Proof of Theorem 7.3

We will prove Theorem 7.3 after stating two preliminary lemmas. Lemma 13.1 bounds the same quantities as in Lemmas 9.7-9.9, and its proof is in the next subsection.

Lemma 13.1**.**

Fix $\mathbf{v},\varkappa>0$, $\alpha\in(0,1)$, $\upsilon\in(0,\alpha/9)$, and a bounded interval $\mathcal{I}\subset{\mathbb{R}}$. Define $\mathfrak{p}=\lceil\frac{2\alpha}{\alpha-9\upsilon}\rceil+1$ and $N\equiv N(n):=\lfloor n^{2\alpha/9}\rfloor$ for $n\in\mathbb{N}$. There exists a positive number $\mathbf{c}\equiv\mathbf{c}(\mathcal{I},\mathbf{v},\varkappa,\alpha,\upsilon)$ such that for any $r\in\mathcal{I}$, $n\in\mathbb{N}$, and i.i.d. array of centered random variables \big{\{}X_{h}^{(n)}\big{\}}_{h\in E_{n}} satisfying

1. (I)

\left|\textup{Var}\big{(}X_{h}^{(n)}\big{)}-\kappa^{2}\big{(}\frac{1}{n}+\frac{\eta\log n}{n^{2}}+\frac{r}{n^{2}}\big{)}\right|<\frac{\mathbf{v}}{n^{2+\alpha}}* and* 2. (II)

\mathbb{E}\Big{[}\big{|}X_{h}^{(n)}\big{|}^{2\mathfrak{p}}\Big{]}<\frac{\varkappa}{n^{\mathfrak{p}}},

the following inequalities hold:

1. (i)

\mathbb{E}\Big{[}\big{(}X^{(N,n)}_{e}-\widehat{X}^{N,n}_{e}\big{)}^{2}\Big{]}^{1/2}\,<\,\mathbf{c}\frac{\log(n+1)}{n^{\alpha/3}}* ,* 2. (ii)

\rho_{2}\big{(}\widehat{X}^{N,n}_{e},\mathbf{\widehat{X}}^{N,n}_{e}\big{)}\,<\,\frac{\mathbf{c}}{n^{4\alpha/9+\upsilon}}\displaystyle* ,* 3. (iii)

\rho_{2}\big{(}\mathbf{\widehat{X}}_{e}^{N,n},\mathbf{\widetilde{X}}_{e}^{(N)}\big{)}\,<\,\frac{\mathbf{c}}{n^{8\alpha/9}}\,,\displaystyle**

where \big{\{}X^{(N,n)}_{e}\big{\}}_{e\in E_{N}} is the $N^{th}$ generation layer of the $\mathcal{Q}$-pyramidic array generated from \big{\{}X_{h}^{(n)}\big{\}}_{h\in E_{n}}, and $\{\widehat{X}^{N,n}_{e}\}_{e\in E_{N}}$, $\{\mathbf{\widehat{X}}_{e}^{N,n}\}_{e\in E_{N}}$, $\{\mathbf{\widetilde{X}}_{e}^{(N)}\}_{e\in E_{N}}$ are defined as in Definition 9.5 with $\mathfrak{m}:=\frac{21}{2\log b}$.

Remark 13.2**.**

In Lemma 13.1, any value of $\mathfrak{m}$ greater than $\frac{21}{2\log b}$ yields the same result.

Recall that the random variables in the array \big{\{}\mathbf{X}_{h}^{(n)}\big{\}}_{h\in E_{n}} from Theorem 6.16 with parameter $r\in{\mathbb{R}}$ have $m^{th}$ moment given by $R^{(m)}(r-n)$, where the function $R^{(2)}\equiv R$ is characterized in Lemma 2.3 and the functions $R^{(m)}$ for $m\geq 3$ are characterized in Theorem 2.4. The following trivial lemma implies that the conditions of Lemma 13.1 are satisfied by $\{\mathbf{X}_{h}^{(n)}\}_{h\in E_{n}}$ for all $n\in\mathbb{N}$ and all $r$ in a bounded interval $\mathcal{I}$ when $\mathbf{v},\varkappa>0$ are large enough.

Lemma 13.3**.**

Fix $\alpha\in(0,1)$, $\mathfrak{p}\in\{2,3,\ldots\}$, and a bounded interval $\mathcal{I}\subset{\mathbb{R}}$. There exist $\mathbf{v},\varkappa>0$ such that (I)-(II) below hold for all $r\in\mathcal{I}$ and $n\in\mathbb{N}$.

1. (I)

\left|R(r-n)-\kappa^{2}\big{(}\frac{1}{n}+\frac{\eta\log n}{n^{2}}+\frac{r}{n^{2}}\big{)}\right|<\frac{\mathbf{v}}{n^{2+\alpha}}** 2. (II)

$R^{(2\mathfrak{p})}(r-n)<\frac{\varkappa}{n^{\mathfrak{p}}}$**

Proof.

The inequalities (I)-(II) above hold for large enough $\mathbf{v},\varkappa>0$ and all $r\in\mathcal{I}$ and $n\in\mathbb{N}$ as a consequence of the asymptotics in (II) of Lemma 2.3 and (II) of Theorem 2.4, respectively. ∎

Proof of Theorem 7.3.

Let $\mathbf{v}$, $\varkappa$, $\alpha$, $\upsilon$, $\mathcal{I}$, $\mathfrak{p}$, and $N$ be as in Lemma 13.1. By Lemma 13.1, there is a $\mathbf{c}\equiv\mathbf{c}(\mathcal{I},\mathbf{v},\varkappa,\alpha,\upsilon)$ such that if $r\in\mathcal{I}$, $n\in\mathbb{N}$, and \big{\{}X^{(n)}_{h}\big{\}}_{h\in E_{n}} is an array of i.i.d. centered random variables satisfying conditions (I)-(II) in Theorem 7.3, then

[TABLE]

where for the third inequality we have used that $n^{-8\alpha/9}$ is \mathit{O}\big{(}n^{-4\alpha/9-\upsilon}\big{)} as $n\rightarrow\infty$ since $\upsilon<\alpha/9$. By the same reasoning as in parts (a)-(c) of the proof of Theorem 6.23, there are i.i.d. families of pair couplings \big{\{}\big{(}\widehat{X}^{N,n}_{e},\mathbf{\widehat{X}}^{N,n}_{e}\big{)}\big{\}}_{e\in E_{N}} and \big{\{}\big{(}\mathbf{\widehat{X}}_{e}^{N,n},\mathbf{\widetilde{X}}_{e}^{(N)}\big{)}\big{\}}_{e\in E_{N}} such that the first two inequalities below hold:

[TABLE]

where $C>0$ arises from an application of Proposition 9.1. For \mathbf{C}:=\mathbf{c}C\big{(}2+\sup_{u\in\mathbb{N}}\frac{\log(u+1)}{u^{\alpha/9-\upsilon}}\big{)}, the third inequality simply uses that $N:=\lfloor n^{2\alpha/9}\rfloor$.

By (13.1) the Wasserstein-2 distance between X^{(0,n)}=\mathcal{Q}^{n}\big{\{}X^{(n)}_{h}\big{\}}_{h\in E_{n}} and \mathcal{Q}^{N}\big{\{}\mathbf{\widetilde{X}}_{e}^{(N)}\big{\}}_{e\in E_{N}} is bounded by a multiple $\mathbf{C}\equiv\mathbf{C}(\mathcal{I},\mathbf{v},\varkappa,\alpha,\upsilon)$ of $n^{-\upsilon}$ for any i.i.d. array \big{\{}X^{(n)}_{h}\big{\}}_{h\in E_{n}} satisfying properties (I)-(II) in the statement of Theorem 7.3. Let the array of random variables \big{\{}\mathbf{X}^{(n)}_{h}\big{\}}_{h\in E_{n}} be defined as in Theorem 6.16 for parameter $r$. By property (III) in Theorem 6.16, the $m^{th}$ positive integer moment of $\mathbf{X}^{(n)}_{h}$ is $R^{(m)}(r-n)$, and thus by Lemma 13.3 the array \big{\{}\mathbf{X}^{(n)}_{h}\big{\}}_{h\in E_{n}} satisfies conditions (I)-(II) of Lemma 13.1 for all $r\in\mathcal{I}$ and $n\in\mathbb{N}$ with possibly larger values of $\mathbf{v},\varkappa>0$. By substituting \big{\{}\mathbf{X}^{(n)}_{h}\big{\}}_{h\in E_{n}} for \big{\{}X^{(n)}_{h}\big{\}}_{h\in E_{n}} in our above analysis, we get that the Wasserstein-$2$ distance between \mathbf{X}=\mathcal{Q}^{n}\big{\{}\mathbf{X}^{(n)}_{h}\big{\}}_{h\in E_{n}} and \mathcal{Q}^{N}\big{\{}\mathbf{\widetilde{X}}_{e}^{(N)}\big{\}}_{e\in E_{N}} is bounded by a multiple $\mathbf{C^{\prime}}\equiv\mathbf{C^{\prime}}(\mathcal{I},\alpha,\upsilon)$ of $n^{-\upsilon}$ for all $n\in\mathbb{N}$ and $r\in\mathcal{I}$. By the triangle inequality, we thus have the bound that we sought for the Wasserstein-2 distance between $X^{(0,n)}$ and $\mathbf{X}$.∎

13.2 Proof of Lemma 13.1

Recall that there are steps in each of the proofs of Lemmas 9.7-9.9 in which we respectively identified sequences $\{\xi_{N}(n)\}_{n\in\mathbb{N}}$, $\{\xi_{N}^{\prime}(n)\}_{n\in\mathbb{N}}$, $\{\xi_{N}^{\prime\prime}(n)\}_{n\in\mathbb{N}}$ that vanish as $n\rightarrow\infty$ for each fixed $N\in\mathbb{N}$ and for which the inequalities ($\textup{i}^{\prime}$)-($\textup{iii}^{\prime}$) below hold for some $\mathfrak{c}>0$ and all $N,n$ with $n\geq 2\mathbf{n}$.

1. ($\textup{i}^{\prime}$)

\mathbb{E}\Big{[}\big{(}X^{(N,n)}_{e}-\widehat{X}^{N,n}_{e}\big{)}^{2}\Big{]}\,\leq\,\mathfrak{c}\frac{\log^{2}(N+1)}{N^{3}}\,+\,\xi_{N}(n) 2. ($\textup{ii}^{\prime}$)

\rho_{1}\big{(}\widehat{X}^{N,n}_{e},\mathbf{\widehat{X}}^{N,n}_{e}\big{)}\,\leq\,\mathfrak{c}\frac{\log^{-\frac{1}{2}}(N+1)}{N^{\mathfrak{m}\log b}}\,+\,\xi_{N}^{\prime}(n)\displaystyle 3. ($\textup{iii}^{\prime}$)

\rho_{2}\big{(}\mathbf{\widehat{X}}_{e}^{N,n},\mathbf{\widetilde{X}}_{e}^{(N)}\big{)}\,\leq\,\frac{\mathfrak{c}}{N^{\frac{\mathfrak{m}}{3}\log b+\frac{1}{2}}}\,+\,\xi_{N}^{\prime\prime}(n)\displaystyle

The inequalities ($\textup{i}^{\prime}$)-($\textup{iii}^{\prime}$) are from (11.4), (11.21), & (11.29). Also recall that the proofs of ($\textup{ii}^{\prime}$) & ($\textup{iii}^{\prime}$) rely on bounds from Lemma 11.3. The following lemma states analogous results to those in Lemma 11.3 under the conditions (I)-(II) of Lemma 13.1, and its proof is in Section 13.3. In the statement of Lemma 13.4, the random variables $Y^{N,n}_{f}$ and $Z^{N,n}_{f}$ are defined as in (9.5) & (9.6), $\sigma_{N,n}^{2}$ is defined as in (11.1), and $\varsigma_{N,n}^{2},\varsigma_{N}^{2}$ are defined as in Lemma 13.4.

Lemma 13.4**.**

Fix $\mathbf{v},\varkappa>0$, $\alpha\in(0,1)$, $\mathfrak{p}\in\{2,3,\ldots\}$, and a bounded interval $\mathcal{I}\subset{\mathbb{R}}$. For $n\in\mathbb{N}$, define $N:=\lfloor n^{2\alpha/9}\rfloor$. There exist positive numbers $\mathbf{c}\equiv\mathbf{c}(\mathcal{I},\mathbf{v},\varkappa,\alpha,\mathfrak{p})$ and $\lambda\equiv\lambda(\mathcal{I},\mathbf{v},\alpha)$ such that for any $r\in\mathcal{I}$, $n\in\mathbb{N}$, and i.i.d. array of centered random variables \big{\{}X_{h}^{(n)}\big{\}}_{h\in E_{n}} satisfying conditions (I)-(II) of Lemma 13.1, the inequalities below hold for the random variables $Y^{N,n}_{f}$, $Z^{N,n}_{f}$, $\widehat{X}^{N,n}_{e}$, $\mathbf{\widehat{X}}_{e}^{N,n}$ and the variances \sigma_{N,n}^{2}:=\textup{Var}\big{(}X_{e}^{(N,n)}\big{)} & \varsigma_{N,n}^{2}:=\textup{Var}\big{(}Z^{N,n}_{f}\big{)} defined through the $\mathcal{Q}$-pyramidic array \big{\{}X_{a}^{(*,n)}\big{\}}_{a\in E_{*}} generated from \big{\{}X_{h}^{(n)}\big{\}}_{h\in E_{n}}.

1. (i)

$\sigma_{N,n}^{2}$* is bounded from above by $\frac{\mathbf{c}}{N}$, and $\sigma_{N,n}^{2}$ is bounded from below by $\frac{\mathbf{c}^{-1}}{N}$ provided that $n>\lambda$.* 2. (ii)

$\varsigma_{N,n}^{2}$* is bounded from above by $\mathbf{c}\frac{\log(N+1)}{N^{2}}$ and satisfies the inequality*

[TABLE] 3. (iii)

The fourth moments of the random variables $Y^{N,n}_{f}$ and $Z^{N,n}_{f}$ are bounded by $\frac{\mathbf{c}}{N^{2}}$ and $\mathbf{c}\frac{\log^{2}(N+1)}{N^{4}}$, respectively. 4. (iv)

The $(2\mathfrak{p})^{th}$ moments of the random variables $X^{(\mathbf{n},n)}_{g}$, $\widehat{X}^{N,n}_{e}$, and $\mathbf{\widehat{X}}_{e}^{N,n}$ are bounded by $\frac{\mathbf{c}}{N^{\mathfrak{p}}}$.

The lemma below states that analogs of the inequalities ($\textup{i}^{\prime}$)-($\textup{iii}^{\prime}$) hold for large enough $\mathfrak{c}\equiv\mathfrak{c}(\mathcal{I},\mathbf{v},\varkappa,\alpha)>0$ when $X^{(N,n)}_{e}$, $\widehat{X}^{N,n}_{e}$, and $\mathbf{\widehat{X}}_{e}^{(N)}$ are defined in terms of an array of random variables \big{\{}X_{h}^{(n)}\big{\}}_{h\in E_{n}} satisfying the conditions of Lemma 13.1. If we were only concerned with having a counterpart to the inequality ($\textup{i}^{\prime}$), then the constant $\mathfrak{c}$ would only depend on the bounded interval $\mathcal{I}$ because the derivation of ($\textup{i}^{\prime}$) in the proof of Lemma 9.7 is entirely based on properties of the function $R$ from Lemma 2.3. The counterparts to ($\textup{ii}^{\prime}$) & ($\textup{iii}^{\prime}$) can be shown by following the steps in the proofs of ($\textup{ii}^{\prime}$) & ($\textup{iii}^{\prime}$) and replacing each application of (i)-(iv) from Lemma 11.3 by an application of (i)-(iv) from Lemma 13.4. Thus we omit the proof of Lemma 13.5, which is a lengthy near-repetition of our previous line of arguments establishing ($\textup{i}^{\prime}$)-($\textup{iii}^{\prime}$) in Section 11.

