On the Ultrametricity Property in Random Field Ising Models

TL;DR

This paper demonstrates that the ultrametricity property, a hierarchical structure, persists in random field Ising models under small perturbations of the external field, extending understanding of their geometric organization.

Contribution

It establishes the stability of ultrametricity in random field Ising models with independent disorder when the field strength is a small perturbation.

Findings

Ultrametricity remains valid under small perturbations.

The property holds for models with independent disorder.

Results extend the understanding of the geometric structure of these models.

Abstract

In this paper we show that the ultrametricity property remains valid in random field Ising models with independent disorder whenever the field strength is a small perturbation.

Equations

$$\displaystyle{G}(\{\sigma\})={1\over Z}\exp\Bigg{(}\beta\sum_{\langle xy\rangle}\sigma_{x}\sigma_{y}+(\mu-h)\sum_{x\in V_{n}}g_{x}\sigma_{x}\Bigg{)}\,,$$

$$\displaystyle h\mbox{$\;\longrightarrow_{{\small n}}\;$~{}}\mu;\quad(\mu-h)\sqrt{|V_{n}|}\mbox{$\;\longrightarrow_{{\small n}}\;$~{}}\infty;\quad{1\over|V_{n}|}\sum_{x\in V_{n}}\mathbb{E}\big{(}|g_{x}|^{3}:|g_{x}|\geqslant\varepsilon(\mu-h)^{-1}\big{)}\mbox{$\;\longrightarrow_{{\small n}}\;$~{}}0\,,$$

$$\beta \sum _{⟨xy⟩}{\sigma }_x{\sigma }_y+(\mu -h)\sum _{x\in V_n}{g}_x{\sigma }_x+H_{n;\mathbf{\alpha }}^{\mathrm{per}}(\sigma ),$$

$$H_{n;\mathbf{\alpha }}^{\mathrm{per}}(\sigma )=c_n\sum _{p⩾2}{\alpha }_p{2}^{-p}{H}_{n;p}(\sigma );H_{n;p}(\sigma )=\frac{1}{|V_n{|}^{(p-1)/2}}\sum _{x_1,\dots ,x_p}{\xi }_{x_1,\dots ,x_p}{\sigma }_{x_1}\cdots {\sigma }_{x_p};$$

$$⟨f{⟩}_{\mathbf{\alpha }}=⟨f({\sigma }^1,\dots ,{\sigma }^m){⟩}_{\mathbf{\alpha }}$$

$${\nu }_{\mathbf{\alpha }}(f):=\mathbb{E}⟨f{⟩}_{\mathbf{\alpha }}=\int ⟨f{⟩}_{\mathbf{\alpha };g=u,\xi =v}\text{d}\gamma (u,v),$$

$$R_{\ell ,{\ell }^{\prime }}:=\frac{1}{|V_n|}\sum _{x\in V_n}{\sigma }_x^{\ell }{\sigma }_x^{{\ell }^{\prime }},\forall \ell ,{\ell }^{\prime }⩾1.$$

$$\displaystyle\nu_{\bm{\alpha}}(f\psi(R_{1,m+1}))-\frac{1}{m}\,\nu_{\bm{\alpha}}(f)\nu_{\bm{\alpha}}(\psi(R_{1,2}))-\frac{1}{m}\sum_{\ell=2}^{m}\nu_{\bm{\alpha}}(f\psi(R_{1,\ell}))\mbox{$\;\longrightarrow_{{\small n}}\;$~{}}0\,.$$

$$|\mathbb{E}yf(y)-\mathbb{E}{f}^{\prime }(y)|⩽\frac{2(\Vert f^{(k-1)}{\Vert }_{\infty }+\Vert f^{(k)}{\Vert }_{\infty })}{(k-1)!}\mathbb{E}(|y{|}^k:|y|⩾K)+\frac{(k+1)K}{k!}\Vert f^{(k)}{\Vert }_{\infty }\mathbb{E}|y{|}^k.$$

$$\displaystyle\biggl{|}$$

$$\displaystyle\leqslant{2\over(k-1)!}\left({\biggl{\|}{\partial^{k-1}f\over\partial y^{k-1}}\biggr{\|}_{\infty}}+{\biggl{\|}{\partial^{k}f\over\partial y^{k}}\biggr{\|}_{\infty}}\right)\mathbb{E}(|x|:|x|\geqslant K_{1})\,\mathbb{E}(|y|^{k}:|y|\geqslant K_{2})$$

$$\displaystyle\quad+{2\over(k-1)!}\biggl{(}{\biggl{\|}{\partial^{k}f\over\partial x^{k-1}\partial y}\biggr{\|}_{\infty}}+{\biggl{\|}{\partial^{k+1}f\over\partial x^{k}\partial y}\biggr{\|}_{\infty}}\biggr{)}\mathbb{E}(|x|^{k}:|x|\geqslant K_{2})$$

