Formal expansions in stochastic model for wave turbulence 2: method of diagram decomposition (complete version)

TL;DR

This paper develops a diagram decomposition method to analyze formal series solutions of the damped cubic NLS equation with random forcing, focusing on wave turbulence limits where amplitude diminishes and domain size grows.

Contribution

It introduces a new diagram decomposition approach for formal series in stochastic wave turbulence analysis, extending previous work on the damped cubic NLS equation.

Findings

Formal series solutions are analyzed under wave turbulence limits.

The diagram decomposition method clarifies the structure of solutions.

Results contribute to understanding wave turbulence in stochastic PDEs.

Abstract

In this paper we continue to study small amplitude solutions of the damped cubic NLS equation, driven by a random force (the study was initiated in our previous work [A.Dymov, S.Kuksin, Comm. Math. Phys.'2021] and continued in [A.Dymov, S.Kuksin, A.Maiocchi, S.Vladuts, arXiv:2104.11967]). We write solutions of the equation as formal series in the amplitude and discuss the behaviour of this series under the wave turbulence limit, when the amplitude goes to zero, while the space-period goes to infinity.

Equations

$$u(x)=L^{-d/2}{\sum }_{s\in {\mathbb{Z}}_L^d}{v}_s{e}^{2\pi is\cdot x},{\mathbb{Z}}_L^d=L^{-1}{\mathbb{Z}}^d.$$

$$v=\mathcal{F}(u),v_s=L^{-d/2}{\int }_{{\mathbb{T}}_L^d}u(x)e^{-2\pi is\cdot x}dx\text{for}s\in {\mathbb{Z}}_L^d.$$

$$\frac{{{\partial}}}{{{\partial}}t}u+i\Delta u-i\lambda\,\big{(}|u|^{2}-2\|u\|^{2}\big{)}u=0,\qquad x\in{\mathbb{T}}^{d}_{L},$$

$$\frac{{{\partial}}}{{{\partial}}t}u+i\Delta u-i\lambda\,\big{(}|u|^{2}-2\|u\|^{2}\big{)}u=-\nu\mathfrak{A}(u)+\sqrt{\nu}\frac{{{\partial}}}{{{\partial}}t}\eta^{\omega}(t,x).$$

$$\mathfrak{A}(u(x))=L^{-d/2}{\sum }_{s\in {\mathbb{Z}}_L^d}{\gamma }_s{v}_s{e}^{2\pi is\cdot x},v=\mathcal{F}(u),{\gamma }_s={\gamma }^0(|s{|}^2).$$

$${\eta }^{\omega }(t,x)=L^{-d/2}{\sum }_{s\in {\mathbb{Z}}_L^d}b(s){\beta }_s^{\omega }(t)e^{2\pi is\cdot x},$$

$$\begin{split}\dot{u}+i\nu^{-1}\Delta u-i\rho\,\big{(}|u|^{2}-2\|u\|^{2}\big{)}u&=-\mathfrak{A}(u)+\dot{\eta}^{\omega}(\tau,x),\\ \eta^{\omega}(\tau,x)&=L^{-d/2}\,{\sum}_{s\in\mathbb{Z}^{d}_{L}}b(s)\beta^{\omega}_{s}(\tau)e^{2\pi is\cdot x},\end{split}$$

$$L\to \infty \text{and}\nu \to 0,$$

$$n_s(\tau ):=\mathbb{E}|v_s(\tau ){|}^2,s\in {\mathbb{Z}}_L^d,\text{ where }v=\mathcal{F}(u).$$

$$\dot{v}_{s}-i\nu^{-1}|s|^{2}v_{s}+\gamma_{s}v_{s}=i\rho L^{-d}\Big{(}{\sum}_{s_{1},s_{2},s_{3}\in\mathbb{Z}^{d}_{L}}\delta^{\prime 12}_{3s}v_{s_{1}}v_{s_{2}}\bar{v}_{s_{3}}-|v_{s}|^{2}v_{s}\Big{)}+b(s)\dot{\beta}_{s}\,,$$

$$\delta^{\prime 12}_{3s}={\delta^{\prime}}^{s_{1}s_{2}}_{s_{3}s}=\left\{\begin{array}[]{ll}1,&\text{ if $s_{1}+s_{2}=s_{3}+s$ and $\{s_{1},s_{2}\}\neq\{s_{3},s\}$}\,,\\ 0,&\text{otherwise}.\end{array}\right.$$

$${\delta }_{3s}^{\prime 12}=1$$

$${\sum }_{s_1,s_2,s_3\in {\mathbb{Z}}_L^d}{{\delta }^{\prime }}_{3s}^{12}=:{\sum }_{s_1}{\sum }_{s_2}{{\delta }^{\prime }}_{3s}^{12}.$$