Lemma 13.5**.**

Fix $\mathbf{v},\varkappa,\mathfrak{m}>0$, $\alpha\in(0,1)$, and a bounded interval $\mathcal{I}$. Define $N=\lfloor n^{2\alpha/9}\rfloor$. There exists a positive number $\mathfrak{c}\equiv\mathfrak{c}(\mathcal{I},\mathbf{v},\varkappa,\alpha,\mathfrak{m})$ such that for any $r\in\mathcal{I}$, $n\in\mathbb{N}$, and i.i.d. array of centered random variables \big{\{}X_{h}^{(n)}\big{\}}_{h\in E_{n}} satisfying conditions (I)-(II) of Lemma 13.1 for $\mathfrak{p}=2$, then the inequalities ($i^{\prime}$)-($iii^{\prime}$) above hold, where \big{\{}X^{(N,n)}_{e}\big{\}}_{e\in E_{N}} is the $N^{th}$ generation layer of the $\mathcal{Q}$-pyramidic array generated from \big{\{}X_{h}^{(n)}\big{\}}_{h\in E_{n}}, and $\{\widehat{X}^{N,n}_{e}\}_{e\in E_{N}}$, $\{\mathbf{\widehat{X}}_{e}^{N,n}\}_{e\in E_{N}}$, $\{\mathbf{\widetilde{X}}_{e}^{(N)}\}_{e\in E_{N}}$ are defined as in Definition 9.5.

Lemma 13.6 offers some control for the rate of convergence in the $m=2$ case of Lemma 6.15 under the $\alpha$-sharp regularity condition on the variance of the random variables in the generating array \big{\{}X_{h}^{(n)}\big{\}}_{h\in E_{n}}. The proof, which is placed in Section 15.3, borrows a technical result from [10].

Lemma 13.6**.**

Fix $\mathbf{v}>0$, $\alpha\in(0,1)$, and a bounded interval $\mathcal{I}\subset{\mathbb{R}}$. There exists a positive number $C_{\mathcal{I},\mathbf{v},\alpha}$ such that for any $r\in\mathcal{I}$, $n\in\mathbb{N}$, and i.i.d. array of centered random variables \big{\{}X_{h}^{(n)}\big{\}}_{h\in E_{n}} satisfying condition (I) of Lemma 13.1, the inequality below holds for all $k\in\{0,1,\ldots,n\}$:

[TABLE]

where \big{\{}X_{a}^{(k,n)}\big{\}}_{a\in E_{k}} is the $k^{th}$ generation layer of the $\mathcal{Q}$-pyramidic array generated from \big{\{}X_{h}^{(n)}\big{\}}_{h\in E_{n}}.

Proof of Lemma 13.1.

Fix $\mathbf{v},\varkappa>0$, $\alpha\in(0,1)$, $\upsilon\in(0,\alpha/9)$, and a bounded interval $\mathcal{I}$. Define $N:=\lfloor n^{2\alpha/9}\rfloor$, $\mathfrak{p}:=\lceil\frac{2\alpha}{\alpha-9\upsilon}\rceil+1$, and $\mathfrak{m}:=\frac{21}{2\log b}$. Let \big{\{}X_{h}^{(n)}\big{\}}_{h\in E_{n}} be an i.i.d. array of random variables satisfying conditions (I)-(II) for $\mathbf{v}$, $\varkappa$, $\alpha$, $\mathfrak{p}$, and some $r\in\mathcal{I}$. Since $\mathfrak{p}\geq 2$, Jensen’s inequality and condition (II) imply that

[TABLE]

Thus \big{\{}X_{h}^{(n)}\big{\}}_{h\in E_{n}} satisfies condition (II) with $\mathfrak{p}\mapsto 2$ and $\varkappa\mapsto\max(1,\varkappa)$. By Lemma 13.5, there is $\mathfrak{c}\equiv\mathfrak{c}(\mathcal{I},\mathbf{v},\varkappa,\alpha,\mathfrak{m})$ such that the inequalities ($\textup{i}^{\prime}$)-($\textup{iii}^{\prime}$) hold. In parts (i)-(iii) below we will start from the inequalities ($\textup{i}^{\prime}$)-($\textup{iii}^{\prime}$), respectively, and focus on bounding the terms $\xi_{N}(n)$, $\xi_{N}^{\prime}(n)$, $\xi_{N}^{\prime\prime}(n)$.

Part (i): By inequality ($\textup{i}^{\prime}$),

[TABLE]

where $\xi_{N}(n)$ is the error of the approximation of (11.2) by the expression in (11.3), i.e.,

[TABLE]

The second inequality in (13.3) holds for some $\mathfrak{c}^{\prime}>0$ since $N:=\lfloor n^{2\alpha/9}\rfloor$. In the analysis below, we will show that $\xi_{N}(n)$ is bounded by a multiple of $\frac{\log(n+1)}{n^{11\alpha/9}}$, and consequently that the $L^{2}$ distance between $X^{(N,n)}_{e}$ and $\widehat{X}^{N,n}_{e}$ is bounded by a multiple of $\frac{\log(n+1)}{n^{\alpha/3}}$ by (13.3).

Define the polynomial $S(x):=M(x)-x$, in other words, as $M$ with the linear term removed. As in the proof of Lemma 9.7, we can use telescoping sums to write

[TABLE]

where we have used the identities $M(\sigma_{k,n}^{2})=\sigma_{k-1,n}^{2}$ and M\big{(}R(r-k)\big{)}=R(r-k+1). Thus $\xi_{N}(n)$ can be written as

[TABLE]

It follows that

[TABLE]

By Lemma 13.6, there is a $C_{\mathcal{I},\mathbf{v},\alpha}>0$ such that for all $r\in\mathcal{I}$ and $n\in\mathbb{N}$

[TABLE]

The lowest-order nonzero term in the polynomial $S(x)$ is quadratic, and thus the following is finite:

[TABLE]

Since $\frac{1}{n^{\alpha}}\leq 1$ for $n\in\mathbb{N}$, (13.5) implies that the distance between S\big{(}\sigma_{k,n}^{2}\big{)} and S\big{(}R(r-k)\big{)} is bounded by

[TABLE]

By applying (13.6) to (13.4) and using that $R$ is an increasing function, we get that

[TABLE]

The supremum above is finite because $R(s)\sim\frac{\kappa^{2}}{-s}$ for $s\gg 1$ by Lemma 2.3. Since $N:=\lfloor n^{2\alpha/9}\rfloor$ and $\mathbf{n}:=N+\lfloor 2\mathfrak{m}\log N\rfloor$, the inequality (13.7) implies that \big{|}\xi_{N}(n)\big{|} is bounded by a multiple of $\frac{\log(n+1)}{n^{11\alpha/9}}$.

Part (ii): Since \xi^{\prime}_{N,n}:=\sqrt{\frac{\pi}{2}}\big{|}\frac{\varsigma_{N,n}^{2}}{\varsigma_{N}}-\varsigma_{N}\big{|}, the first inequality below is ($\textup{i}^{\prime\prime}$):

[TABLE]

The second inequality holds for some $C>0$ by part (ii) of Lemma 13.4 for the second term and since $N:=\lfloor n^{2\alpha/9}\rfloor$ and $\mathfrak{m}:=\frac{21}{2\log b}$ for the first term.

As in the proof of Lemma 9.8, we will use Lemma 11.5 to bound the Wasserstein-$2$ distance using the Wasserstein-$1$ distance. Applying Lemma 11.5 with $m=2\mathfrak{p}-1$ yields

[TABLE]

The second inequality uses that $N=\lfloor n^{2\alpha/9}\rfloor$. Note that the exponent $\frac{5\alpha}{9}-\frac{4\alpha}{9(2\mathfrak{p}-1)}$ is strictly greater than $\frac{4\alpha}{9}+\upsilon$ since $\mathfrak{p}:=\lceil\frac{2\alpha}{\alpha-9\upsilon}\rceil+1$, and thus the above shows that the Wassertstein-$2$ distance between $\widehat{X}^{N,n}_{e}$ and $\mathbf{\widehat{X}}^{N,n}_{e}$ is bounded by a multiple of $n^{-4\alpha/9-\upsilon}$.

Part (iii): Since \xi^{\prime\prime}_{N,n}:=\mathfrak{c}\big{|}\sigma_{\mathbf{n},n}-\sqrt{R(r-\mathbf{n})}\big{|}, the inequality ($\textup{iii}^{\prime}$) gives us that

[TABLE]

Since $\mathfrak{m}:=\frac{21}{2\log b}$ and $N=\lfloor n^{2\alpha/9}\rfloor$, the first term on the right side of (13.9) is bounded by a multiple of $n^{-8\alpha/9}$. By Lemma 13.6, we have the second inequality below:

[TABLE]

Since $\mathbf{n}=N+\lfloor 2\mathfrak{m}\log N\rfloor$ and $N=\lfloor n^{2\alpha/9}\rfloor$, the above is bounded by a multiple of $n^{-8\alpha/9}$. ∎

13.3 Proof of Lemma 11.3

The following is an analog of Proposition 12.1 that provides bounds for the moments of the random variables in a $\mathcal{Q}$-pyramidic array \big{\{}X_{a}^{(k,n)}\big{\}}_{a\in E_{k}} generated from an i.i.d. array \big{\{}X_{h}^{(n)}\big{\}}_{h\in E_{n}} satisfying the conditions of Lemma 13.1. The proof uses techniques from [10] and is placed in Section 15.4.

Proposition 13.7**.**

Fix $\mathbf{v},\varkappa>0$, $\alpha\in(0,1)$, $\mathfrak{p}\in\{2,3,\ldots\}$, and a bounded interval $\mathcal{I}\subset{\mathbb{R}}$. There exists a positive number $C\equiv C(\mathcal{I},\mathbf{v},\varkappa,\alpha,\mathfrak{p})$ such that for any $r\in\mathcal{I}$, $n\in\mathbb{N}$, and i.i.d. array of centered random variables \big{\{}X_{h}^{(n)}\big{\}}_{h\in E_{n}} satisfying conditions (I)-(II) of Lemma 13.1, the inequality below holds for all $k\in\{0,1,\ldots,n\}$:

[TABLE]

where \big{\{}X_{a}^{(k,n)}\big{\}}_{a\in E_{k}} is the $k^{th}$ generation layer of the $\mathcal{Q}$-pyramidic array generated from \big{\{}X_{h}^{(n)}\big{\}}_{h\in E_{n}}.

Proof of Lemma 11.3.

Part (i): By Lemma 13.6, there is a $C_{\mathcal{I},\mathbf{v},\alpha}>0$ such that

[TABLE]

holds for any $r\in\mathcal{I}$, $n\in\mathbb{N}$, and i.i.d. array of random variables \big{\{}X_{h}^{(n)}\big{\}}_{h\in E_{n}} with $n\geq\mathbf{n}$ and satisfying condition (I) of Lemma 13.1, where we have used that $N:=\lfloor n^{2\alpha/9}\rfloor$ for the second inequality. Since $R(s)\sim-\frac{\kappa^{2}}{s}$ when $-s\gg 1$ and $\mathbf{n}\sim N$ for $N\gg 1$, the supremum and infimum of $R(r-\mathbf{n})$ for $r\in\mathcal{I}$ are respectively bounded from above and below by positive multiples $C_{\mathcal{I}}$ and $c_{\mathcal{I}}$ of $\frac{1}{N}$:

[TABLE]

Thus $\sigma^{2}_{\mathbf{n},n}$ is bounded from above by a constant multiple of $\frac{1}{N}$. When N>\lambda:=\big{(}\frac{2C_{\mathcal{I},\mathbf{v},\alpha}}{c_{\mathcal{I}}}\big{)}^{2/7}, then $\sigma^{2}_{\mathbf{n},n}$ is bounded from below by $\frac{c_{\mathcal{I}}}{2N}$.

Part (ii): Define the polynomial $S(x):=M(x)-x$, in other terms, as $M$ with the linear term removed. We can write $\varsigma_{N,n}^{2}$ and $\varsigma_{N}^{2}$ in the forms below:

[TABLE]

The first equality on the top line above uses (12.6) and that $\sigma_{k-1,n}^{2}=M(\sigma_{k,n}^{2})$, and the first equality on the second line uses that M\big{(}R(s)\big{)}=R(s+1) by part (I) of Lemma 2.3.

We will first prove the bound for $\frac{1}{\varsigma_{N}}|\varsigma_{N,n}^{2}-\varsigma_{N}^{2}|$. By the same reasoning as in (13.6), there is a $\mathbf{C}\equiv\mathbf{C}(\mathcal{I},\mathbf{v},\alpha)$ such that the inequality below holds

[TABLE]

From the relations (13.11) and (13.12), we get the first inequality below,

[TABLE]

The supremum above is finite since the lowest-order nonzero term in the polynomial $S$ is quadratic and $R(s)\sim-\frac{\kappa^{2}}{s}$ as $s\rightarrow-\infty$. The above shows that $\frac{1}{\varsigma_{N}}|\varsigma_{N,n}^{2}-\varsigma_{N}^{2}|$ is bounded by a constant multiple of $n^{-\alpha}\log^{1/2}(n+1)$ since $\mathbf{n}-\mathbf{\widehat{n}}\approx\mathfrak{m}\log N\approx\frac{2\mathfrak{m}}{9}\log n$.

Next we show that $\varsigma_{N,n}^{2}$ is bounded by a constant multiple of $\frac{\log(N+1)}{N^{2}}$. By the triangle inequality and (13.12), we have that

[TABLE]

Note that $R(r-\mathbf{n})\propto\frac{\kappa^{2}}{N}$ by (II) of Lemma 2.3 since $\mathbf{n}\sim N$ as $N\gg 1$. Thus, since the lowest-order nonzero term of the polynomial $S$ is quadratic, the first term on the right side of (13.13) is bounded by a constant multiple of $\frac{\log(N+1)}{N^{2}}$. The second term on the right side of (13.13) is bounded by a constant multiple of $\frac{\log(N+1)}{N^{11/2}}$ because $n^{\alpha}\sim N^{9/2}$. Thus $\varsigma_{N,n}^{2}$ has the stated bound.

Part (iii): The proof follows through the same steps as the proof of part (iii) of Lemma 11.3 with each application of Proposition 12.1 replaced by an application of Proposition 13.7.

Part (iv): The bound for the $(2\mathfrak{p})^{th}$ moment of $X_{g}^{(\mathbf{n},n)}$ follows from Proposition 13.7 since $\mathbf{n}\geq N$. We will only prove the bound for the $(2\mathfrak{p})^{th}$ moment of $\widehat{X}^{N,n}_{e}$ since the analysis for $\mathbf{\widehat{X}}^{N,n}_{e}$ is similar. By (9.2) the random variable $\widehat{X}_{e}^{N,n}$ can be written in the form

[TABLE]

It suffices to bound the $(2\mathfrak{p})^{th}$ moment of each of the terms (a) and (b) by a multiple of $N^{-\mathfrak{p}}$.

(a): Fix $k\in\{0,\ldots,\mathbf{n}-N\}$, and let $\mathbf{a}\in E_{\mathbf{n}-k}$. Since \mathcal{L}^{k}\big{\{}X_{g}^{(\mathbf{n},n)}\big{\}}_{g\in\mathbf{a}\cap E_{\mathbf{n}}} is an i.i.d. sum of random variables $\frac{1}{b^{k}}X_{g}^{(\mathbf{n},n)}$ indexed by $g\in\mathbf{a}\cap E_{\mathbf{n}}$, the Marcinkiewicz-Zygmund inequality gives us the first inequality below for some universal constant $B_{\mathfrak{p}}>0$.

[TABLE]

The third inequality uses that $\mathbf{n}\geq N$. Applying the inequality above with $k=\mathbf{n}-N$ yields the sought-after bound for the $(2\mathfrak{p})^{th}$ moment of (a).