$$\displaystyle\quad+{2(k+1)K_{1}\over k!}\left(K_{2}{\biggl{\|}{\partial^{k}f\over\partial y^{k}}\biggr{\|}_{\infty}}\mathbb{E}|y|^{k}+{\biggl{\|}{\partial^{k+1}f\over\partial x^{k}\partial y}\biggr{\|}_{\infty}}\mathbb{E}|x|^{k}\right)$$

$$\displaystyle\quad+{(k+1)K_{1}\over k!}\left({\biggl{\|}{\partial^{k}f\over\partial y^{k}}\biggr{\|}_{\infty}}\mathbb{E}(|y|^{k+1}:|y|\geqslant K_{2})+{\biggl{\|}{\partial^{k+1}f\over\partial x^{k}\partial y}\biggr{\|}_{\infty}}\mathbb{E}|x|^{k}\right)$$

$$\displaystyle\quad+{(k+1)\over k!}\left(K_{2}{\biggl{\|}{\partial^{k}f\over\partial y^{k}}\biggr{\|}_{\infty}}\mathbb{E}(|x|:|x|\geqslant K_{1})\,\mathbb{E}|y|^{k}+{\biggl{\|}{\partial^{k+1}f\over\partial x^{k}\partial y}\biggr{\|}_{\infty}}\mathbb{E}(|x|^{k+1}:|x|\geqslant K_{1})\right).$$

$$F_n:=\log Z_n;{\psi }_n:=\frac{F_n}{|V_n|};p_n:=\mathbb{E}{\psi }_n,$$

$$\mathrm{Var}(F_n)⩽C_n|V_n|+o(|V_n|).$$

$${\gamma }_n(s):=\mathbb{E}[{\mathbb{E}}^{\ast }{F}_n(G){]}^2,\text{with}{F}_n(G)=\log Z_n(G),$$

$$\frac{\mathrm{d}{\gamma }_n}{\mathrm{d}s}(s)$$

$$\displaystyle\biggl{|}\mathbb{E}\biggl{[}{g_{x}\over\sqrt{s}}\,\mathbb{E}^{*}F_{n}(G)\mathbb{E}^{*}{\partial F_{n}(G)\over\partial G_{x}}\biggr{]}$$

$$⩽\frac{C_1}{\sqrt{s}}\mathbb{E}(|g_x{|}^2:|g_x|⩾\epsilon (\mu -h{)}^{-1})+\frac{C_2}{\sqrt{s}}\epsilon (\mu -h{)}^{-1}$$

$$\displaystyle\biggl{|}{\mathbb{E}^{*}F_{n}(G)}\mathbb{E}^{*}{g_{x}^{*}\over\sqrt{1-s}}{\partial F_{n}(G)\over\partial G_{x}}$$

$$⩽\frac{C_3}{\sqrt{1-s}}\mathbb{E}(|g_x{|}^2:|g_x|⩾\epsilon (\mu -h{)}^{-1})+\frac{C_4}{\sqrt{1-s}}\epsilon (\mu -h{)}^{-1},$$

$$\displaystyle\biggl{|}{{\rm d}\gamma_{n}\over{\rm d}s}(s)-\sum_{x}\mathbb{E}\left[{1\over\sqrt{s}}\,{\partial\over\partial g_{x}}\mathbb{E}^{*}F_{n}(G)\,\mathbb{E}^{*}{\partial F_{n}(G)\over\partial G_{x}}-{\mathbb{E}^{*}F_{n}(G)\over\sqrt{1-s}}\,\mathbb{E}^{*}{\partial\over\partial g_{x}^{*}}{\partial F_{n}(G)\over\partial G_{x}}\right]\biggl{|}$$

$$\displaystyle\leqslant\biggl{(}{C_{1}\over\sqrt{s}}+{C_{3}\over\sqrt{1-s}}\biggr{)}\sum_{x}\mathbb{E}(|g_{x}|^{2}:|g_{x}|\geqslant\varepsilon(\mu-h)^{-1})+\biggl{(}{C_{2}\over\sqrt{s}}+{C_{4}\over\sqrt{1-s}}\!\biggr{)}|V_{n}|\varepsilon(\mu-h)^{-1}$$

$$=:\Theta (n,s,\epsilon ,\mu ).$$

$$\frac{\partial F_n(G)}{\partial g_x}=\sqrt{s}\frac{\partial F_n(G)}{\partial G_x};\frac{\partial F_n(G)}{\partial g_x^{\ast }}=\sqrt{1-s}\frac{\partial F_n(G)}{\partial G_x},$$