$$v_s(-T)=0\text{mbox}foralls\in {\mathbb{Z}}_L^d.$$

$$v_s=\sum _{i=0}^{\infty }{\rho }^i{v}_s^{(i)},v_s^{(i)}(-T)=0,$$

$${\dot{v}}_s^{(0)}(\tau )-i{\nu }^{-1}|s{|}^2{v}_s^{(0)}(\tau )+{\gamma }_s{v}_s^{(0)}(\tau )=b(s){\dot{\beta }}_s(\tau ),v_s^{(0)}(-T)=0,$$

$$\dot{v}^{(1)}_{s}-i\nu^{-1}|s|^{2}v^{(1)}_{s}+\gamma_{s}v^{(1)}_{s}=iL^{-d}\Big{(}{\sum}_{s_{1},s_{2},s_{3}}\delta^{\prime 12}_{3s}v^{(0)}_{s_{1}}v^{(0)}_{s_{2}}\bar{v}^{(0)}_{s_{3}}-|v_{s}^{(0)}|^{2}v_{s}^{(0)}\Big{)},$$

$$\chi_{d}^{N}(\nu)=\left\{\begin{array}[]{cl}-\ln\nu&\quad\mbox{if}\quad(N,d)=(3,2)\mbox{ or }(2,1)\\ 1&\quad\mbox{otherwise}\end{array}\right.$$

$$C^{\#}(x)\le c_n(1+|x|{)}^{-n}\forall x\in {\mathbb{R}}^k,$$

$$\big{|}{\mathbb{E}}v_{s}^{(m)}(\tau_{1})\bar{v}_{s}^{(n)}(\tau_{2})\big{\rvert}\leq C^{\#}(s)\big{(}\nu^{-2}L^{-2}+\nu^{\min(\lceil N/2\rceil,d)}\chi_{d}^{N}(\nu)\big{)},$$

$${\tilde{J}}_s({\tau }_1,{\tau }_2;\mathfrak{F})={\int }_{{\mathbb{R}}^N}dl{\int }_{{\mathbb{R}}^{dN}}dz{F}_s^{\mathfrak{F}}({\tau }_1,{\tau }_2;l,z)e^{i{\nu }^{-1}{\Omega }^{\mathfrak{F}}(l,z)}$$

$$|{\tilde{J}}_s(\mathfrak{F})|\le C^{\#}(s){\nu }^{\min (\lceil N/2\rceil ,d)}{\chi }_d^N(\nu )\forall \mathfrak{F},$$

$$L\ge {\nu }^{-2-\delta }\text{mbox}forsome\delta >0,$$

$${\tilde{v}}_s(\tau ):=v_s^{(0)}(\tau )+\rho v_s^{(1)}(\tau )+{\rho }^2{v}_s^{(2)}(\tau ),s\in {\mathbb{Z}}_L^d,$$

$${\tilde{n}}_s(\tau )=n_s^{(0)}(\tau )+\rho n_s^{(1)}(\tau )+{\rho }^2{n}_s^{(2)}(\tau )+{\rho }^3{n}_s^{(3)}(\tau )+{\rho }^4{n}_s^{(4)}(\tau ),$$

$$n_s^{(k)}=\sum _{\begin{array}{c}0\le k_1,k_2\le 2:\\ k_1+k_2=k\end{array}}\mathbb{E}{v}_s^{(k_1)}{\bar{v}}_s^{(k_2)}.$$

$$\rho =\sqrt{\epsilon }{\nu }^{-1/2},\text{mbox}sothat{\rho }^2\nu =\epsilon .$$

$${\tilde{n}}_s=n_s^{(0)}+\epsilon {\tilde{n}}_s^{(2)}(\epsilon ),\text{mbox}where{\tilde{n}}_s^{(2)}={\nu }^{-1}{n}_s^{(2)}+O(\epsilon )\sim 1$$

$$\dot{m}(\tau ,s)=-2{\gamma }_{s}m(\tau ,s)+2b(s{)}^2+\epsilon K(m(\tau ,\cdot ))(s),s\in {\mathbb{R}}^d;$$

$$2\pi {\int }_{{}^s{\Sigma }_{\ast }}\frac{d{s}_{1}d{s}_2{\mid }_{{}^s{\Sigma }_{\ast }}{m}_1{m}_2{m}_3{m}_s}{\sqrt{|s_1-s{|}^2+|s_2-s{|}^2}}\left(\frac{1}{m_s}+\frac{1}{m_3}-\frac{1}{m_1}-\frac{1}{m_2}\right),$$

Full text

1. Introduction and results

We continue the study of the nonlinear cubic Shrödinger equation (NLS) on a torus of large period, dumped by a viscosity and driven by a random force, initiated in our papers [9, 10]. Results of this work are crucially used in [9, 10] and are of independent interest for further study of wave turbulence (WT) in the equation which we consider and in similar ones. In particular, our approach applies to the NLS equations with higher order nonlinearities.