(b): For $\ell\in\{1,\ldots,\mathbf{n}-N\}$, define the array \big{\{}Y^{\ell,\mathbf{n},n}_{a}\big{\}}_{a\in e\cap E_{\mathbf{n}-\ell}}:=\mathcal{E}\mathcal{L}^{\ell-1}\big{\{}X_{g}^{(\mathbf{n},n)}\big{\}}_{g\in e\cap E_{\mathbf{n}}}. By the triangle inequality,

[TABLE]

We will show that the maximum above is bounded by a multiple of $N^{-2\mathfrak{p}}$, which suffices to show that the $(2\mathfrak{p})^{th}$ moment of (b) has order $N^{-\mathfrak{p}}$ since $\mathbf{n}-N\approx\mathfrak{m}\log N$. Applying the Marcinkiewicz-Zygmund and Jensen inequalities as in (13.15) yields the following inequality for a representative $a\in e\cap E_{\mathbf{n}-\ell}$:

[TABLE]

where $T_{\mathfrak{p}}(y_{2},\ldots,y_{2\mathfrak{p}})$ is a linear combination of monomials $y_{j_{1}}y_{j_{2}}\cdots y_{j_{m}}$ with $j_{1}+\cdots+j_{m}\geq 4\mathfrak{p}$. It follows that (13.17) is bounded by a multiple of $N^{-2\mathfrak{p}}$ for all $\ell\in\{1,\ldots,\mathbf{n}-N\}$ since an application of Jensen’s inequality and (13.15) yields

[TABLE]

∎

14 The site-disorder model

The goal of this section is to prove Theorem 3.1. As mentioned in Remark 3.3, the proof involves showing that \widehat{W}_{n}^{\omega}\big{(}\widehat{\beta}_{n,r}\big{)} has a vanishing $L^{2}$ distance from a reduced partition function, \widetilde{W}_{n}^{\omega}\big{(}\widehat{\beta}_{n,r}\big{)}, for which the disorder variables corresponding to vertices of generation less than $\log n$ have been integrated out (Lemma 14.2). Moreover, \widetilde{W}_{n}^{\omega}\big{(}\widehat{\beta}_{n,r}\big{)} is the peak of a $\mathcal{Q}$-pyramidic array of random variables with $\lfloor\log n\rfloor$ layers (Proposition 14.1). Lemmas 14.3 & 14.4 respectively verify the conditions (II) and (III) in Definition 6.12 for the large-$n$ behavior of the variance and higher moments of the random variables in the base layer of the $\mathcal{Q}$-pyramidic array. We can then apply Theorem 6.23 to conclude that \widetilde{W}_{n}^{\omega}\big{(}\widehat{\beta}_{n,r}\big{)}—and consequently also \widehat{W}_{n}^{\omega}\big{(}\widehat{\beta}_{n,r}\big{)}—converges in distribution to $\mathbf{W}_{r}$ as $n\rightarrow\infty$.

14.1 Proof of Theorem 3.1

We will prove Theorem 3.1 after stating the technical lemmas used in its proof. The proofs of the lemmas are placed in the next four subsections.

Recall that $V_{n-1}$ is canonically identifiable with a subset of $V_{n}$ and that under this identification $V_{n}\backslash V_{n-1}$ is referred to as the set of generation-$n$ vertices. Thus, for $k\leq n$, the set $V_{n}\backslash V_{k}$ is all vertices on the diamond graph $D_{n}$ of generation greater than $k$. The elementary proposition below, whose proof is in Section 14.2, states that the conditional expectation of the site-disorder partition function $\widehat{W}_{n}^{\omega}(\beta)$ with respect to the $\sigma$-algebra generated by $\omega_{a}$ for $a\in V_{n}\backslash V_{k}$ can be expressed in terms of the array map $\mathcal{Q}$.

Proposition 14.1**.**

Let $k,n\in\mathbb{N}_{0}$, and assume $k\leq n$. Define the $\sigma$-algebra \mathcal{F}_{n}^{k}:=\sigma\big{\{}\omega_{a}\,\big{|}\,a\in V_{n}\backslash V_{k}\big{\}}. The conditional expectation of $\widehat{W}_{n}^{\omega}(\beta)$ with respect to $\mathcal{F}_{n}^{k}$ can be written in the form

[TABLE]

where \big{\{}X_{h}(\beta)\big{\}}_{h\in E_{k}} is an array of independent copies of $\widehat{W}_{n-k}^{\omega}(\beta)\,-\,1$.

Lemma 14.2 states that the partition function \widehat{W}_{n}^{\omega}\big{(}\widehat{\beta}_{n,r}\big{)} is not changed much by integrating out the disorder variables labeled by vertices of generation less than $\log n$ when $n$ is large. The proof is in Section 14.4.

Lemma 14.2**.**

For fixed $r\in{\mathbb{R}}$, let the sequence $\{\widehat{\beta}_{n,r}\}_{n\in\mathbb{N}}$ have the large $n$ asymptotics (3.3). The $L^{2}$ distance between $\widehat{W}_{n}^{\omega}(\widehat{\beta}_{n,r})$ and \widetilde{W}_{n}^{\omega}(\widehat{\beta}_{n,r}):=\mathbb{E}\big{[}\widehat{W}_{n}^{\omega}(\widehat{\beta}_{n,r})\,\big{|}\,\mathcal{F}_{n}^{\lfloor\log n\rfloor}\big{]} vanishes as $n\rightarrow\infty$.

It follows from Proposition 14.1 and Lemma 14.2 that the $L^{2}$ distance between \widehat{W}_{n}^{\omega}\big{(}\widehat{\beta}_{n,r}\big{)} and 1+\mathcal{Q}^{\lfloor\log n\rfloor}\big{\{}X_{h}(\widehat{\beta}_{n,r})\big{\}}_{h\in E_{\lfloor\log n\rfloor}} converges to zero as $n\rightarrow\infty$, where \big{\{}X_{h}(\widehat{\beta}_{n,r})\big{\}}_{h\in E_{\lfloor\log n\rfloor}} is an array of independent copies of \widehat{W}_{n-\lfloor\log n\rfloor}^{\omega}\big{(}\widehat{\beta}_{n,r}\big{)}-1. The following lemma verifies the variance asymptotics in condition (II) of Definition 6.12—with $n$ replaced by $\lfloor\log n\rfloor$—for the sequence in $n\in\mathbb{N}$ of $\mathcal{Q}$-pyramidic arrays generated from the edge-labeled arrays \big{\{}X_{h}(\widehat{\beta}_{n,r})\big{\}}_{h\in E_{\lfloor\log n\rfloor}}. Our proof, which is in Section 14.3, refines an argument from the proof of [1, Lemma 5.16].

Lemma 14.3**.**

The variance of \widehat{W}_{n-\lfloor\log n\rfloor}^{\omega}\big{(}\widehat{\beta}_{n,r}\big{)} has the large $n$ asymptotics

[TABLE]

Lemma 14.4 verifies the vanishing higher moment condition (III) of Definition 6.12 for random variables in the array \big{\{}X_{h}\big{(}\widehat{\beta}_{n,r}\big{)}\big{\}}_{h\in E_{\lfloor\log n\rfloor}}. The proof is in Section 14.5.

Lemma 14.4**.**

For each $m\in\mathbb{N}$, the $m^{th}$ centered moment of \widehat{W}_{n-\lfloor\log n\rfloor}^{\omega}\big{(}\widehat{\beta}_{n,r}\big{)} vanishes as $n\rightarrow\infty$.

Proof of Theorem 3.1.

For \big{\{}X_{h}\big{(}\widehat{\beta}_{n,r}\big{)}\big{\}}_{h\in E_{\lfloor\log n\rfloor}} defined as in Proposition 14.1, the $L^{2}$ distance between the generation-$n$ vertex-disorder partition function \widehat{W}_{n}^{\omega}\big{(}\widehat{\beta}_{n,r}\big{)} and the effectively generation-$\lfloor\log n\rfloor$ edge-disorder partition function given by

[TABLE]

vanishes with large $n$ by Lemma 14.2, where the second equality above holds by Proposition 14.1. In particular, the Wasserstein-$2$ distance between \widehat{W}_{n}^{\omega}\big{(}\widehat{\beta}_{n,r}\big{)}-1 and \mathcal{Q}^{\lfloor\log n\rfloor}\big{\{}X_{h}\big{(}\widehat{\beta}_{n,r}\big{)}\big{\}}_{h\in E_{\lfloor\log n\rfloor}} vanishes as $n\rightarrow\infty$. Thus it suffices to prove that the Wasserstein-$2$ distance between \mathcal{Q}^{\lfloor\log n\rfloor}\big{\{}X_{h}\big{(}\widehat{\beta}_{n,r}\big{)}\big{\}}_{h\in E_{\lfloor\log n\rfloor}} and $\mathbf{X}_{r}\stackrel{{\scriptstyle d}}{{=}}\mathbf{W}_{r}-1$ converges to zero with large $n$.

Notice that the statements (I)-(III) below hold.

1. (I)

By Proposition 14.1, the random variables in the array \big{\{}X_{h}\big{(}\widehat{\beta}_{n,r}\big{)}\big{\}}_{h\in E_{\lfloor\log n\rfloor}} are independent copies of \widehat{W}_{n-\lfloor\log n\rfloor}^{\omega}\big{(}\widehat{\beta}_{n,r}\big{)}-1. 2. (II)

By Lemma 14.3 the variance of the random variable \widehat{W}_{n-\lfloor\log n\rfloor}^{\omega}\big{(}\widehat{\beta}_{n,r}\big{)} has the large $n$ asymptotics

[TABLE] 3. (III)

By Lemma 14.4, the $m^{th}$ centered moment of \widehat{W}_{n-\lfloor\log n\rfloor}^{\omega}\big{(}\widehat{\beta}_{n,r}\big{)} vanishes as $n\rightarrow\infty$ for each $m\in\{4,6,\ldots\}$.

Statements (I)-(III) imply that the sequence in $n\in\mathbb{N}$ of edge-labeled arrays \big{\{}X_{h}\big{(}\widehat{\beta}_{n,r}\big{)}\big{\}}_{h\in E_{\lfloor\log n\rfloor}} satisfies the conditions (I)-(III) in Definition 6.12. Thus, by Theorem 6.23, the Wasserstein-$2$ distance between $\mathbf{X}_{r}$ and \mathcal{Q}^{\lfloor\log n\rfloor}\big{\{}X_{h}\big{(}\widehat{\beta}_{n,r}\big{)}\big{\}}_{h\in E_{\lfloor\log n\rfloor}} vanishes with large $n$.141414Although the definition of a “regular” sequence of $\mathcal{Q}$-pyramidic arrays formulated in Definition 6.12 assumes that the generation, $\mathfrak{g}_{n}\in\mathbb{N}$, of the bottom layer of the $n^{th}$ $\mathcal{Q}$-pyramidic array is $\mathfrak{g}_{n}=n$, the conclusions of Theorem 6.23 remain valid when $(\mathfrak{g}_{n})_{n\in\mathbb{N}}$ is any sequence that diverges to $\infty$, such as $\mathfrak{g}_{n}=\lfloor\log n\rfloor$. Therefore, \widehat{W}_{n}^{\omega}\big{(}\widehat{\beta}_{n,r}\big{)} converges in law to $\mathbf{W}_{r}$ as $n\rightarrow\infty$. ∎

14.2 Proof of Proposition 14.1

As a preliminary, we will extend our observations and notations relating to the structure of the diamond hierarchical graphs. For $k\leq n$ recall that $V_{n}\backslash V_{k}$ is the set of vertices on the diamond graph $D_{n}$ of generation greater than $k$.

1. (I)

From the construction of the sequence of diamond graphs outlined in Section 2.1, we can see that $D_{n}$ has $b^{2k}$ embedded copies of $D_{n-k}$, which are in canonical one-to-one correspondence with elements of $E_{k}$. The vertices in $V_{k}$—viewed as a subset of $V_{n}$—are roots of the embedded copies of $D_{n-k}$, and the remaining vertices in $V_{n}\backslash V_{k}$ are internal (non root) to the embedded copies of $D_{n-k}$. We denote that set of internal vertices on the copy of $D_{n-k}$ associated with $h\in E_{k}$ by $h\cap V_{n}$.151515This abuse of notation is similar to our previous use of $h\cap E_{n}$ to denote a subset of $E_{n}$. The collection $\{h\cap V_{n}\,|\,h\in E_{k}\}$ is a partition of the set $V_{n}\backslash V_{k}$. 2. (II)

For $h\in E_{k}$, let $\Gamma_{n}^{h}$ denote the set of functions $\mathbf{q}:\{1,\ldots,b^{n-k}\}\rightarrow h\cap E_{n}$ that are directed paths crossing the embedded copy of $D_{n-k}$ corresponding to $h$. Thus each $\Gamma_{n}^{h}$ is a copy of $\Gamma_{n-k}$. 3. (III)

For $a\in V_{n}$ and $\mathbf{q}\in\Gamma_{n}^{h}$, we write $a\boldsymbol{\in}\mathbf{q}$ when $a$ sits internally (non endpoint) along the path $\mathbf{q}$, i.e., when $\mathbf{q}(j)\in h\cap E_{n}$ is incident to $a$ for some $j\in\{2,\dots,b^{n-k}-1\}$. A vertex $a\in V_{n}$ is an element of $V_{n}\backslash V_{k}$ if and only if there is an $h\in E_{k}$ and a $\mathbf{q}\in\Gamma_{n}^{h}$ such that $a\boldsymbol{\in}\mathbf{q}$.161616This is equivalent to the remark in (I) that $a\in V_{n}\backslash V_{k}$ iff $a$ is an internal vertex to one of the subcopies of $D_{n-k}$. 4. (IV)

There is a canonical one-to-one correspondence between $\Gamma_{n}$ and the union of $b^{k}$-fold product sets given by $\bigcup_{q\in\Gamma_{k}}\prod_{\ell=1}^{b^{k}}\Gamma_{n}^{q(\ell)}$. In this association, each $p\in\Gamma_{n}$ has a generation-$k$ coarse-graining $q\in\Gamma_{k}$ and the component $\mathbf{q}_{\ell}\in\Gamma_{n}^{q(\ell)}$ in the $b^{k}$-tuple $(\mathbf{q}_{1},\ldots,\mathbf{q}_{b^{k}})$ is the trajectory of $p$ through the embedded copy of $D_{n-k}$ corresponding to $q(\ell)\in E_{k}$.

The following defines a restricted partition function $\widehat{W}_{n}^{(h)}(\beta)$ for the embedded copy of $D_{n-k}$ within $D_{n}$ that corresponds to $h\in E_{k}$.

Definition 14.5**.**

Let $k,n\in\mathbb{N}_{0}$, and assume $k\leq n$. For $h\in E_{k}$, define the random variable

[TABLE]

where the set $\Gamma_{n}^{h}$ and the relation $\boldsymbol{\in}$ are defined as in (II) and (III) above, respectively.

Remark 14.6**.**

The random variable $\widehat{W}_{n}^{(h)}(\beta)$ in Definition 14.5 is equal in distribution to $\widehat{W}_{n-k}(\beta)$.

Proof of Proposition 14.1.

Taking the conditional expectation of $\widehat{W}_{n}^{\omega}(\beta)$ with respect to $\mathcal{F}_{n}^{k}$ is equivalent to integrating out the variables $\omega_{a}$ with $a\in V_{k}$:

[TABLE]

The last equality is equivalent to what we proved in Proposition 6.5. ∎

14.3 Proof of Lemma 14.3

For $k\in\mathbb{N}_{0}$ and $\beta>0$, let $\hat{\varrho}_{k}(\beta)$ denote the variance of the partition function $\widehat{W}_{k}(\beta)$. As a consequence of the distributional identity (3.2), the sequence of variances \big{\{}\hat{\varrho}_{k}(\beta)\big{\}}_{k\in\mathbb{N}_{0}} satisfies the recursive equation

[TABLE]

where the map $\widehat{M}_{V}:[0,\infty)\rightarrow[0,\infty)$ is defined by

[TABLE]

Of course, $\widehat{M}_{V}$ reduces to the map M(x)=\frac{1}{b}\big{[}(1+x)^{b}-1\big{]} when $V=0$.

The inverse temperature scaling (3.3) results in the following variance scaling:171717A short computation at the end of Appendix A verifies (14.4) starting from (3.3).

[TABLE]

It will be convenient to write $V_{n,r}$ in the form $V_{n,r}=\frac{\widehat{\kappa}^{2}}{\mathbf{n}_{n,r}^{2}}=\frac{b}{b-1}\frac{\pi^{2}\kappa^{2}}{4\mathbf{n}_{n,r}^{2}}$ for \mathbf{n}_{n,r}:=\frac{\pi\kappa}{2}\big{(}\frac{b}{b-1}\big{)}^{1/2}V_{n,r}^{-1/2}, which has the large $n$ asymptotics

[TABLE]

Proof of Lemma 14.3.

We separate the proof into parts (a)-(h).