$$\displaystyle\biggl{|}{{\rm d}\gamma_{n}\over{\rm d}s}(s)-\sum_{x}\mathbb{E}\Big{(}\mathbb{E}^{*}{\partial F_{n}(G)\over\partial G_{x}}\Big{)}^{2}\biggl{|}\leqslant\Theta(n,s,\varepsilon,\mu)\,.$$

$$\frac{\mathrm{d}{\gamma }_n}{\mathrm{d}s}(s)⩽{\mu }^2|V_n|+\Theta (n,s,\epsilon ,\mu ).$$

$$\mathrm{Var}(F_n)⩽{\mu }^2|V_n|+{\int }_0^1\Theta (n,s,\epsilon ,\mu )\text{d}s$$

$$\displaystyle=\big{[}\mu^{2}+(C_{2}+C_{4})\varepsilon(\mu-h)^{-1}\big{]}|V_{n}|+(C_{1}+C_{3})\sum_{x}\mathbb{E}(|g_{x}|^{2}:|g_{x}|\geqslant\varepsilon(\mu-h)^{-1})$$

$$=:C_n(\epsilon ,\mu )|V_n|+o(|V_n|),$$

$${\Delta }_n:=\frac{1}{|V_n|}\sum _{x\in V_n}{g}_x{\sigma }_x$$

$$\displaystyle\nu(\Delta_{n})\mbox{$\;\longrightarrow_{{\small n}}\;$~{}}{\partial p\over\partial\mu}(\beta,\mu);\quad\mathbb{E}|\langle\Delta_{n}\rangle-\nu(\Delta_{n})|\mbox{$\;\longrightarrow_{{\small n}}\;$~{}}0\,.$$

$$\nu\big{(}|\Delta_{n}-\nu(\Delta_{n})|\big{)}\mbox{$\;\longrightarrow_{{\small n}}\;$~{}}0\,.$$

$$\displaystyle\biggl{|}{\partial\langle\sigma_{x};\sigma_{y}\rangle\over\partial g_{y}}\biggr{|}\leqslant 2\mu;\quad\biggl{|}{\partial^{2}\langle\sigma_{x};\sigma_{y}\rangle\over\partial g_{x}\partial g_{y}}\biggr{|}\leqslant 6\mu^{2};\quad\biggl{|}{\partial^{3}\langle\sigma_{x};\sigma_{y}\rangle\over\partial g_{x}^{2}\partial g_{y}}\biggr{|}\leqslant 24\mu^{3}\,.$$

Full text

1. Introduction

In statistical mechanics, the random field Ising model (RFIM) [13, 16] is considered one of the simplest non-trivial models that belongs to a class of disordered systems in which the disorder is coupled to the order parameter of the system. This model is under intensive investigation and has been studied from several aspects. For example, it is expected that many properties, as the Parisi ultrametricity (see [22, 24]) and the (extended) Ghirlanda-Guerra identities (see [1, 12, 19]), in disordered spin models should not depend on the particular distribution of the coupling constants. These properties are known to hold in several mean-field spin glass models, such as the Sherrington-Kirkpatrick model [26] and generic mixed $p$-spin models, see, e.g., [3]. The ultrametricity property was predicted by Parisi in [24] as an attempt to describe the expected behavior of the model and it still remains an unsolved mathematical problem. On the other hand, in [12] it was proven rigorously that the Ghirlanda-Guerra identities hold (in the infinite volume limit) in some approximate sense; for some specific choice of perturbed parameters [27]. Results involving the ultrametricity property in spin glass models can be found, for example, in [3, 18, 20, 22, 28].

The main goal of this paper is to remove the hypothesis of Gaussian disorder and to show that the Parisi ultrametricity is valid in RFIMs under mild assumptions on the disorder. To this end, results similar to the ones in Chatterjee (2015) [7] are proven for the RFIM with independent disorder in the case that the field strength is a small perturbation. Afterward, we added to the random field asymptotically vanishing non-Gaussian perturbations, and for this generalized model, we derived the extended Ghirlanda-Guerra identities, for the random models containing $p$-spin terms, for all $p\geqslant 1$. These identities combined with the main theorem of Panchenko (2011) [22] establishes ultrametricity.

This paper is organized as follows. In Section 2, we present the random models and state our main result. Sections 3 and 4, are dedicated to prove this paper’s main result. In Appendix, we provide a proof for the integration by parts formula used in this paper.

2. Statement of the result

Given $n\geqslant 1$, let $V_{n}=\mathbb{Z}^{d}\cap[1,n]^{d}$, $d\geqslant 1$, be a finite subset of vertices of $d$-dimensional hypercubic lattice with cardinality denoted by $|V_{n}|$. The (random) Gibbs measure of the ferromagnetic RFIM on the set of spin configurations $\{\pm 1\}^{V_{n}}$ is given by

[TABLE]

where $\langle xy\rangle$ denotes the set of ordered pairs in $V_{n}$ of nearest neighbors, $\beta>0$ and $\mu-h>0$ (with $\mu>0$), called inverse temperature and field strength, respectively. The partition function $Z$ appearing in the definition of ${G}$ is a normalizing factor and $g_{x}$’s are independent random variables (that collectively are called the disorder) with zero-mean and unit-variance. Additionally, we assumed that the field strength is a small perturbation with the following decay rate,

[TABLE]

for any $\varepsilon>0$. RFIMs with presence of disorder and field strength satisfying condition (2) appeared in Auffinger and Chen (2016) [3]. In the reference [25] the authors have studied the behaviour of RFIM with disorder having similar decay rate but different from ours. Additionally, the field strength used in [25] remains unchanged with respect to the volume.

Moreover, as in Panchenko (2013) [23] and Talagrand (2011) [28], we add asymptotically vanishing perturbations to the Hamiltonian corresponding to Gibbs measure in (1) to define the random Gibbs measure ${G}_{\bm{\alpha}}$ on $\{\pm 1\}^{V_{n}}$ with the following perturbing Hamiltonian

[TABLE]

with

[TABLE]

where the right-sided summation is over all $(x_{1},\ldots,x_{p})\in\otimes_{i=2}^{p}V_{n}$. Here, the sequence of numbers $(c_{n})$ is such that c_{n}\mbox{;\longrightarrow_{{\small n}};~{}}0, the sequence $\bm{\alpha}=(\alpha_{p})$ is given and satisfies $|\alpha_{p}|\leqslant 1$, and the disorder $(\xi_{x_{1},\ldots,x_{p}})$ consists of i.i.d. real-valued random variables $\xi_{x_{1},\ldots,x_{p}}$, for $p\geqslant 2$, with zero-mean and unit-variance. Note that when $\bm{\alpha}=\bm{0}$, ${G}_{\bm{\alpha}}(\sigma)={G}(\sigma)$ for all $\sigma\in\{\pm 1\}^{V_{n}}$.

For a function $f:(\{\pm 1\}^{V_{n}})^{m}\to\mathbb{R}$, $m\geqslant 1$, we define

[TABLE]

The randomness of the $g_{x}$’s and $\xi_{x_{1},\ldots,x_{p}}$’s will be represented by the measure $\gamma$ on $\mathbb{R}^{V_{n}}\times\mathbb{R}^{\otimes_{i\geqslant 2}V_{n}}$. Following the notation of Talagrand (2003) [29], we write

[TABLE]

averaging on the realizations of the disorder, where $\langle\,\cdot\,\rangle_{\bm{\alpha};g=u,\xi=v}$ is the Gibbs expectation defined by setting $g_{x}$ and $\xi_{x_{1},\ldots,x_{p}}$ in $\langle\,\cdot\,\rangle_{\bm{\alpha}}$ to be $u_{x}$ and $v_{x_{1},\ldots,x_{p}}$ for each $p$, respectively, for each $x,x_{i}\in V_{n}$. Since $\sqrt{\mathbb{E}H^{2}_{n;p}(\sigma)}\leqslant\sqrt{|V_{n}|}$, where $H_{n,p}$ is as in (4), it follows from Lemma $3.6{{}^{\prime\prime}}$ of Loève (1951) [17] that the series $c_{n}\sum_{p\geqslant 2}\alpha_{p}2^{-p}H_{n;p}(\sigma)$ converges almost surely. Therefore the Hamiltonian $c_{n}\alpha_{p}2^{-p}H_{n;p}(\sigma)$ is well-defined almost surely.

A collection $\sigma^{1},\sigma^{2},\ldots$ of configurations which are independent and identically distributed with respect to the Gibbs measure (3) are known as replicas. The spin overlap between two replicas $\sigma^{\ell}$, $\sigma^{\ell^{\prime}}$ is defined as

[TABLE]

Note that $|R_{\ell,\ell^{\prime}}|\leqslant 1$, $R_{\ell,\ell}=1$ and that the infinite random array $R=(R_{\ell,\ell^{\prime}})_{\ell,\ell^{\prime}\geqslant 1}$ is symmetric, non-negative definite and weakly exchangeable. Following [11], an infinite random array $R$ with such properties is known as Gram-de Finetti matrix. The array $R$ is said to satisfy the extended Ghirlanda-Guerra identities (see [1, 12, 19]) if for any $m\geqslant 2$, any bounded measurable function f=f\big{(}(R_{\ell,\ell^{\prime}})_{1\leqslant\ell,\ell^{\prime}\leqslant m}\big{)}, and for any bounded measurable function $\psi:\mathbb{R}\to\mathbb{R}$ of one overlap,

[TABLE]

In order to lighten the notation, we will omit the subscript $\bm{\alpha}$, when $\bm{\alpha}=\bm{0}$.