(a) An approximation for the variance map: Since the variance \hat{\varrho}_{k}\big{(}\widehat{\beta}_{n,r}\big{)} of \widehat{W}_{k}\big{(}\widehat{\beta}_{n,r}\big{)} satisfies the recursive equation (14.2) in $k\in\mathbb{N}_{0}$, we have that

[TABLE]

Let $\widetilde{M}_{n,r}:[0,\infty)\rightarrow[0,\infty)$ be defined through an approximation of the expression for $\widehat{M}_{n,r}(x)$ in (14.3) around $(x,V_{n,r})=(0,0)$ that is third-order in $x$ and first-order in $V_{n,r}$:

[TABLE]

Define $\mathscr{E}(x,\mathbf{n}_{n,r}):=\widehat{M}_{n,r}(x)-\widetilde{M}_{n,r}(x)$, in other terms, the error of the approximation of $\widehat{M}_{n,r}$ by $\widetilde{M}_{n,r}$. The error term has the bound below for some $\mathbf{c}>0$ and all $n\in\mathbb{N}$ and $0\leq x\leq 1$:

[TABLE]

The above inequality follows by foiling the expression (14.3) in $x$ & $V$ and then applying Young’s inequality to the cross-terms, of which the lowest-order cross-term is $xV_{n,r}\propto x/\mathbf{n}_{n,r}^{2}$.

(b) Transforming the variables: For $r\in{\mathbb{R}}$ and $n\in\mathbb{N}$, define the sequence \big{\{}\mathbf{r}_{k}^{(n,r)}\big{\}}_{k\in\mathbb{N}_{0}} of numbers in the interval $[0,1)$ as

[TABLE]

Note that $\mathbf{r}_{0}^{(n,r)}=0$ since $\widehat{M}_{n,r}^{0}(0)=0$. For notational neatness, we will identify $\mathbf{r}_{k}^{(n,r)}\equiv\mathbf{r}_{k}$, i.e., suppress the dependence on the superscript variables. The sequence \big{\{}\mathbf{r}_{k}^{(n,r)}\big{\}}_{k\in\mathbb{N}_{0}} converges monotonically to $1$ as $k\rightarrow\infty$, and it will suffice for us to show that

[TABLE]

To see the equivalence between (14.8) and (14.1), note that for large $n$—and thus small $1-\mathbf{r}_{n-\lfloor\log n\rfloor}$—we get the second equality below through second-order Taylor expansions of f_{1}(x)=\sin\big{(}\frac{\pi}{2}x\big{)} and f_{2}(x)=\cos\big{(}\frac{\pi}{2}x\big{)} at $x=1$:

[TABLE]

Finally, recall from (14.5) that $\mathbf{n}_{n,r}=n+\mathit{O}(\log n)$ for large $n$. Thus we only need to prove (14.8).

(c) Rewriting the increments of $\{\mathbf{r}_{k}\}_{k\in\mathbb{N}_{0}}$ using Taylor’s theorem: By writing \widehat{M}_{n,r}^{k+1}(0)=\widehat{M}_{n,r}\big{(}\widehat{M}_{n,r}^{k}(0)\big{)} and splitting $\widehat{M}_{n,r}$ into a sum of $\widetilde{M}_{n,r}$ and the error term $\mathscr{E}$, we get the equality

[TABLE]

With (14.7), we can rewrite the equation above in terms of the variables $\mathbf{r}_{k}$ and $\mathbf{r}_{k+1}$ as below, where the bracketed expressions have combined to form the $\sec^{2}$ term.

[TABLE]

If $\mathbf{r}_{k}<1-1/\mathbf{n}_{n,r}$, Taylor’s theorem applied to the function g(x)=\tan\big{(}\frac{\pi}{2}x\big{)} around the point $x=\mathbf{r}_{k}$ with second-order error implies there is an $\mathbf{r}_{k}^{*}\in[\mathbf{r}_{k},\mathbf{r}_{k}+1/\mathbf{n}_{n,r})$ such that

[TABLE]

Define $\Delta_{k}$ as the difference between the terms $(\mathbf{II})$ and $(\mathbf{I})$:

[TABLE]

By Taylor’s theorem applied to the function $h(x)=\frac{2}{\pi}\tan^{-1}(x)$ around the point x=\tan\big{(}\frac{\pi}{2}\mathbf{r}_{k+1}\big{)}, there is an $\mathbf{r}_{k}^{**}$ between $\mathbf{r}_{k+1}$ and $\mathbf{r}_{k}+1/\mathbf{n}_{r,n}$ such that

[TABLE]

(d) Bounds for the various terms in (14.11): The inequalities below hold for some $C>0$ and all $k\in\mathbb{N}_{0}$ and $n\in\mathbb{N}$ such that $1-\mathbf{r}_{k}\geq\frac{\log n}{2n}>1/\mathbf{n}_{n,r}$.181818The lower bound of $1-\mathbf{r}_{k}$ by $1/\mathbf{n}_{n,r}$ ensures that $\mathbf{r}_{k}^{*}$ is well-defined by (14.10). When $n$ is sufficiently large, $\frac{\log n}{2n}>1/\mathbf{n}_{n,r}$ holds as a consequence of (14.5).

1. (i)

0\,\leq\,\mathbf{n}_{n,r}\mathscr{E}\left(\frac{\pi\kappa^{2}}{2\mathbf{n}_{n,r}}\tan\big{(}\frac{\pi}{2}\mathbf{r}_{k}\big{)},\mathbf{n}_{n,r}\right)\,\leq\,\frac{C}{n^{3}(1-\mathbf{r}_{k})^{4}}\,+\,\frac{C}{n^{5/3}} 2. (ii)

$|\Delta_{k}|\,\leq\,\frac{C}{n^{2}(1-\mathbf{r}_{k})^{3}}\,+\,\frac{C}{n^{5/3}}$ 3. (iii)

\Big{|}\Delta_{k}-\frac{2\eta}{\pi n^{2}(1-\mathbf{r}_{k})^{3}}\Big{|}\,\leq\,\frac{C}{n^{3}(1-\mathbf{r}_{k})^{4}}\,+\,\frac{C}{n^{5/3}} 4. (iv)

\big{|}\mathbf{r}_{k}+\frac{1}{\mathbf{n}_{n,r}}-\mathbf{r}_{k+1}\big{|}\,\leq\,\frac{C}{n^{2}(1-\mathbf{r}_{k})}\,+\,\frac{C}{n^{5/3}} 5. (v)

\Big{|}\frac{2}{\pi}\Delta_{k}^{2}\sin\big{(}\frac{\pi}{2}\mathbf{r}_{k}^{**}\big{)}\cos^{3}\big{(}\frac{\pi}{2}\mathbf{r}_{k}^{**}\big{)}\Big{|}\,\leq\,\frac{C}{n^{4}(1-\mathbf{r}_{k})^{3}}\,+\,\frac{C}{n^{10/3}} 6. (vi)

\Big{|}\frac{2}{\pi}\Delta_{k}\cos^{2}\big{(}\frac{\pi}{2}\mathbf{r}_{k+1}\big{)}\,-\,\frac{\eta}{n}\log\big{(}\frac{1-\mathbf{r}_{k}}{1-\mathbf{r}_{k+1}}\big{)}\Big{|}\,\leq\,\frac{C}{n^{3}(1-\mathbf{r}_{k})^{2}}\,+\,\frac{C}{n^{5/3}}

The terms $\frac{C}{n^{5/3}}$ above arise from (14.6) and are less important than the first bounding terms. Note that (vi) approximates the second term on the right side of (14.11) by an expression that conveniently telescopes when summed over $k$, and (v) bounds the last term on the right side of (14.11).

The bound (i) follows from (14.6), that $\mathbf{n}_{n,r}\sim n$ for $n\gg 1$ by (14.5), and the estimates below for $0\leq 1-x\ll 1$:

[TABLE]

The bound (ii) follows from (iii), so we will focus on (iii) next. The inequality $\mathbf{r}_{k}^{*}-\mathbf{r}_{k}<1/\mathbf{n}_{n,r}$ and (14.5) imply the equalities below.

[TABLE]

It follows from (14.13) and (14.5) that the difference between $\frac{2\eta}{\pi n^{2}(1-\mathbf{r}_{k})^{3}}$ and the braced expression (III) in part (d) is bounded by

[TABLE]

The last inequality holds by another application of Young’s inequality to get $\frac{\log(n+1)}{n^{3}(1-\mathbf{r}_{k})^{3}}\leq\frac{\log^{4}(n+1)}{4n^{3}}+\frac{3}{4n^{3}(1-\mathbf{r}_{k})^{4}}$ and since $\frac{\log^{4}(n+1)}{4n^{3}}\ll\frac{1}{n^{5/3}}$. Finally, (iii) follows by combining (14.14) with (i).

Note that (iii) implies that $\Delta_{k}$ is positive for all $k$ with $1-\mathbf{r}_{k}\geq\frac{\log n}{2n}$ when $n$ is sufficiently large. Thus (14.3)-(14.11) imply that $\mathbf{r}_{k}\leq\mathbf{r}_{k+1}\leq\mathbf{r}_{k}^{**}\leq\mathbf{r}_{k}+1/\mathbf{n}_{n,r}$. The bound (iv) follows from applying (ii) to (14.11) and using that $\mathbf{r}_{k+1}$ and $\mathbf{r}_{k}^{**}$ are within a distance of $1/\mathbf{n}_{n,r}\sim 1/n$ from $\mathbf{r}_{k}$. The bounds (v) & (vi) follow from (ii) and (iii), respectively, using basic calculus estimates.

(e) A consequence of (iv): Before going to the estimates in part (f) below, we will point out an easy consequence of the bound (iv) in (d): if $\ell\in\mathbb{N}_{0}$ satisfies $1-\mathbf{r}_{\ell}\geq\frac{\log n}{2n}>1/\mathbf{n}_{n,r}$, then the spacing between the terms in the sequence $\{\mathbf{r}_{k}\}_{k=0}^{\ell}$ has the large $n$ form

[TABLE]

where the errors, \mathit{O}\big{(}\frac{1}{n\log^{2}n}\big{)}, are uniformly bounded by a multiple of $\frac{1}{n\log^{2}n}$ for all $0\leq k<\ell$ and $n\gg 1$. The second equality above holds since \mathbf{n}_{n,r}=n+\mathit{O}\big{(}1/\log n\big{)}. A Riemann sum approximation thus gives us

[TABLE]

(f) Applying the bounds to a key telescoping sum: Assume that $\ell\in\mathbb{N}$ satisfies $\ell\leq n$ and that $1-\mathbf{r}_{\ell}\geq\frac{\log n}{2n}>1/\mathbf{n}_{n,r}$ holds so that (14.15) and the inequalities in part (d) are applicable. Since $\mathbf{r}_{0}=0$, the equality below results from a telescoping sum:

[TABLE]

(g) How we can make use of (14.16): We will temporarily assume that $1-\mathbf{r}_{n-\lfloor\log n\rfloor}\geq\frac{\log n}{2n}$ holds for sufficiently large $n$ to show that the asymptotics (14.8) follows. If $1-\mathbf{r}_{n-\lfloor\log n\rfloor}\geq\frac{\log n}{2n}$, then the equality (14.16) holds with $\ell=n-\lfloor\log n\rfloor$, which gives us

[TABLE]

Note that (14.8) holds provided that the bracketed term is $\mathit{o}(1/n)$ for $n\gg 1$. Since $1-\mathbf{r}_{n-\lfloor\log n\rfloor}\geq\frac{\log n}{2n}$, we can get an upper bound for $1\,-\,\mathbf{r}_{n-\lfloor\log n\rfloor}$ by substituting $\frac{\log n}{2n}$ in place of $1-\mathbf{r}_{n-\lfloor\log n\rfloor}$ on the right side of (14.17):

[TABLE]

Thus $1\,-\,\mathbf{r}_{n-\lfloor\log n\rfloor}$ is bounded from above and below by constant multiples of $\frac{\log n}{n}$ for $n\gg 1$. It follows that the bracketed term in (14.17) is $\mathit{O}(1/n)$, and hence we can conclude from (14.17) that 1-\mathbf{r}_{n-\lfloor\log n\rfloor}=\frac{\log n}{n}\big{(}1+\mathit{o}(1)\big{)}. Plugging this asymptotics for $1-\mathbf{r}_{n-\lfloor\log n\rfloor}$ back into the right side of (14.17), however, yields that the bracketed term in (14.17) is \mathit{o}\big{(}1/n), which proves (14.8) under the assumption that $1-\mathbf{r}_{n-\lfloor\log n\rfloor}\geq\frac{\log n}{2n}$.

(h) Establishing the validity of (14.16) when $\ell=n-\lfloor\log n\rfloor$: It remains to show that $1-\mathbf{r}_{n-\lfloor\log n\rfloor}\geq\frac{\log n}{2n}$ holds for large enough $n$. Let $\ell^{*}\equiv\ell^{*}(n,r)$ be the smallest $\ell^{*}\in\mathbb{N}$ such that

[TABLE]

Since $1-\mathbf{r}_{\ell^{*}-1}>\frac{3\log n}{4n}$ and $\mathbf{r}_{\ell^{*}}-\mathbf{r}_{\ell^{*}-1}=\frac{1}{n}+\mathit{o}(\frac{1}{n})$ by (iv) in part (d), we have the inequality $1-\mathbf{r}_{\ell^{*}}\geq\frac{\log n}{2n}$ for large enough $n$. Thus the equality (14.16) will hold with $\ell=\ell^{*}$ when $n\gg 1$:

[TABLE]

Since $1\,-\,\mathbf{r}_{\ell^{*}}$ is bounded by $\frac{3\log n}{4n}$ and the braced term on the right side of (14.3) is greater than $\frac{3\log n}{4n}$ for large $n$, the first term on the right side of (14.3) must be negative when $n\gg 1$, and therefore $\ell^{*}>n-\lfloor\log n\rfloor$. It follows that $1-\mathbf{r}_{n-\lfloor\log n\rfloor}\geq\frac{\log n}{2n}$ for large $n$. ∎

14.4 Proof of Lemma 14.2

Since the random variables \mathbb{E}\big{[}\widehat{W}_{n}^{\omega}\big{(}\widehat{\beta}_{n,r}\big{)}\,\big{|}\,\mathcal{F}_{n}^{\lfloor\log n\rfloor}\big{]} and \widehat{W}_{n}^{\omega}\big{(}\widehat{\beta}_{n,r}\big{)}-\mathbb{E}\big{[}\widehat{W}_{n}^{\omega}\big{(}\widehat{\beta}_{n,r}\big{)}\,\big{|}\,\mathcal{F}_{n}^{\lfloor\log n\rfloor}\big{]} are uncorrelated, the square of the $L^{2}$ distance between \widehat{W}_{n}^{\omega}\big{(}\widehat{\beta}_{n,r}\big{)} and \mathbb{E}\big{[}\widehat{W}_{n}^{\omega}\big{(}\widehat{\beta}_{n,r}\big{)}\,\big{|}\,\mathcal{F}_{n}^{\lfloor\log n\rfloor}\big{]} is equal to

[TABLE]

where the random variables $X_{h}^{n,r}$ are independent copies of \widehat{W}_{n-\lfloor\log n\rfloor}^{\omega}\big{(}\widehat{\beta}_{n,r}\big{)}. The equalities above use (14.2), Proposition 14.1, and (i) of Remark 6.6. It follows that Lemma 14.2 is a corollary of the following:

Lemma 14.7**.**

The difference between $\widehat{M}_{n,r}^{n}(0)$ and M^{\lfloor\log n\rfloor}\big{(}\widehat{M}_{n,r}^{n-\lfloor\log n\rfloor}(0)\big{)} vanishes as $n\rightarrow\infty$.

Remark 14.8**.**

Note that M^{\lfloor\log n\rfloor}\big{(}\widehat{M}_{n,r}^{n-\lfloor\log n\rfloor}(0)\big{)} converges to $R(r)$ as $n\rightarrow\infty$. This follows from Lemma 2.3 since $\widehat{M}_{n,r}^{n-\lfloor\log n\rfloor}(0)$, which is equal to the variance of \widehat{W}_{n-\lfloor\log n\rfloor}\big{(}\widehat{\beta}_{n,r}\big{)}, has the large-$n$ asymptotics (14.1) by Lemma 14.3.

In the proof of Lemma 14.7, we will use Lemma 14.9 below, which is a result from [10, Lemma 2.2(iv)]. Notice that applying the chain rule to the $k$-fold composition of M(x)=\frac{1}{b}\big{[}(1+x)^{b}-1\big{]} yields

[TABLE]

where the function $D_{k}:[0,\infty)\rightarrow[0,\infty)$ is defined by

[TABLE]

In the above, $M^{-\ell}$ denotes the $\ell$-fold composition of the function inverse of the map $M$. The following lemma gives us uniform bounds for the sequence in $k\in\mathbb{N}_{0}$ of functions $D_{k}$.