We are finally ready to state our main result.

Theorem 1**.**

Under the assumption (2), the following hold:

$R_{2,3}\geqslant\min\{R_{1,2},R_{1,3}\}$* almost surely w.r.t. the infinite volume limit of $\nu$;*

• 2)

$R_{2,3}\geqslant\min\{R_{1,2},R_{1,3}\}$* almost surely w.r.t. the infinite volume limit of $\nu_{\bm{\alpha}}$, for all $p\geqslant 1$ in (4).*

Therefore for RFIMs, defined by (1)-(2) and (2)-(4), the array $R$ is ultrametric.

The major step of the proof of Theorem 1 requires a generalization of the Gaussian integration by parts, as in [3, 6] and [10].

Our first tool will be the following proposition. Its proof appears in Auffinger and Chen (2016) [3], Lemma 2.2.

Proposition 2.1**.**

Let $y$ be a random variable such that its first $k\geqslant 2$ moments match those of a Gaussian random variable. Suppose that $f\in C^{k+1}(\mathbb{R})$. For any $K\geqslant 1$;

[TABLE]

Our second tool will be the following proposition. Its proof is presented in Appendix. This result is new and can be seen as a generalization of Proposition 2.1 for the bivariate case.

Proposition 2.2**.**

Let $x,y$ be two independent random variables such that their first $k\geqslant 2$ moments match those of a Gaussian random variable. Suppose that $f\in C^{k+2}(\mathbb{R}^{2})$. For any $K_{1},K_{2}\geqslant 1$;

[TABLE]

The rest of this paper is devoted to the proof of Theorem 1.

3. Proof of Item 1

In this section we show the validity of the ultrametricity property by proving lack of replica symmetry breaking in the RFIM defined by (1)-(2). In other words, we show that the degree to which the values of a spin overlap differ from its expectation value vanishes in the thermodynamic limit of the RFIM with weak disorder (2). Results involving absence of replica symmetry breaking in the RFIM and related models can be found, for example, in [7, 14, 15, 25].

Using the same notation as in Chatterjee (2015) [7], for each $(\beta,\mu)\in(0,\infty)^{2}$, let us define

[TABLE]

where $\psi_{n}$ is known as the free-energy density.

The proof of the next result was inspired in the proofs of Lemmas 4.1 and 2.5 of references [25] and [15], respectively.

Lemma 3.1**.**

For each $(\beta,\mu)\in(0,\infty)^{2}$ and any $\varepsilon>0$ there is $C_{n}(\varepsilon,\mu)=O(\sqrt{|V_{n}|})$ so that

[TABLE]

Proof.

We will use the same notation as in Lemma 4.1 of [25]. Given two disorders $g=(g_{x})$ and $g^{*}=(g_{x}^{*})$ consisting of independent random variables, for each $s\in[0,1]$, we define a new random field $G=(G_{x})$ as follows: $G_{x}\coloneqq\sqrt{s}\,g_{x}+\sqrt{1-s}\,g_{x}^{*},\ x\in V_{n}.$ Moreover, we also consider the following generating function

[TABLE]

where $\mathbb{E}$ and $\mathbb{E}^{*}$ denote expectation over $g$ and $g^{*}$, respectively.

For each $s\in(0,1)$, a simple computation shows that

[TABLE]

Using the integration by parts (see Proposition 2.1), with $k=2$ and $K=\varepsilon(\mu-h)^{-1}$, for any $\varepsilon>0$, we obtain

[TABLE]

and

[TABLE]

where $C_{1},C_{2},C_{3}$ and $C_{4}$ are positive constants.

By combining the last two inequalities with (9), we get the following estimate

[TABLE]

On other hand, by using the identities

[TABLE]

in (3), for each $s\in(0,1)$, we have

[TABLE]

Since $\mathbb{E}(\mathbb{E}^{*}{\partial F_{n}(G)\over\partial G_{x}})^{2}\leqslant\mu^{2}$, it follows from the above inequality that

[TABLE]

Combining the relation $\mathrm{Var}(F_{n})=\gamma_{n}(1)-\gamma_{n}(0)=\int_{0}^{1}{{\rm d}\gamma_{n}\over{\rm d}s}(s)\,\text{d}s$ with the above inequality we obtain

[TABLE]

thus completing the proof. ∎

For any $n\geqslant 1$, let

[TABLE]

be the part of the energy function (1) due to the disorder. Let ${\mathcal{A}}$ be the countable set of all $(\beta,\mu)\in(0,\infty)^{2}$ such that $\textstyle{\partial p\over\partial\mu^{-}}(\beta,\mu)\neq{\partial p\over\partial\mu^{+}}(\beta,\mu).$

Proposition 3.2**.**

For any $(\beta,\mu)\in{\mathcal{A}}^{c}$, we have

[TABLE]

Proof.

Note that the same result was proved in Lemma 2.7 of Chatterjee (2015) [7] regardless of distribution of the disorder. Three ingredients are fundamental to prove this result: first, the convexity of $\psi_{n}$. Second, the variance of $F_{n}$ does not grow faster than $\sqrt{|V_{n}|}$, which is guaranteed by Proposition 3.1. Third, the limit $p=\lim_{n\rightarrow\infty}p_{n}$ exists and is differentiable at $\mu$, which is guaranteed by Lemmas 2.1 and 2.7 in [7]. Therefore, the proof follows. ∎

The Proposition 3.2 plays an important role in the proof of the next result.

Lemma 3.3**.**

Under the hypothesis of Theorem 1, for any $(\beta,\mu)\in{\mathcal{A}}^{c}$, we have

[TABLE]

Proof.

Let $\langle\sigma_{x};\sigma_{y}\rangle\coloneqq\langle\sigma_{x}\sigma_{y}\rangle-\langle\sigma_{x}\rangle\langle\sigma_{y}\rangle$ be the truncated two-point correlation. A straightforward computation shows that

[TABLE]

Let $\langle\,\cdot\,\rangle_{g_{x}=u,g_{y}=v}$ be the Gibbs expectation defined by setting $g_{x}$ and $g_{y}$ in $\langle\,\cdot\,\rangle$ to be $u$ and $v$ respectively, and $F_{x,y}(u,v)\coloneqq\langle\sigma_{x};\sigma_{y}\rangle_{g_{x}=u,g_{y}=v}$. Integrating by parts (see Proposition 2.2) with $f_{x,y}(u,v)=\mathbb{E}F_{x,y}(u,v)$, $k=2$ and $K_{1}=K_{2}=\varepsilon(\mu-h)^{-1}$, for any $\varepsilon>0$, gives

[TABLE]

Dividing this inequality by $|V_{n}|^{2}$ and summing over all $x,y\in V_{n}$, the triangle inequality gives

[TABLE]

Combining the last estimate with the following inequality

[TABLE]

it follows from (2), and the fact that $\varepsilon$ is arbitrary, that \nu\big{(}|\Delta_{n}-\langle\Delta_{n}\rangle|\big{)}\mbox{;\longrightarrow_{{\small n}};~{}}0. Finally, the proof follows from triangle inequality and Proposition 3.2. ∎

Proposition 3.4**.**

Under the hypothesis of Theorem 1, for any $(\beta,\mu)\in{\mathcal{A}}^{c}$, the following ergodic property holds:

[TABLE]

Proof.

Let

[TABLE]

be the Hamiltonian corresponding to the Gibbs measure (1) of our RFIM. It is well-known that

[TABLE]

see [21, 23]. Therefore follows from the second triangular inequality that

[TABLE]

By combining this inequality with (12), (2), and Lemma 3.3, the proof of the ergodic property follows. ∎

Our next step is to prove the Ghirlanda-Guerra identities for our model, which is precisely stated in the next lemma. In order to obtain these identities the Lemma 3.3 plays an important role.

Lemma 3.5**.**

Given $m\geqslant 2$, let $f:\mathbb{R}^{m(m-1)/2}\rightarrow[-1,1]$ be a bounded measurable function of the overlaps (6) that not change with $n$. Then, under assumption (2), the Ghirlanda-Guerra identities with $\psi\equiv{\rm Id}$ and $\bm{\alpha}=\bm{0}$ in (7) are satisfied at almost all $(\beta,\mu)$. That is, if $f$ is as above,

[TABLE]

Proof.

Since $\|f\|_{\infty}\leqslant 1$, we have

[TABLE]

where $\Delta_{n}$ is as in (11).

On the other hand, let $\langle\,\cdot\,\rangle_{g_{x}=u}$ be the Gibbs expectation defined by setting $g_{x}$ in $\langle\,\cdot\,\rangle$ to be $u$ and $F_{x}(u)\coloneqq\langle\,\sigma_{x}^{1}\,f\,\rangle_{g_{x}=u}$. Using (5), a straightforward computation shows that

[TABLE]

We remark that the above formula has also appeared in Chen (2019) [10].