Lemma 14.9**.**

The sequence of functions $\{D_{k}\}_{k\in\mathbb{N}_{0}}$ converges uniformly over any bounded subinterval of $[0,\infty)$ to a limit function $D$. In particular, $F(L):=\sup_{k\in\mathbb{N}_{0}}\sup_{x\in[0,L]}D_{k}(x)$ is finite for any $L>0$.

Proof of Lemma 14.7.

Define $A_{n,r}:=\widehat{M}_{n,r}^{n-\lfloor\log n\rfloor}(0)$. By Remark 14.8, $M^{\lfloor\log n\rfloor}(A_{n,r})$ converges to $R(r)$ as $n\rightarrow\infty$. For any $\ell\in\{0,\ldots,\lfloor\log n\rfloor\}$, the definition of $A_{n,r}$ implies that

[TABLE]

We will return to (14.23) after obtaining bounds for the terms (I) and (II).

Bound for (I): The difference between the functions $\widehat{M}_{n,r}$ and $M$ has the bound,

[TABLE]

where the inequality holds for large enough $n$ since $V_{n,r}$ is vanishing. Thus for large $n$

[TABLE]

Bound for (II): By the chain rule, the derivative of $\widehat{M}_{n,r}^{k}$ can be written in the form

[TABLE]

where the equality uses the definition (14.21) of the function $D_{k}:[0,\infty)\rightarrow[0,\infty)$. An application of (14.26) to the term $\mathbf{(b)}$ gives us

[TABLE]

where the second inequality again uses that $\widehat{M}_{n,r}(x)\geq M(x)$ for all $x\geq 0$.

Returning to (14.23): Applying (14.25) and (14.27) to (14.23) gives us the first inequality below for all $0\leq\ell\leq\lfloor\log n\rfloor$ when $n$ is large enough.

[TABLE]

Let $\ell^{*}_{n,r}\in\mathbb{N}$ be the minimum of $\ell=\lfloor\log n\rfloor$ and the largest $\ell$ such that $\widehat{M}_{n,r}^{\ell}(A_{n,r})\leq 2M^{\ell}(A_{n,r})$. Applying (14.28) with $\ell=\ell^{*}_{n,r}$ yields

[TABLE]

We can apply (14.29) to get the inequality below

[TABLE]

However, since $M^{\ell^{*}_{n,r}+1}(A_{n,r})\geq A_{n,r}\geq\frac{\kappa^{2}}{\lfloor\log n\rfloor}$ for large $n$, the inequality (14.30) precludes the possibility that $\widehat{M}_{n,r}^{\ell^{*}_{n,r}+1}(A_{n,r})>2M^{\ell^{*}_{n,r}+1}(A_{n,r})$ when $n$ is large. It follows from the definition of $\ell^{*}_{n,r}$ that $\ell^{*}_{n,r}:=\lfloor\log n\rfloor$, and thus (14.29) implies that the difference between $\widehat{M}_{n,r}^{\lfloor\log n\rfloor}(A_{n,r})$ and $M^{\lfloor\log n\rfloor}(A_{n,r})$ vanishes with large $n$. ∎

14.5 Proof of Lemma 14.4

Proof.

It suffices to show that the (uncentered) positive integer moments of \widehat{W}_{n-\lfloor\log n\rfloor}\big{(}\widehat{\beta}_{n,r}\big{)} all converge to one as $n\rightarrow\infty$. For $m\in\{2,3,\ldots\}$, $n\in\mathbb{N}$, $r\in{\mathbb{R}}$, and $k\in\mathbb{N}_{0}$ define

[TABLE]

Note that $\mu_{n,r}^{(m)}(0)=1$ since $\widehat{W}_{0}(\widehat{\beta}_{n,r})=1$ by definition, and $\mu_{n,r}^{(m)}(k),\nu_{n,r}^{(m)}\geq 1$ by Jensen’s inequality. We obtain the following recursive equation in $k\in\mathbb{N}$ by evaluating the $m^{th}$ moment of both sides of the distributional equality (3.2):

[TABLE]

where $\mathbf{P}_{m}(y_{2},\ldots,y_{m-1})$ is a polynomial with nonnegative coefficients that sum to $1-1/b^{m-1}$. In particular, $\mathbf{P}_{m}(y_{2},\ldots,y_{m-1})=1-1/b^{m-1}$ when evaluated at $(y_{2},\ldots,y_{m-1})=(1,\ldots,1)$. Moreover, $1-1/b^{m-1}$ is a lower bound for \mathbf{P}_{m}\Big{(}\big{(}\mu_{n,r}^{(\ell)}(k)\big{)}^{b}\big{(}\nu_{n,r}^{(\ell)}\big{)}^{b}\,;\,\,\ell\in\{2,\ldots,m-1\}\Big{)} since $\mu_{n,r}^{(m)}(k),\nu_{n,r}^{(m)}\geq 1$.

We will use induction to prove that \max_{0\leq k\leq n-\lfloor\log n\rfloor}\big{|}\mu_{n,r}^{(m)}(k)-1\big{|} vanishes as $n\rightarrow\infty$ for each $m\in\{2,3,\ldots\}$. As a consequence of Lemma 14.3, \mu_{n,r}^{(2)}\big{(}n-\lfloor\log n\rfloor\big{)} converges to one as $n\rightarrow\infty$. Since \big{\{}\mu_{n,r}^{(2)}(k)\big{\}}_{k\in\mathbb{N}_{0}} is an increasing sequence and $\mu_{n,r}^{(2)}(0)=1$, it follows that \max_{0\leq k\leq n-\lfloor\log n\rfloor}\big{|}\mu_{n,r}^{(2)}(k)-1\big{|} vanishes as $n\rightarrow\infty$. Suppose for the purpose of a strong induction argument that

[TABLE]

for each $\ell\in\{2,\ldots,m\}$. Note that $\nu_{n,r}^{(\ell)}$ converges to one as $n\rightarrow\infty$ for each $\ell\in\mathbb{N}$ since $\widehat{\beta}_{n,r}$ vanishes with large $n$. Fix some $\epsilon\in(0,1)$. Since $\mathbf{P}_{m+1}$ is continuous and $\mathbf{P}_{m+1}(1,\ldots,1)=1-1/b^{m}$, we can choose $n\in\mathbb{N}$ large enough such that

[TABLE]

Let $k^{*}_{n,r,\epsilon}$ be the minimum of $k=n-\lfloor\log n\rfloor$ and the smallest $k\in\mathbb{N}$ such that

[TABLE]

By (14.31) and the definition of $k^{*}_{n,r,\epsilon}$, we have the recursive inequality in $k\in\{0,\ldots,k^{*}_{n,r,\epsilon}-1\}$ below.

[TABLE]

Applying (14.35) $k^{*}_{n,r,\epsilon}$ times and using that $\mu_{n,r}^{(m+1)}(0)=1$ yields

[TABLE]

The bracketed term converges to $1-1/b^{m}$ as $n\rightarrow\infty$ by the same reasoning as for (14.33). We will show that $k^{*}_{n,r,\epsilon}=n-\lfloor\log n\rfloor$ holds for large enough $n$ by showing that the condition (14.34) cannot hold for $k\leq n-\lfloor\log n\rfloor$ when $n\gg 1$. Notice that

[TABLE]

Moreover, since $m\geq 2$, the following inequality holds for small $\epsilon>0$:

[TABLE]

Thus $k^{*}_{n,r,\epsilon}$ does not satisfy (14.34) when $n$ is large, and therefore $k^{*}_{n,r,\epsilon}=n-\lfloor\log n\rfloor$ for large $n$. Going back to (14.36) with $k^{*}_{n,r,\epsilon}=n-\lfloor\log n\rfloor$, we get

[TABLE]

Since $\epsilon>0$ is arbitrarily and $\mu_{n,r}^{(m+1)}(k)\geq 1$, the sequence \big{\{}\max_{0\leq k\leq n-\lfloor\log n\rfloor}\big{|}\mu_{n,r}^{(m+1)}(k)-1\big{|}\big{\}}_{n\in\mathbb{N}} is vanishing. Therefore, by induction, \max_{0\leq k\leq n-\lfloor\log n\rfloor}\big{|}\mu_{n,r}^{(m)}(k)-1\big{|} converges to zero for each $m\in\{2,3,\ldots\}$, which completes the proof. ∎

15 Miscellaneous proofs from Sections 12 & 13

15.1 Proof of Proposition 12.1

To prepare for the proof of Proposition 12.1, we will define some additional notation related to the recursive formulas governing the positive integer moments of random variables in a $\mathcal{Q}$-pyramidic array generated from an i.i.d. array of random variables and cite a bound (Lemma 15.7) from [10].

Let \big{\{}X_{h}^{(n)}\big{\}}_{h\in E_{n}} be an i.i.d. array of centered random variables with finite $m^{th}$ absolute moment for some $m\in\{2,3,\ldots\}$ and $\{X^{(*,n)}_{a}\}_{a\in E_{*}}$ be the $\mathcal{Q}$-pyramidic array generated from it. For $k\in\{0,\ldots,n\}$ and $a\in E_{k}$, we will use the notation

[TABLE]

and condense subscripts when $k=n$ as follows: $\sigma^{(m)}_{n,n}\equiv\sigma^{(m)}_{n}$. For $m=2$ note that $\sigma^{(2)}_{k,n}$ is interchangeable with our previous notation $\sigma^{2}_{k,n}$ from (11.1). By (i) of Remark 6.6, the recursive relation $\{X^{(k-1,n)}_{a}\}_{a\in E_{k-1}}:=\mathcal{Q}\{X^{(k,n)}_{a}\}_{a\in E_{k}}$ implies that $M(\sigma^{2}_{k,n})=\sigma^{2}_{k-1,n}$ for the polynomial $M(x)=\frac{1}{b}[(1+x)^{b}-1]$. More generally, the multilinear form of the map $\mathcal{Q}$ implies that the vector of higher moments \big{(}\sigma^{(3)}_{k,n},\ldots,\sigma^{(m)}_{k,n}\big{)} obeys a recursive equation with $\sigma^{(2)}_{k,n}$ as an additional input:

[TABLE]

where $\vec{P}_{m}:{\mathbb{R}}^{m-1}\rightarrow{\mathbb{R}}^{m-2}$ is a vector of polynomials $P_{j}:{\mathbb{R}}^{j-1}\rightarrow{\mathbb{R}}$:191919The polynomials $P_{j}$ are the same as those in (I) of Theorem 2.4.

[TABLE]

In the above, the variables $y_{j}$ are indexed according to the number $j$ of the moment, $\sigma^{(j)}_{k,n}$, that they correspond to. The polynomials $P_{j}$ have nonnegative coefficients and are thus nondecreasing in each variable $y_{i}$ for $i\in\{2,\ldots,j\}$ on the subdomain $[0,\infty)^{j-1}$; see Lemma 15.13 for some additional properties of these polynomials.

Let $\vec{H}_{m}:(0,\infty)\rightarrow{\mathbb{R}}^{m-2}$ be defined as below for the limiting moment functions $R^{(j)}:{\mathbb{R}}\rightarrow[0,\infty)$ from Theorem 2.4:

[TABLE]

where $x>0$ and $R^{-1}$ is the inverse of the variance function $R\equiv R^{(2)}$. In other terms, $\vec{H}_{m}$ determines the vector of limiting higher moments with $3\leq j\leq m$ from the variance $x$. In Definition 15.1, we use the functions $\vec{P}_{m}$ and $M$ to construct functions $\vec{H}^{(k)}_{m}(x;y)$ from $(0,\infty)\times{\mathbb{R}}^{m-2}$ to ${\mathbb{R}}^{m-2}$ that converge pointwise with large $k\in\mathbb{N}$ to $\vec{H}_{m}(x)$ when $y$ has small enough norm by [10, Lemma 3.2]. For the purpose of proving Proposition 12.1, the relevant properties of the functions $\vec{H}^{(k)}_{m}$ are the identities in Remarks 15.2 & 15.3 below and the bound on their derivatives in Lemma 15.7. As before, $M^{-k}$ denotes the $k$-fold composition of the function inverse of $M$.

Definition 15.1**.**

For $m\in\{3,4,\ldots\}$, let $\vec{P}_{m}:{\mathbb{R}}^{m-1}\rightarrow{\mathbb{R}}^{m-2}$ be the vector of polynomials determined by (15.2). Given $x>0$ and $k\in\mathbb{N}$, define $F^{(x)}_{k}:{\mathbb{R}}^{m-2}\rightarrow{\mathbb{R}}^{m-2}$ such that for $y=(y_{3},\ldots,y_{m})\in{\mathbb{R}}^{m-2}$

[TABLE]

Define $\vec{H}^{(k)}_{m}:(0,\infty)\times{\mathbb{R}}^{m-2}\rightarrow{\mathbb{R}}^{m-2}$ through the $k$-fold composition of the maps $F^{(x)}_{j}$ given by

[TABLE]

We denote the $(m-2)$-by-$(m-2)$ matrix of first-order derivatives of $\vec{H}^{(k)}_{m}(x;y)$ with respect to the variables $y_{j}$ for $j\in\{3,\ldots,m\}$ by $\mathbf{D}\vec{H}^{(k)}_{m}(x;y)$.

Remark 15.2**.**

Let $n\in\mathbb{N}$ and $k\in\{0,\ldots,n\}$. Since $\sigma^{2}_{k,n}:=M^{n-k}(\sigma^{2}_{n})$, the recursive relation (15.2) implies the identity

[TABLE]

Remark 15.3**.**

Note that R(r-k)=M^{n-k}\big{(}R(r-n)\big{)} by part (I) of Lemma 2.3. Hence part (I) of Theorem 2.4 implies that

[TABLE]

We will use the following simple vector notation.

Notation 15.4**.**

For $d\in\mathbb{N}$ let $y=(y_{1},\ldots,y_{d})$ and $y^{\prime}=(y_{1}^{\prime},\ldots,y_{d}^{\prime})$ be elements of ${\mathbb{R}}^{d}$ and $A$ be a $d\times d$ real-valued matrix.

1. (i)

We write $y\leq y^{\prime}$ if the inequality holds component-wise, i.e., $y_{j}\leq y_{j}^{\prime}$ for all $j\in\{1,\ldots,d\}$. 2. (ii)

$\|y\|_{\infty}$* denotes the max norm of $y$, i.e., $\|y\|_{\infty}=\max_{1\leq j\leq d}|y_{j}|$.* 3. (iii)

$\|A\|$* is the operator norm with respect to the max norm on ${\mathbb{R}}^{d}$, i.e., $\|A\|=\max_{1\leq i\leq d}\sum_{j=1}^{d}|A_{i,j}|$.*

Remark 15.5**.**

In the sense of (i) in Notation 15.4, we will refer to a function $f:{\mathbb{R}}^{d}\rightarrow{\mathbb{R}}^{d}$ as being nondecreasing on a subdomain $D\subset{\mathbb{R}}^{d}$ if $f(y_{1})\leq f(y_{2})$ holds for all $y_{1},y_{2}\in D$ with $y_{1}\leq y_{2}$.

Remark 15.6**.**

Since the polynomials $P_{j}$ have nonnegative coefficients, $\vec{P}_{m}$ is nondecreasing on $[0,\infty)^{m-1}$. Since $M$ is increasing, it follows from the construction in Definition 15.1 that $\vec{H}^{(k)}_{m}(x;y)$ is also nondecreasing on $[0,\infty)^{m-1}$.

The lemma below from [10, Eqn. 3.8] implies that the function $\vec{H}^{(k)}_{m}(x;y)$ is essentially independent of $y\in{\mathbb{R}}^{m-2}$ when $k\gg 1$ and $(x;y)\in(0,\infty)\times{\mathbb{R}}^{m-2}$ is restricted to a small region around the origin.

Lemma 15.7**.**

For any $m\in\{3,4,\ldots\}$, there is an $\epsilon\equiv\epsilon(m)>0$ such that for all $k\in\mathbb{N}$:

[TABLE]

Proof of Proposition 12.1.