Since $|{\partial^{j}F_{x}(u)\over\partial u^{j}}|\leqslant(2m\mu)^{j}$ and $\mathbb{E}g_{x}^{2}=1$, the integration by parts formula (Proposition 2.1) with $f_{x}(u)\coloneqq\mathbb{E}F_{x}(u)$, $k=2$ and $K=\varepsilon(\mu-h)^{-1}$, gives

[TABLE]

for any $\varepsilon>0$. Dividing the above inequality by $|V_{n}|$, summing over $x\in V_{n}$, applying the triangle inequality, and using (14) with $j=1$ we get that

[TABLE]

Therefore, from both the assumption (2) and arbitrariness of $\varepsilon$, it follows that

[TABLE]

In the particular case where $f=1$ and $m=1$, we obtain that |\nu(\Delta_{n}(\sigma^{1}))-(\mu-h)\nu(R_{1,1}-R_{1,2})|\mbox{;\longrightarrow_{{\small n}};~{}}0. By combining this with (13) and (15), it follows from Lemma 3.3 that

[TABLE]

∎

The last ingredient of the proof of Item (1) of Theorem 1 is the following result by Auffinger and Chen (2016) [3] which proves the self-averaging of the spin overlap.

Proposition 3.6** ([3]).**

Under assumption (2), for any $(\beta,\mu)\in(0,\infty)^{2}$,

[TABLE]

where $m(\sigma)=\sum_{x}\sigma_{x}/|V_{n}|$ define the magnetization.

Proof of Item 1 of Theorem 1. At this point, we already have established the validity of the key ingredients: the self-averaging of the overlap (Proposition 3.6) and the Ghirlanda-Guerra identities (Lemma 3.5). By a quite standard argument (see, e.g., Chatterjee (2015) [7]) these two mentioned facts imply the replica symmetry breaking does not occur in the RFIM. In other words, the spin overlap is concentrated at its expectation value and then the ultrametricity property follows. ∎

4. Proof of Item 2

To prove this item, which is the ultrametricity property, the strategy is to combine the results of the previous section with the following known results in the literature, concerning to mixed $p$-spin models for $p\geqslant 2$:

• •

the main theorem of Panchenko (2010) [21];

• •

the universality of Ghirlanda-Guerra identities in mixed $p$-spin models; see Chen (2019) [10];

• •

and the main theorem of Panchenko (2011) [22].

For any $n\geqslant 1$, let $\widetilde{\xi}_{x_{1},\ldots,x_{p}}\coloneqq g_{x}\delta_{p,1}+{\xi}_{x_{1},\ldots,x_{p}}(1-\delta_{p,1})$, where $\delta$ is the Kronecker delta function, and

[TABLE]

the part of the energy function (3) due to the disorder. When $p=1$, $\Delta_{n;p}$ coincides with the random function $\Delta_{n}$ given in (11).

Next, we prove a very important and technical result used to obtain the extended Ghirlanda-Guerra identities, which is the following limit

[TABLE]

This result is established in Panchenko (2010) [21] and Auffinger and Chen (2018) [2]. Actually, in [2] a stronger result is obtained but at the cost of requiring a stronger hypothesis than the one used in [21]. To be more precise, in [21] Panchenko obtain the above limit under the following assumptions:

• a)

|V_{n}|^{-1}\,\mathbb{E}\big{|}\log Z_{\bm{\alpha}}(t)-\mathbb{E}\log Z_{\bm{\alpha}}(t)\big{|}\mbox{;\longrightarrow_{{\small n}};~{}}0,

• b)

|V_{n}|^{-1}\,\mathbb{E}\log Z_{\bm{\alpha}}(t)\mbox{;\longrightarrow_{{\small n}};~{}}P(t)  in some neighborhood of  $t$, and

• c)

$P(t)$  is differentiable at  $t$.

On the other hand, this limit is obtained in Auffinger and Chen’s work [2] under the following assumptions:

• i)

there exists a nonrandom function $P:I=(t-\varepsilon,t+\varepsilon)\to\mathbb{R}$, for some $\varepsilon>0$, such that for any $t^{\prime}\in I$, |V_{n}|^{-1}\log Z_{\bm{\alpha}}(t^{\prime})\mbox{;\longrightarrow_{{\small n}};~{}}P(t^{\prime}) almost surely, and

• ii)

$P(t)$ is differentiable at $t$,

We can show that conditions i) and ii) implies a), b) and c). But, as observed before, the conclusion in [2] is stronger than the one in [21]. However, in our setting is more convenient to establish the validity of conditions a), b) and c).

Proposition 4.1**.**

For any $p\geqslant 1$

[TABLE]

Proof.

For almost all $(\beta,\mu)$, the convergence in (16) is proved in Lemma 3.3 for $p=1$. For $p\geqslant 2$, let $Z_{\bm{\alpha}}(t)$ be the partition function associated to the Hamiltonian $c_{n}\alpha_{p}2^{-p}H_{n;p}(\sigma)$ with $\alpha_{p}=t$. Under the condition of two matching moments of the disorder $({\xi}_{x_{1},\ldots,x_{p}})$, Lemma 8 of Carmona and Hu (2006) [6] shows that the limit of the free-energy function $|V_{n}|^{-1}\log Z_{\bm{\alpha}}(t)$ does not depend on the particular distribution of environment. Therefore this limit is everywhere differentiable in $t$, for fixed $t$. By using the martingale difference argument by Burkholder’s inequality [5] and integration by parts formula, Chen (2019) [10] has proved that

[TABLE]

Therefore, the hypotheses a), b), and c) of the main theorem in [21] are satisfied for the mixed $p$-spin model with $p\geqslant 2$. Consequently, (16) is also valid for all $p\geqslant 2$.

∎

Proof of Item 2 of Theorem 1. In order to derive the extended Ghirlanda-Guerra identities (7), we can assume, without loss of generality, that $|\psi|\leqslant 1$. Since the space of real-valued compactly supported continuous functions on $[-1,1]$ is dense in the space of Lebesgue integrable functions on $[-1,1]$ and since any continuous function can be uniformly approximated on $[-1,1]$ by a polynomial, it is sufficient to prove the extended Ghirlanda-Guerra identities (7) for any higher moments of the overlap, that is, for all $p\geqslant 1$,

[TABLE]

for any bounded measurable function f=f\big{(}(R_{\ell,\ell^{\prime}})_{1\leqslant\ell,\ell^{\prime}\leqslant m}\big{)}:\mathbb{R}^{m(m-1)/2}\rightarrow[-1,1].

Indeed, firstly we consider the case $p=1$. In this case, the validity of identities (17), follows from Proposition 4.1 and Lemma 3.5. Secondly, the validity of (17) for $p\geqslant 2$, is consequence of Proposition 4.1 and the integration by parts formula (as in the proof of Theorem 2.1-Step 2, in Chen (2019) [10]). Therefore, in the infinite volume limit, the Gibbs measure of the RFIM defined in (2)-(4) satisfies the extended Ghirlanda-Guerra identities (7) for any $p\geqslant 1$.

Finally, since $R=(R_{\ell,\ell^{\prime}})_{\ell,\ell^{\prime}\geqslant 1}$ is a Gram-de Finetti matrix [11] and since the extended Ghirlanda-Guerra identities (7) are satisfied, the main theorem of Panchenko (2011) [22] establishes the ultrametricity. ∎

Acknowledgements

We would like to thank S. Chatterjee for raising the question on the ultrametricity property in the RFIM in general non-Gaussian disorders. We would like to offer our special thank to L. Cioletti for many valuable comments and careful reading of this manuscript. This study was financed in part by the Coordenação de Aperfeiçoamento de Pessoal de Nível Superior

• Brasil (CAPES) - Finance Code 001. Jamer Roldan was supported by CNPq.

Appendix

Proof of Proposition 2.2.

Let $g(x,y)\coloneqq xf(x,y)$. By Taylor expanding the function $g$ at $(k-1)$-th and $k$-th orders, we have

[TABLE]

Similarly, considering the Taylor expansion of ${\partial g(x,y)\over\partial y}$ up to its $(k-1)$-th order, we get

[TABLE]

where $a(y),b(y),c(y)$ are some functions depending only on $y$. From (18) and (20) it follows that

[TABLE]

where $h_{n}(y)\coloneqq({y^{n+1}\over n!}-{y^{n-1}\over(n-1)!})$. From (19) and (20), we get

[TABLE]

Expanding ${\partial f(x,y)\over\partial y}$ up to $(k-1)$-th and $k$-th orders, we have

[TABLE]

Again, expanding ${\partial^{2}f(x,y)\over\partial x\partial y}$ up to $(k-1)$-th orders, we obtain

[TABLE]

where $\widetilde{a}(x),\widetilde{b}(x),\widetilde{c}(x)$ are some functions depending only on $x$. From (23) and (25),

[TABLE]

From (24) and (25) we have

[TABLE]

Summing (21) and (26) one get that

[TABLE]

By adding up (22) and (27) we get

[TABLE]

Defining

[TABLE]

we obtain from (28) and (29) the following two inequalities:

[TABLE]

and

[TABLE]

Taking expectation of $|I_{1}|$ on $D\coloneqq\{|x|\geqslant K_{1},|y|\geqslant K_{2}\}$, with $K_{1},K_{2}\geqslant 1$, for the first inequality, we get

[TABLE]

Now, by taking expectation of $|I_{2}|$ on the set $\{|x|\leqslant K_{1}\}$ we obtain

[TABLE]

Similarly, by taking expectation of $|I_{2}|$ on the set $\{|y|\leqslant K_{2}\}$ we see that

[TABLE]

Since $x,y$ are two random variables such that their first $k\geqslant 2$ moments match those of a Gaussian random variable, it follows that $\mathbb{E}h_{n}(x)=\mathbb{E}h_{n}(y)=0$, $n=1,\ldots,k-1$. Then,

[TABLE]

Finally, combining the above inequality with (30), (31) and (Proof of Proposition 2.2.), we conclude the proof.

∎