Part (i): Pick any $r_{\downarrow},r_{\uparrow}\in{\mathbb{R}}$ with $r_{\downarrow}<r<r_{\uparrow}$. Since \sigma_{n}^{2}=\textup{Var}\big{(}X_{h}^{(n)}\big{)} has the large $n$ asymptotics (6.2) and $R(s)$ has the asymptotics in (II) of Lemma 2.3 as $s\rightarrow-\infty$, we have the following inequality for all $n$ larger than some $\widetilde{n}>0$

[TABLE]

Thus for any $n>\widetilde{n}$ and $k\in\{0,\ldots,n\}$ the relations below hold:

[TABLE]

where we have used (I) of Lemma 2.3, the definition \sigma_{k,n}^{2}:=M^{n-k}\big{(}\sigma_{n}^{2}\big{)}, and that $M$ is increasing. Since $R(s)\sim\frac{\kappa^{2}}{-s}$ for $-s\gg 1$ and $R$ takes values in $(0,\infty)$, the terms $R(r_{\uparrow}-k)$ and $R(r_{\downarrow}-k)$ are respectively bounded from above and below by positive multiples, $c_{\uparrow}$ and $c_{\downarrow}$, of $\frac{1}{k+1}$. Thus we have $\frac{c_{\downarrow}}{k+1}<\sigma_{k,n}^{2}<\frac{c_{\uparrow}}{k+1}$ for all $n>\widetilde{n}$ and $k\in\{0,\ldots,n\}$. Since there are only finitely many $k,n\in\mathbb{N}_{0}$ with $0\leq k\leq n\leq\widetilde{n}$, the inequalities $\frac{c_{\downarrow}}{k+1}<\sigma_{k,n}^{2}<\frac{c_{\uparrow}}{k+1}$ can be extended to all $k,n$ by choosing the constants $c_{\uparrow},c_{\downarrow}$ to be larger/smaller if needed.

Part (ii): Let $\epsilon\in(0,1/2)$ be small enough to satisfy the conclusion of Lemma 15.7 with $m=4$, and fix any $r^{\uparrow}\in(r,\infty)$. Let $\widetilde{n}\in\mathbb{N}$ be large enough such that statements (a)-(c) below hold for all $n\in\mathbb{N}$ with $n>\widetilde{n}$.

1. (a)

$\sigma^{(2)}_{k,n}\leq R(r^{\uparrow}-k)$ for all $k\in\{0,\ldots,n\}$, 2. (b)

\max_{m\in\{3,4\}}\big{|}\sigma^{(m)}_{n}\big{|}<\epsilon, and 3. (c)

$\max_{m\in\{2,3,4\}}R^{(m)}(r^{\uparrow}-n)<\epsilon$.

To see that $\widetilde{n}$ exists, notice the following: statement (a) holds for large $n$ by the reasoning leading to (15.4); statement (b) holds for large enough $n$ as a consequence of our minimal regularity assumption that the fourth moments $\sigma^{(4)}_{n}$ vanish as $n\rightarrow\infty$; statement (c) holds for large enough $n$ since $R^{(m)}(s)$ vanishes as $s\searrow-\infty$ for each $m\in\{2,3,4\}$ by (II) of Theorem 2.4; .

Since there are only finitely many terms $\sigma^{(4)}_{k,n}$ with $n\leq\widetilde{n}$, we can focus on the case that $n>\widetilde{n}$. For $n>\widetilde{n}$, let $k^{*}$ be the smallest element of $\{0,\ldots,n\}$ such that $R^{(4)}(r^{\uparrow}-k^{*})<\epsilon$. Note that $k^{*}$ much exist as a consequence of (c). Since $\sigma^{(4)}_{k,n}$ converges to $R^{(4)}(r-k)$ as $n\rightarrow\infty$ for each $k\in\mathbb{N}$ by (III) of Lemma 6.15, the following is finite:

[TABLE]

Thus it suffices for us to assume that $k>k^{*}$ in the remainder of the proof.

Let $n>\widetilde{n}$ and $k\in\{k^{*},\ldots,\lfloor n/2\rfloor\}$. The equality below is the $m=4$ case of the identity in Remark 15.2.

[TABLE]

The inequality above holds by statement (a) and Remark 15.6. Since statements (b) and (c) imply that $R(r^{\uparrow}-k)<\epsilon$, \big{\|}\big{(}\big{|}\sigma^{(3)}_{n}\big{|},\sigma^{(4)}_{n}\big{)}\big{\|}_{\infty}<\epsilon, and \big{\|}\big{(}R^{(3)}(r^{\uparrow}-n),R^{(4)}(r^{\uparrow}-n)\big{)}\big{\|}_{\infty}<\epsilon, we can apply Lemma 15.7 to get the first inequality below.

[TABLE]

By Remark 15.3, the bracketed term is equal to \Big{(}R^{(3)}(r^{\uparrow}-k),R^{(4)}(r^{\uparrow}-k)\Big{)}, and thus the inequality (15.7) implies that

[TABLE]

where $(1,1)$ refers to the vector in ${\mathbb{R}}^{2}$. Combining the vector inequalities (15.6) and (15.8) yields the following for the second components of the vectors:

[TABLE]

For the second inequality, we have used that $\epsilon<1/2$ and that $k\leq\lfloor n/2\rfloor$. Since $R^{(4)}(s)$ is \mathit{O}\big{(}\frac{1}{s^{2}}\big{)} for $-s\gg 1$ by part (II) of Theorem 2.4, $R^{(4)}(r^{\uparrow}-k)$ is bounded by a constant multiple of $\frac{1}{(k+1)^{2}}$ for all $k\in\mathbb{N}_{0}$. Also, \big{(}\frac{b+1}{2b}\big{)}^{n/2}, which decays exponentially in $n$, is bounded from above by a multiple of $\frac{1}{(k+1)^{2}}$ for all $k\leq n$. Thus we have the desired inequality when $n>\widetilde{n}$ and $k\in\{k^{*},\ldots,\lfloor n/2\rfloor\}$, which completes the proof. ∎

15.2 Proof of Lemma 12.2

Let $\{x_{a}\}_{a\in E_{n}}$ be an array of centered random variables with finite fourth moments, and define $Y_{\ell}:=\mathcal{L}^{\ell-1}\mathcal{E}\mathcal{L}^{n-\ell}\{x_{a}\}_{a\in E_{n}}$ for $\ell\in\{1,\ldots,n\}$. Recall that Lemma 12.2 states that \mathbb{E}\big{[}\big{(}\sum_{\ell=1}^{n}Y_{\ell}\big{)}^{4}\big{]} is bounded by a constant multiple of n\sum_{\ell=1}^{n}\mathbb{E}\big{[}Y_{\ell}^{4}\big{]}.

Notation 15.8**.**

For distinct $a_{1},a_{2}\in E_{n}$, let $\gamma(a_{1},a_{2})$ denote the smallest value of $k\in\{1,\ldots,n\}$ such that there exist distinct $\mathbf{b}_{1},\mathbf{b}_{2}\in E_{k}$ with $a_{1}\in\mathbf{b}_{1}$ and $a_{2}\in\mathbf{b}_{2}$. When $a_{1}=a_{2}$, we define $\gamma(a_{1},a_{2})=\infty$.

Remark 15.9**.**

Let $S\subset E_{n}$ and $\mathbf{b}\in E_{k}$ for $1\leq k<n$. If $\mathbf{b}\cap S\neq\emptyset$ and $\gamma(a_{1},a_{2})>k$ for all $a_{1},a_{2}\in S$, then $S\subset\mathbf{b}$.

Remark 15.10**.**

Let $S_{1},S_{2}\subset E_{n}$. If $\gamma(a_{1},a_{2})$ is independent of $a_{1}\in S_{1}$ and $a_{2}\in S_{2}$, then we define $\gamma(S_{1},S_{2}):=\gamma(a_{1},a_{2})$ for $a_{1}\in S_{1}$ and $a_{2}\in S_{2}$.

Proof of Lemma 12.2.

Let $\sigma^{2}$ denote the variance of the variables $x_{a}$, $a\in E_{n}$. By foiling, we get

[TABLE]

By applying Young’s inequality, $|xy|\leq\frac{1}{p}|x|^{p}+\frac{1}{q}|x|^{q}$, to the bracketed products above with $(p,q)=(\frac{4}{3},4)$ and $(p,q)=(2,2)$, respectively, we can bound the second and third terms on the right side of (15.2) by multiples of n\sum_{\ell=1}^{n}\mathbb{E}\big{[}Y_{\ell}^{4}\big{]}. In the analysis below, we will show that \mathbb{E}\big{[}Y_{\ell}Y_{l_{1}}Y_{l_{2}}Y_{l_{3}}\big{]}=0 when $\ell<l_{1},l_{2},l_{3}$ and thus that the last term on the right side of (15.2) is zero. We will also show that \mathbb{E}\big{[}Y_{\ell}^{2}Y_{l_{1}}Y_{l_{2}}\big{]} is bounded by a constant multiple of $(\sigma^{2})^{4}b^{-(l_{1}+l_{2})}$ for all $\ell,l_{1},l_{2}\in\mathbb{N}$ with $\ell<l_{1}<l_{2}$, which implies that there are $C,C^{\prime}>0$ such that the inequalities below hold for all $n\in\mathbb{N}$.

[TABLE]

The third inequality holds since M(x):=\frac{1}{b}\big{[}(1+x)^{b}-1\big{]}\geq x+\frac{b-1}{2}x^{2} for $x\geq 0$ and $b\geq 2$. The equality holds by Remark 6.6 since $Y_{1}:=\mathcal{E}\mathcal{L}^{n-1}\{x_{a}\}_{a\in E_{n}}$, and the last inequality is Jensen’s. It follows that the fourth term on the right side of (15.2) is easily bounded by a constant multiple of n\sum_{\ell=1}^{n}\mathbb{E}\big{[}Y_{\ell}^{4}\big{]}.

For $1\leq\ell\leq n$ and $\mathbf{a}\in E_{\ell}$, define $x_{\mathbf{a}}^{(\ell)}:=\mathcal{L}^{n-\ell}\{x_{a}\}_{a\in\mathbf{a}\cap E_{n}}=\frac{1}{b^{n-\ell}}\sum_{a\in\mathbf{a}\cap E_{n}}x_{a}$. The random variable $Y_{\ell}$ can be written in the forms

[TABLE]

From (15.2) we see that $Y_{\ell}$ is a degree-$b$ multilinear polynomial in the variables $\{x_{a}\}_{a\in E_{n}}$ consisting of a linear combination of monomials $\prod_{a\in B}x_{a}$ for subsets $B$ of $E_{n}$ satisfying

1. (I)

$|B|\geq 2$ and 2. (II)

$\gamma(a_{1},a_{2})=\ell$ for any distinct $a_{1},a_{2}\in B$.

For numbers $k_{\epsilon}\in\{1,\ldots,n\}$ indexed by $\epsilon\in\{1,2,3,4\}$, let $B_{\epsilon}$ be a subset of $E_{n}$ satisfying (I)-(II) for $\ell=k_{\epsilon}$. The product of the monomials $\prod_{a\in B_{\epsilon}}x_{a}$ can be written as

[TABLE]

where the exponent $\lambda(a)\in\{1,2,3,4\}$ is defined by \lambda(a):=\big{|}\big{\{}j\in\{1,2,3,4\}\,\big{|}\,a\in B_{j}\big{\}}\big{|}. The expectation of (15.11) is zero if $\lambda(a)=1$ for some $a\in\cup_{\epsilon}B_{\epsilon}$. The first case below implies \mathbb{E}\big{[}Y_{\ell}Y_{l_{1}}Y_{l_{2}}Y_{l_{3}}]=0 when $\ell<l_{1},l_{2},l_{3}$.

Case $\mathbf{k_{1}<k_{2},\,k_{3},\,k_{4}}$: To reach a contradiction, suppose that $k_{1}<k_{2},\,k_{3},\,k_{4}$ and $\lambda(a)\geq 2$ for all $a\in\cup B_{\epsilon}$. Since $B_{1}$ satisfies properties (I)-(II) with $\ell=k_{1}$, there must be distinct $a_{1},a_{2}\in B_{1}$ and distinct $\mathbf{b}_{1},\mathbf{b}_{2}\in E_{k_{1}}$ such that $a_{1}\in\mathbf{b}_{1}$ and $a_{2}\in\mathbf{b}_{2}$. By our assumption that $k_{1}<k_{\epsilon}$ for $\epsilon\in\{2,3,4\}$, property (II) for $B_{\epsilon}$ implies that for each $\epsilon\in\{2,3,4\}$ we have $\mathbf{b}_{1}\cap B_{\epsilon}=\emptyset$ or $\mathbf{b}_{2}\cap B_{\epsilon}=\emptyset$ (since otherwise there exist distinct $c_{1},c_{2}\in B_{\epsilon}$ with $\gamma(c_{1},c_{2})<k_{\epsilon}$). In particular, $\mathbf{b}_{1}$ or $\mathbf{b}_{2}$ is disjoint from the sets $B_{\epsilon}$ for at least two values of $\epsilon\in\{2,3,4\}$. Without loss of generality, we can assume that $\mathbf{b}_{1}\cap(B_{3}\cup B_{4})=\emptyset$ and consequently that $a_{1}\notin B_{3}\cup B_{4}$. Since $a_{1}\notin B_{3}$ and $a_{1}\notin B_{4}$, we must have $a_{1}\in B_{2}$ to ensure that $\lambda(a_{1})\geq 2$. Thus $\mathbf{b}_{1}\cap B_{2}\neq\emptyset$. Note that $B_{2}\subset\mathbf{b}_{1}$, by Remark 15.9, because $\mathbf{b}_{1}\cap B_{2}\neq\emptyset$, and $B_{2}$ satisfies property (II) with $\ell=k_{2}$ and $k_{2}>k_{1}$. By properties (I)-(II) for $B_{2}$, there exists $b\in B_{2}$ with $b\neq a_{1}$ and $\gamma(a_{1},b)=k_{2}$. Since $a_{1}\in B_{1}$ and $\gamma(a_{1},b)=k_{2}>k_{1}$, it follows from property (II) for $B_{1}$ that $b\notin B_{1}$. Also $b\notin B_{3}\cup B_{4}$ since $b\in B_{2}\subset\mathbf{b}_{1}$ and $\mathbf{b}_{1}\cap(B_{3}\cup B_{4})=\emptyset$. To summarize, $b\in B_{1}$, but $b\notin B_{\epsilon}$ for all $\epsilon\in\{2,3,4\}$. Therefore, $\lambda(b)=1$, which is a contradiction.

Remark 15.11**.**

To summarize the above contradiction proof, both $\mathbf{b}_{1}$ and $\mathbf{b}_{2}$ need to have $B_{\epsilon}\subset\mathbf{b}_{1}$ for at least two values of $\epsilon\in\{2,3,4\}$ to avoid having $b\in\cup_{\epsilon}B_{\epsilon}$ with $\lambda(b)=1$, however, this is inconsistent with $\mathbf{b}_{1},\mathbf{b}_{2}\in E_{k_{1}}$ being distinct and thus disjoint when viewed as subsets of $E_{n}$.

Case $\mathbf{k_{1}=k_{2}<k_{3}<k_{4}}$: Let $B_{\epsilon}$ for $\epsilon\in\{1,2,3,4\}$ satisfy properties (I)-(II) above respectively for $\ell=k_{\epsilon}$ with $k_{1}=k_{2}<k_{3}<k_{4}$. There are two special types—see ($\textup{I}^{\prime}$)-($\textup{II}^{\prime}$) below—of configurations of the sets $B_{\epsilon}$ such that $\lambda(a)\geq 2$ for all $a\in\cup_{\epsilon}B_{\epsilon}$. For both types, $|B_{1}|=|B_{2}|$ and $|B_{3}|=|B_{4}|=2$.

• ($\textup{I}^{\prime}$)

There exists $\mathbf{a}\in E_{k_{1}-1}$ and distinct $\mathbf{b}_{1},\mathbf{b}_{2}\in\mathbf{a}\cap E_{k_{1}}$ such that $B_{3}\subset\mathbf{b}_{1}$ and $B_{4}\subset\mathbf{b}_{2}$. The sets in the collection \mathscr{P}:=\big{\{}B_{\epsilon}\cap B_{\delta}\,\big{|}\,\epsilon\in\{1,2\},\,\delta\in\{3,4\}\big{\}} are pairwise disjoint, have cardinality one, and their union is equal to $B_{3}\cup B_{4}=B_{1}\Delta B_{2}$. In particular, $B_{3}\cap B_{4}=\emptyset$ and $|B_{1}\cap B_{2}|=|B_{1}|-2$.

• ($\textup{II}^{\prime}$)

There exists $\mathbf{a}\in E_{k_{3}-1}$ and $\mathbf{b}\in\mathbf{a}\cap E_{k_{3}}$ such that $B_{3}\subset\mathbf{a}$ and $B_{4}\subset\mathbf{b}$. The sets $B_{1}\backslash B_{2}$, $B_{2}\backslash B_{1}$, $B_{3}\backslash B_{4}$, $B_{4}\backslash B_{3}$ have cardinality one and $B_{1}\Delta B_{2}=B_{3}\Delta B_{4}$. In particular, $|B_{3}\cap B_{4}|=1$ and $|B_{1}\cap B_{2}|=|B_{1}|-1$.

The types ($\textup{I}^{\prime}$) and ($\textup{II}^{\prime}$) correspond to the cases of $\gamma(B_{3},B_{4})=k_{1}$ and $\gamma(B_{3},B_{4})>k_{1}$, respectively. The possibility $\gamma(B_{3},B_{4})<k_{1}$ can be excluded because it results in multiple $b\in\cup_{\epsilon}B_{\epsilon}$ with $\lambda(b)=1$ by simpler reasoning than in the case $k_{1}<k_{2},k_{3},k_{4}$ discussed above.

To understand the type-($\textup{I}^{\prime}$) configuration, notice that the intersections $B_{\epsilon}\cap B_{\delta}$ for $\epsilon\in\{1,2\}$ and $\delta\in\{3,4\}$ contain at most one element since $B_{\epsilon}$ and $B_{\delta}$ satisfy property (II) for $\ell=k_{1}=k_{2}$ and $\ell>k_{1}$, respectively. Thus $B_{1}$ and $B_{2}$ can each contribute at most one to each of the sums $\sum_{a\in B_{3}}\lambda(a)$ and $\sum_{a\in B_{4}}\lambda(a)$. Since $B_{3}\cap B_{4}=\emptyset$ (because $B_{3}\subset\mathbf{b}_{1}$ and $B_{4}\subset\mathbf{b}_{2}$ for distinct $\mathbf{b}_{1},\mathbf{b}_{2}\in E_{k_{1}}$), it is only possible that $\lambda(a)\geq 2$ for all $a\in B_{3}\cup B_{4}$ if $|B_{3}|=|B_{4}|=2$ and the collection \mathscr{P}:=\big{\{}B_{\epsilon}\cap B_{\delta}\,\big{|}\,\epsilon\in\{1,2\},\,\delta\in\{3,4\}\big{\}} is a partition of $B_{3}\cup B_{4}$ comprised of single-element sets. Similarly, $B_{1}\Delta B_{2}:=(B_{1}\backslash B_{2})\cup(B_{2}\backslash B_{1})$ must be a subset of $B_{3}\cup B_{4}$ to avoid having an $a\in B_{1}\cup B_{2}$ with $\lambda(a)=1$. Since the sets in $\mathscr{P}$ are disjoint and have union equal to $B_{3}\cup B_{4}$, it follows that $B_{1}\Delta B_{2}=B_{3}\cup B_{4}$. Finally, $|B_{1}|=|B_{2}|$ and $|B_{1}\cap B_{2}|=|B_{1}|-2$ since sets in $\mathscr{P}$ have cardinality one.

To derive the type-($\textup{II}^{\prime}$) configuration, suppose that there is a single $\mathbf{b^{\prime}}\in E_{k_{1}}$ such that $B_{3},B_{4}\subset\mathbf{b^{\prime}}$. Since $B_{1}$ and $B_{2}$ satisfy property (II) with $\ell=k_{1}=k_{2}$, the sets $B_{1}\cap\mathbf{b^{\prime}}$ and $B_{2}\cap\mathbf{b^{\prime}}$ contain at most one element. It follows that $B_{1}$ and $B_{2}$ can each contribute at most one to the sum $\sum_{a\in B_{3}\Delta B_{4}}\lambda(a)$. Since $B_{3}$ and $B_{4}$ satisfy property (II) respectively for $\ell=k_{3}$ and $\ell=k_{4}$ with $k_{3}<k_{4}$, the set $B_{3}\cap B_{4}$ has at most one element. Under these constraints, it is only possible that $\lambda(a)\geq 2$ for all $a\in B_{3}\cup B_{4}$ if $|B_{3}|=|B_{4}|=2$, $|B_{3}\cap B_{4}|=1$, and the sets $(B_{3}\Delta B_{4})\cap B_{1}$ and $(B_{3}\Delta B_{4})\cap B_{2}$ have cardinality one and are disjoint. Since $B_{3},B_{4}\subset\mathbf{b^{\prime}}\in E_{k_{1}}$ and $B_{1},B_{2}$ satisfy property (II) with $\ell=k_{1}=k_{2}$, the sets $B_{3}$, $B_{4}$ jointly contribute at most one to each of the sums $\sum_{a\in B_{1}\backslash B_{2}}\lambda(a)$ and $\sum_{a\in B_{2}\backslash B_{1}}\lambda(a)$. In order for $\lambda(a)\geq 2$ for all $a\in B_{1}\Delta B_{2}$, it must be that $|B_{2}\backslash B_{1}|=1$ and $|B_{1}\backslash B_{2}|=1$. Hence $|B_{1}\cap B_{2}|=|B_{1}|-1=|B_{2}|-1$. Since $B_{3}$ and $B_{4}$ satisfy property (II) respectively for $\ell=k_{3}$ and $\ell=k_{4}>k_{3}$ with $B_{3}\cap B_{4}\neq\emptyset$, there exists $\mathbf{a}\in E_{k_{3}-1}$ and $\mathbf{b}\in E_{k_{3}}$ such that $B_{3}\subset\mathbf{a}$ and $B_{4}\subset\mathbf{b}\subset\mathbf{a}$.

Next we bound the expectation of $Y_{k_{1}}Y_{k_{2}}Y_{k_{3}}Y_{k_{4}}$ when $k_{1}=k_{2}<k_{3}<k_{4}$. Using the formula (15.2), we can write

[TABLE]

where the second equality holds by foiling the product over $\epsilon\in\{1,2,3,4\}$ by our observations above. The type-($\textup{I}^{\prime}$) and type-($\textup{II}^{\prime}$) contributions to (15.12) both yield multiples of $b^{-(k_{3}+k_{4})}$. The cases are similar, so we will discuss only the type-($\textup{I}^{\prime}$) case.

When the product over $\epsilon\in\{1,2,3,4\}$ inside the expectation in (15.12) is foiled, only the terms with $\mathbf{a}_{1}=\mathbf{a}_{2}$, $i_{1}=i_{2}$, $A_{1}=A_{2}$ can be of type-($\textup{I}^{\prime}$) or type-($\textup{II}^{\prime}$) and thus nonzero. In the type-($\textup{I}^{\prime}$) case, there are distinct $j,J\in A_{1}$ such that $\mathbf{a}_{3}\in(\mathbf{a}_{1}{\times}(i_{1},j))\cap E_{k_{3}}$ and $\mathbf{a}_{4}\in(\mathbf{a}_{1}{\times}(i_{1},J))\cap E_{k_{4}}$, where $\mathbf{a}_{1}{\times}(i_{1},j)$ and $\mathbf{a}_{1}{\times}(i_{1},J)$ have the roles of $\mathbf{b}_{1}$ and $\mathbf{b}_{2}$, respectively, in the statement of ($\textup{I}^{\prime}$). The type-($\textup{I}^{\prime}$) contribution has the form

[TABLE]

where we interpret $|A_{2}|:=|A_{1}|$ inside the product $\prod_{\epsilon=1}^{4}$, and $\eta(\phi_{1},\phi_{2},\phi_{3},\phi_{4})\in\{0,1\}$ is defined as

[TABLE]

Note that the sets $\textup{Rng}(\phi_{3})=\phi_{3}(A_{3})$ and $\textup{Rng}(\phi_{4})=\phi_{4}(A_{4})$ in the definition of $\eta(\phi_{1},\phi_{2},\phi_{3},\phi_{4})$ both contain exactly two elements. There are respectively $|G_{i_{3},A_{3}}^{n,\mathbf{a}_{3}}|=b^{2|A_{3}|(n-k_{3})}=b^{4(n-k_{3})}$ and $|G_{i_{4},A_{4}}^{n,\mathbf{a}_{4}}|=b^{2|A_{4}|(n-k_{4})}=b^{4(n-k_{4})}$ choices for the functions $\phi_{3}$ and $\phi_{4}$. When $\phi_{3}\in G_{i_{3},A_{3}}^{n,\mathbf{a}_{3}}$ and $\phi_{4}\in G_{i_{4},A_{4}}^{n,\mathbf{a}_{4}}$ are given, there are $4b^{2(|A_{1}|-2)(n-k_{1})}$ combinatorial possibilities for the pair of functions $\phi_{1},\phi_{2}\in G_{i_{1},A_{1}}^{n,\mathbf{a}_{1}}$ such that $\eta(\phi_{1},\phi_{2},\phi_{3},\phi_{4})=1$, where the factor of $4$ comes from the assignment choices for $\phi_{1},\phi_{2}$ on the subdomain $\{j,J\}$. For the purpose of evaluating (15.13), it will be convenient to reformulate the sums (ii)-(iii) as

[TABLE]

The summation (15.13) is equal to

[TABLE]

where the sum is independent of a particular choice of $j,J\in\{1,\ldots,b\}$ with $j\neq J$. Moreover, the sum has $2^{b-2}$ terms and the summand is indepedent of $|A_{1}|$ because of the cancellation of $b^{2|A_{1}|(n-k_{1})}$ between the numerator and the denominator. The product above is equal to $2^{b-2}(b-1)^{3}(\sigma^{2})^{4}/b^{k_{3}+k_{4}}$. ∎

15.3 Proof of Lemma 13.6

In this section, we will prove the following lemma, which uses more restrictive assumptions on the asymptotics for $x\equiv x^{n,r}$ in (2.8) to gain more explicit control of the error in the convergence of $M^{n}(x)$ to $R(r)$ as $n\rightarrow\infty$ in Lemma 2.3. Recall from Remark 6.6 that if the random variables in an i.i.d. array \big{\{}X_{h}^{(n)}\big{\}}_{h\in E_{n}} are centered with variance $x$, then the random variables in the array \big{\{}X_{a}^{(k,n)}\big{\}}_{a\in E_{k}}:=\mathcal{Q}^{n-k}\big{\{}X_{h}^{(n)}\big{\}}_{h\in E_{n}} have variance $M^{n-k}(x)$. It follows that Lemma 15.12 below is equivalent to Lemma 13.6.

Lemma 15.12**.**

Fix $\mathbf{v}>0$, $\alpha\in(0,1)$, and a bounded interval $\mathcal{I}\subset{\mathbb{R}}$. There exists $C_{\mathcal{I},\mathbf{v},\alpha}>0$ such that for any $x>0$, $n\in\mathbb{N}$, and $r\in\mathcal{I}$ satisfing the inequality

[TABLE]

the following inequality holds:

[TABLE]

The proof of Lemma 15.12 will rely on an application of Lemma 14.9.

Proof.

Let $\mathbf{v}>0$, $\alpha\in(0,1)$, and $\mathcal{I}$ be a bounded interval in ${\mathbb{R}}$. As a preliminary, note that the asymptotic form for $R(s)$ as $s\rightarrow-\infty$ in (II) of Lemma 2.3 implies that there exists a $C_{\mathcal{I},\alpha}>0$ such that for all $r\in\mathcal{I}$ and $n\in\mathbb{N}$

[TABLE]

Let $x>0$, $n\in\mathbb{N}$, and $r\in\mathcal{I}$ be any values satisfying the condition (15.14), and let $k\in\{0,\ldots,n\}$. By (I) of Lemma 2.3, we can rewrite the difference between $M^{n-k}(x)$ and $R(r-k)$ as

[TABLE]

Since the derivative of $M^{n-k}$ is increasing, the absolute value of (15.17) is bounded by

[TABLE]

Let $k^{*}\equiv k^{*}(x,n,r)$ be the smallest $k\in\{0,\ldots,n\}$ such that

[TABLE]

which exists because (15.19) is satisfied with $k=n$ by (15.14) and (15.16). Note that (15.19) implies that $M^{n-k^{*}}(x)\leq 2R(r)+C_{\mathcal{I},\alpha}+\mathbf{v}$ since $R$ is increasing. Thus with (15.18), for any $x>0$, $n\in\mathbb{N}$, $r\in\mathcal{I}$ satisfying (15.14), we have that

[TABLE]

We will show that $k^{*}=0$ whenever $n\geq N_{\mathcal{I},\mathbf{v},\alpha}$, where $N_{\mathcal{I},\mathbf{v},\alpha}>0$ is defined by

[TABLE]

Suppose to reach a contradiction that $n\geq N_{\mathcal{I},\mathbf{v},\alpha}$ and $k^{*}\equiv k^{*}(x,n,r)>0$ for some $x>0$, $n\in\mathbb{N}$, $r\in\mathcal{I}$ such that (15.14) holds. Using similar reasoning as in (15.18), the difference between $M^{n-k^{*}+1}(x)$ and $R(r-k^{*}+1)$ is bounded by

[TABLE]

where we have applied (15.20) in the second inequality. Since $n\geq N_{\mathcal{I},\mathbf{v},\alpha}$, the above is smaller than $R(r)$. Thus $k:=k^{*}-1$ satisfies \big{|}M^{n-k}(x)\,-\,R(r-k)\big{|}\leq R(r), which contradicts that $k^{*}$ is the smallest element of $\{0,\ldots,n\}$ satisfying (15.19). Therefore, $k^{*}=0$ when $n\geq N_{\mathcal{I},\mathbf{v},\alpha}$.

Since \big{|}M^{n-k}(x)\,-\,R(r-k)\big{|}\leq R(r)+C_{\mathcal{I},\alpha}+\mathbf{v} holds for all $k\in\{0,\ldots,n\}$ when $x>0$, $n\in\mathbb{N}$, $r\in\mathcal{I}$ satisfy (15.14) and $n\geq N_{\mathcal{I},\mathbf{v},\alpha}$, under these conditions on $x$, $n$, $r$ the inequality (15.18) yields

[TABLE]

Thus we have the inequality that we sought under the restriction $n\geq N_{\mathcal{I},\mathbf{v},\alpha}$. The remaining case when $n$ is smaller than $N_{\mathcal{I},\mathbf{v},\alpha}$ is trivial. ∎

15.4 Proof of Proposition 13.7

For $m\in\{2,3,\ldots\}$, let the polynomial $P_{m}:{\mathbb{R}}^{m-1}\rightarrow{\mathbb{R}}$ be defined as in Section 15.1. The following lemma is from [10, Proposition 3.1].

Lemma 15.13**.**

The multivariate polynomial $P_{m}:{\mathbb{R}}^{m-1}\rightarrow{\mathbb{R}}$ satisfies the properties below.

1. (i)

$P_{m}(y_{2},\ldots,y_{m})$* has nonnegative coefficients, no constant term, and its only linear term is $\frac{1}{b^{m-2}}y_{m}$. In other words, there exist polynomials $U_{m}:{\mathbb{R}}^{m-1}\rightarrow{\mathbb{R}}$ and $V_{m}:{\mathbb{R}}^{m-2}\rightarrow{\mathbb{R}}$ with nonnegative coefficients such that*

[TABLE]

where the polynomials $y_{m}U_{m}(y_{2},\ldots,y_{m})$ and $V_{m}(y_{2},\ldots,y_{m-1})$ have no constant or linear terms. 2. (ii)

The polynomial $V_{m}(y_{2},\ldots,y_{m-1})$ is a linear combination of monomials $y_{j_{1}}\cdots y_{j_{\ell}}$ with

[TABLE]

The polynomial $y_{m}U_{m}(y_{2},\ldots,y_{m})$ is a linear combination of monomials with $j_{1}+\cdots+j_{\ell}\geq m+2$.

The next lemma follows easily from (II) of Theorem 2.4.

Lemma 15.14**.**

For any $\mathfrak{p}\in\mathbb{N}$ and bounded interval $\mathcal{I}\subset{\mathbb{R}}$, there is a positive number $C_{\mathcal{I},\mathfrak{p}}$ such that for all $r\in\mathcal{I}$ and $n\in\mathbb{N}$

[TABLE]

We will use the notation \sigma^{(m)}_{k,n}:=\mathbb{E}\big{[}\big{(}X_{a}^{(k,n)}\big{)}^{m}\big{]} and $\sigma^{(m)}_{n,n}\equiv\sigma^{(m)}_{n}$ from (15.1) throughout the following proof. The $m^{th}$ absolute moment of variables in the generating array \big{\{}X_{h}^{(n)}\big{\}}_{h\in E_{n}} will be denoted by $\macc@depth\char 1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a 111{\sigma}^{(m)}_{n}$.

Proof of Proposition 13.7.

Fix $\mathbf{v},\varkappa\geq 1$, $\alpha\in(0,1)$, and a bounded interval $\mathcal{I}\subset{\mathbb{R}}$.212121Without losing any generality we can assume $\mathbf{v},\varkappa\geq 1$ rather than $\mathbf{v},\varkappa>0$. We will use induction in $m\in\{2,3,\ldots\}$ to show that there is a $c_{m}\equiv c_{m}(\mathcal{I},\mathbf{v},\alpha,\varkappa)>0$ such that for any $r\in\mathcal{I}$, $n\in\mathbb{N}$, and i.i.d. array of centered random variables $\{X^{(n)}_{h}\}_{h\in E_{n}}$ satisfying

1. (I)

\Big{|}\sigma_{n}^{2}-\kappa^{2}\big{(}\frac{1}{n}+\frac{\eta\log n}{n^{2}}+\frac{r}{n^{2}}\big{)}\Big{|}<\frac{\mathbf{v}}{n^{2+\alpha}} and 2. (II)

$\macc@depth\char 1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a 111{\sigma}^{(m)}_{n}<\frac{\varkappa}{n^{m/2}}$,

the following inequality holds for all $k\in\{0,\ldots,n\}$

[TABLE]

Notice that the existence of $c_{2}$ follows from Lemma 15.14 with $\mathfrak{p}=1$ and Lemma 13.6. Assume for the purpose of a strong induction argument that there exist constants $c_{m}\equiv c_{m}(\mathcal{I},\mathbf{v},\alpha,\varkappa)>0$ satisfying the statement above for each $m\in\{2,\ldots,\mathbf{m}-1\}$ for some $\mathbf{m}\in\{3,4,\ldots\}$. Let $r\in\mathcal{I}$, $n\in\mathbb{N}$, and $\{X^{(n)}_{h}\}_{h\in E_{n}}$ be an i.i.d. array of centered random variables satisfying (I)-(II) for $m=\mathbf{m}$. Note that for any $m\in\{2,\ldots,\mathbf{m}-1\}$ Jensen’s inequality and condition (II) give us the first two inequalities below:

[TABLE]

The third inequality holds since $\varkappa\geq 1$. Thus $\{X^{(n)}_{h}\}_{h\in E_{n}}$ satisfies condition (II) for each $m\in\{2,\ldots,\mathbf{m}-1\}$, and therefore (15.22) holds for all $m\in\{2,\ldots,\mathbf{m}-1\}$ by our induction assumption. Define $c:=\max_{2\leq m\leq\mathbf{m}-1}c_{m}$.

The last component of the recursive relation (15.2) implies that

[TABLE]

Since (15.22) holds for all $m\in\{2,\ldots,\mathbf{m}-1\}$, the term \Big{|}V_{\mathbf{m}}\Big{(}\sigma_{k,n}^{(2)},\ldots,\sigma_{k,n}^{(\mathbf{m}-1)}\Big{)}\Big{|} has the bound

[TABLE]

where $c^{\prime}\equiv c^{\prime}(\mathcal{I},\mathbf{v},\varkappa,\alpha,\mathbf{m})$ is defined by c^{\prime}=\sup_{\ell\in\mathbb{N}_{0}}\,(\ell+1)^{\frac{\mathbf{m}}{2}}V_{\mathbf{m}}\Big{(}c(\ell+1)^{-1},\ldots,c(\ell+1)^{-\frac{\mathbf{m}-1}{2}}\Big{)}, and we have used that $V_{\mathbf{m}}$ has nonnegative coefficients. The supremum above is finite as a consequence of part (ii) of Lemma 15.13.

Again invoking that (15.22) holds for all $m\in\{2,\ldots,\mathbf{m}-1\}$, the factor \big{|}U_{\mathbf{m}}\big{(}\sigma_{k,n}^{(2)},\ldots,\sigma_{k,n}^{(\mathbf{m})}\big{)}\big{|} in (15.24) has the bound

[TABLE]

The above also uses that the coefficients of the polynomial $U_{\mathbf{m}}$ are nonnegative. Since the polynomial $U_{\mathbf{m}}$ has no constant term by (i) of Lemma 15.13, there is a $\mathbf{c}\equiv\mathbf{c}(\mathcal{I},\mathbf{v},\varkappa,\alpha,\mathbf{m})>0$ such that for all $k\in\mathbb{N}_{0}$ and $y\in[0,1]$

[TABLE]

Define $N_{\varkappa,\mathbf{c}}:=\textup{max}(\varkappa,8\mathbf{c})$. Note that when $n\geq N_{\varkappa,\mathbf{c}}$ the inequalities below are satisfied for $k=n$ as a consequence of assumption (II) with $m=\mathbf{m}$:

[TABLE]

For $n\geq N_{\varkappa,\mathbf{c}}$ define $k^{*}_{n}$ as the smallest $k\in\{0,\ldots,n\}$ satisfying (15.28). Note that for all $k\in\{k^{*}_{n},\ldots,n\}$

[TABLE]

by (15.26)-(15.28) and since $\mathbf{m}\geq 3$ and $b\geq 2$.

Assume $n\geq N_{\varkappa,\mathbf{c}}$. By the bounds (15.24), (15.25), and (15.29), we have the inequality below for all $k\in\{k^{*}_{n},\ldots,n\}$.

[TABLE]

Using (15.30) recursively, it follows that for any $k\in\{k^{*}_{n}-1,\ldots,n\}$

[TABLE]

where $c^{\prime\prime}:=\varkappa 2^{\mathbf{m}/2}+4c^{\prime}$, and we have used the crude bound $k+1\leq 2n$. It follows from (15.31) that $k^{*}_{n}$ is bounded from above by $\widehat{k}\equiv\widehat{k}(\mathcal{I},\mathbf{v},\varkappa,\alpha,\mathbf{m})$ defined by

[TABLE]

If n\geq\max\big{(}\widehat{k},N_{\varkappa,\mathbf{c}}\big{)}, then (15.31) has the form of our desired inequality (15.22) for $m=\mathbf{m}$ and all $k\in\{\widehat{k},\ldots,n\}$. Since there are only finitely many remaining $k\in\{0,\ldots,\widehat{k}-1\}$, we can use the recursive relation \sigma_{k-1,n}^{(\mathbf{m})}=P_{\mathbf{m}}\Big{(}\sigma_{k,n}^{(2)},\ldots,\sigma_{k,n}^{(\mathbf{m})}\Big{)} and our induction assumption to bound the remaining terms by a constant depending only on $\mathcal{I}$, $\mathbf{v}$, $\varkappa$, $\alpha$, and $\mathbf{m}$. Finally, we can pick our constant large enough to extend the inequality to the finitely many $n\in\mathbb{N}$ with n<\max\big{(}\widehat{k},N_{\varkappa,\mathbf{c}}\big{)}. By induction this completes the proof. ∎

Appendix A Inverse temperature scaling

We will outline the calculation verifying that the variance scaling (2.7) determines the inverse temperature scaling $\beta_{n,r}$ in (2.5). In other terms, $V(\beta_{n,r})=V_{n,r}+\mathit{o}(1/n^{2})$ as $n\rightarrow\infty$ for

[TABLE]

Recall that $\tau:=\mathbb{E}[\omega^{3}]$ and $\tau^{\prime}:=\mathbb{E}[\omega^{4}]-3$. Since $\mathbb{E}[e^{\beta\omega}]=1+\frac{1}{2}\beta^{2}+\frac{\tau}{6}\beta^{3}+\frac{\tau^{\prime}+3}{24}\beta^{4}+\mathit{O}(\beta^{5})$ for $0<\beta\ll 1$, a computation shows that

[TABLE]

Another computation using the expansion (A.1) shows that for small $\beta>0$

[TABLE]

Substituting V_{n,r}+\mathit{o}\big{(}\frac{1}{n^{2}}\big{)} in for $V(\beta)$ on the right side of (A.2) yields

[TABLE]

which is the asymptotic form for $\beta_{n,r}$ in (2.5). Alternatively, if we substitute the sharper asymptotic form V_{n,r}+\mathit{O}\big{(}\frac{1}{n^{2+\alpha}}\big{)} in for $V(\beta)$ on the right side of (A.2), then \mathit{o}\big{(}\frac{1}{n^{3/2}}\big{)} can be replaced by \mathit{O}\big{(}\frac{1}{n^{3/2+\alpha}}\big{)} on the right side of (A.3).

For the site-disorder model, the inverse temperature scaling (3.3) results in the variance scaling (14.4) since by (A.1) we have

[TABLE]

Appendix B Variance function consistency check

There is instructional value in implementing a consistency check between properties (I) and (II) in the statement of Lemma 2.3, i.e., between the claim that M\big{(}R(r)\big{)}=R(r+1) and the $-r\gg 1$ asymptotics

[TABLE]

where $\kappa^{2}:=\frac{2}{b-1}$ and $\eta:=\frac{b+1}{3(b-1)}$. Fix some $r$ with $-r\gg 1$ and define $V_{n}=R(r-n)$ for $n\in\mathbb{N}_{0}$. We begin by writing $R(r)$ as a telescoping sum

[TABLE]

We will analyze the expressions (a), (b), and (c) to verify that the right side of (B.2) has the asymptotics (B.1). The expression (c) is $\mathit{O}(1/r^{3})$ since the terms $V_{k}$ are bounded by a constant multiple of $(k-r)^{-1}$ as a consequence of (B.1).

Applying (B.1) to $V_{k}$ in the expression (a) yields

[TABLE]

where we have used a trapezoidal Riemann approximation to get

[TABLE]

and right-hand Riemann approximations to get

[TABLE]

Again applying (B.1) to $V_{k}$, foiling, and using that $\kappa^{-2}=\frac{b-1}{2}$, the expression (b) is equal to

[TABLE]

where we have used the Riemann approximation

[TABLE]

Summing up (a), (b), and (c) gives the desired asymptotics (B.1) as a result of the cancellation $-\frac{1}{(b-1)r^{2}}+\frac{\eta\kappa^{2}}{2r^{2}}+\frac{b-2}{3}\frac{\kappa^{4}}{2r^{2}}=0$ between the bracketed terms above.

Appendix C The zero bias approach to Stein’s method

We will discuss the zero bias variation on Stein’s method introduced in [19], which provides an easy proof of Lemma 11.6 (restated in Lemma C.4).

C.1 Zero bias transformation

Let $X$ be a centered random variable with variance $\sigma^{2}$. The zero bias transformation, $X^{*}$, of $X$ is the distribution satisfying

[TABLE]

for all absolutely continuous functions $f$ on ${\mathbb{R}}$. The right side above can be written as

[TABLE]

Thus if $X$ has distribution measure $\mu$, then $X^{*}$ is constructed by choosing a number $x$ using the measure $\nu(dx)=\frac{x^{2}}{\sigma^{2}}\mu(dx)$ and then picking a number uniformly at random from the interval between [math] and $x$. The normal distribution is the unique fixed point for the zero bias transformation:

Lemma C.1**.**

Let $X$ be a centered random variable with variance $\sigma^{2}$. Then $X\stackrel{{\scriptstyle d}}{{=}}X^{*}$ iff $X\sim\mathcal{N}(0,\sigma^{2})$.

Lemma C.2**.**

Let $X$ be a centered random variable with variance $\sigma^{2}$ and finite absolute moment \varsigma_{n}:=\mathbf{E}\big{[}|X|^{n}\big{]} for some $n\geq 3$. The absolute moment $\varsigma_{n-2}^{*}$ of $X^{*}$ is finite and equal to $\varsigma_{n-2}^{*}=\frac{\varsigma_{n}}{\sigma^{2}(n-1)}$.

Proof.

This follows easily from the definition of $X^{*}$ since

[TABLE]

∎

The lemma below gives a key distributional identity for the zero bias transformation of a finite sum of independent random variables; see, for instance, Lemma 2.2 of [18] for the proof.

Lemma C.3**.**

Let $X_{1},\ldots,X_{n}$ be independent centered random variables with $\textup{Var}(X_{k})=\sigma_{k}^{2}$. Let i be a variable taking values in $\{1,2,\ldots,n\}$ with probability \mathcal{P}\big{[}\textbf{i}=k\big{]}=\frac{\sigma_{k}^{2}}{\sigma_{1}^{2}+\cdots+\sigma_{n}^{2}}. The distribution of $(X_{1}+\cdots+X_{n})^{*}$ has the form

[TABLE]

where i is independent of the random variables $X_{k}$ and $X^{*}_{k}$. In other terms, the $k^{th}$ variable $X_{k}$ in the sum is replaced by $X_{k}^{*}$ with probability $\frac{\sigma_{k}^{2}}{\sigma_{1}^{2}+\cdots+\sigma_{n}^{2}}$.

C.2 Relation to Stein’s method

Recall that \rho_{1}(X,Y):=\sup_{h\in\textup{Lip}_{1}}\mathbb{E}\big{[}h(X)-h(Y)\big{]} for two random variables $X$ and $Y$ with finite first absolute moments. Also, recall that the auxiliary function $f$ for a given $h\in\textup{Lip}_{1}$ in Stein’s method satisfies the differential equation

[TABLE]

and that the first- and second-order derivatives have the bounds $\sup_{x}|f^{\prime}(x)|\leq 1$ and $\sup_{x}|f^{\prime\prime}(x)|\leq 2$. In particular $f^{\prime}$ is absolutely continuous with Lipschitz constant $\leq 2$. If $X$ is a centered random variable with variance $\sigma^{2}$ and \mathcal{X}\sim\mathcal{N}\big{(}0,\sigma^{2}\big{)}, then by definition of $X^{*}$ we have

[TABLE]

Thus, by supremizing over $h\in\textup{Lip}_{1}$ above, we have the bound \rho(X,\mathcal{X})\,\leq\,2\rho\big{(}X,X^{*}\big{)} since $|f^{\prime\prime}|\leq 2$. Therefore, the Wasserstein-$1$ norm between $X$ and the normal random variable $\mathcal{X}$ is smaller than two times the Wasserstein-$1$ norm between $X$ and its zero bias transformation.

Lemma C.4**.**

Let $X_{1}$,…,$X_{n}$ be i.i.d. variables with mean [math] and variance $\sigma^{2}$. For $Y_{n}:=\frac{X_{1}+\cdots+X_{n}}{\sqrt{n}}$, we have the inequality

[TABLE]

for $Y_{n}=\frac{X_{1}+\cdots+X_{n}}{\sqrt{n}}$. Moreover, if \mathbb{E}\big{[}|X_{1}|^{3}\big{]}<\infty and \mathcal{Y}\sim\mathcal{N}\big{(}0,\sigma^{2}\big{)}, then

[TABLE]

Proof.

Let the pairs $(X_{k},X_{k}^{*})$ be i.i.d. couplings of the variables $X_{k}$ and $X_{k}^{*}$ such that

[TABLE]

Then $\rho_{1}(Y_{n},Y_{n}^{*})$ is bounded as follows:

[TABLE]

and the last term is equal to $\frac{1}{\sqrt{n}}\rho_{1}(X_{1},X_{1}^{*})$ by assumption. Next we simply observe that

[TABLE]

where the second inequality is by Lemma C.2. The result then holds because $\rho_{1}(Y_{n},\mathcal{Y})\leq 2\rho_{1}(Y_{n},Y_{n}^{*})$.∎

Proof of Lemma 11.7.

Lemma 11.5 gives us the inequality

[TABLE]

The second inequality above uses that \mathbb{E}\big{[}\macc@depth\char 1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a 111{X}_{n}^{4}\big{]}=3\sigma^{4}\big{(}1-\frac{1}{n}\big{)}+\frac{1}{n}\mathbb{E}\big{[}X_{1}^{4}\big{]} is smaller than 3\mathbb{E}\big{[}X_{1}^{4}\big{]} and 2^{\frac{2}{3}}3^{\frac{1}{3}}\big{(}1+3^{\frac{1}{6}}\big{)}<6.∎