Below we recall the setup, state the main theorem and then discuss its relevance for works [9, 10] and for the theory of WT. The theorem’s proof is outlined in Section 2 and is given in details in Sections 3-7. In Section 8 we establish some lemmas, used in the proof, and discuss a number of related results.

For a general theory of wave turbulence we refer to works [19, 22, 20], and for an additional discussion of the relevance of our results for WT – to the introduction in [9].

1.1. The setting

Let ${\mathbb{T}}^{d}_{L}={\mathbb{R}}^{d}/(L{\mathbb{Z}}^{d})$ be a $d$-dimensional torus, $d\geq 1$, of period $L\geq 1$. We denote by $\|u\|$ the normalized $L_{2}$-norm of a complex function $u$ on $\mathbb{T}_{L}^{d}$, $\|u\|^{2}=L^{-d}\int_{{{\mathbb{T}}}^{d}_{L}}|u(x)|^{2}\,dx\,,$ and write the Fourier series of $u$ as 111 The scaling factor $L^{-d/2}$ is convenient for our calculations.

[TABLE]

Here the vector of Fourier coefficients $v=(v_{s})_{s\in\mathbb{Z}^{d}_{L}}$ is given by the Fourier transform of $u(x)$,

[TABLE]

The Parseval identity reeds $\|u\|^{2}=L^{-d}\,{\sum}_{s\in\mathbb{Z}^{d}_{L}}|v_{s}|^{2}.$ We will study solutions $u(t,x)$ whose norms satisfy $\|u(t,\cdot)\|\sim 1$ as $L\to\infty$.

Our goal is to study the cubic NLS equation with modified nonlinearity

[TABLE]

where $u=u(t,x)$, $\Delta=(2\pi)^{-2}\sum_{j=1}^{d}({{\partial}}^{2}/{{\partial}}x_{j}^{2})$ and $\lambda\in(0,1]$ is a small parameter. The modification of the nonlinearity by the term $2i\lambda\|u\|^{2}u$ keeps the main features of the standard cubic NLS equation, reducing some non-crucial technicalities, see the introduction to [9] for a detailed discussion.

The objective of WT is to study solutions of (1.2) under the wave turbulence limit $L\to\infty$ and $\lambda\to 0$ on long time intervals, usually while ”pumping the energy to low modes and dissipating it in high modes”. To make this rigorous, following Zakharov-L’vov [21], we consider the NLS equation (1.2) dumped by a (hyper) viscosity and driven by a random force:

[TABLE]

Here $\nu\in(0,1/2]$ is a small parameter and $\mathfrak{A}$ is the dissipative linear operator

[TABLE]

The real function $\gamma^{0}(y)\geq 1$ is smooth and has at most a polynomial growth at infinity, together with all its derivatives. 222For example $\mathfrak{A}=(1-\Delta)^{r^{*}}$ for some $r^{*}\geq 0$. The random noise $\eta^{\omega}$ is given by the Fourier series

[TABLE]

where $\{\beta_{s}(t),s\in{{\mathbb{Z}}}^{d}_{L}\}$ are standard independent complex Wiener processes 333 i.e. $\beta_{s}=\beta_{s}^{1}+i\beta_{s}^{2}$, where $\{\beta_{s}^{j},s\in{{\mathbb{Z}}}^{d}_{L},j=1,2\}$ are standard independent real Wiener processes. and $b(x)$ is a Schwartz function on ${{\mathbb{R}}}^{d}\supset\mathbb{Z}^{d}_{L}$. 444Often it is assumed that the intensity $b(x)$ of the noise $\eta^{\omega}$ is non-negative, but we do not impose this condition. Still note that if $b(x)\equiv 0$, then our results become trivial since below we will provide (1.3) with the zero initial conditions. If the function $\gamma^{0}(y)$ grows at infinity as $|y|^{r_{*}}$ with $r_{*}$ sufficiently big, then eq. (1.3) is well posed. In difference with [9, 10], where we assumed $d\geq 2$, in this work we allow $d=1$ and impose less restrictions on the function $\gamma^{0}$.

We will study the equation on time intervals of order $\nu^{-1}$, so it is convenient to pass from $t$ to the slow time $\tau=\nu t$. Then eq. (1.3) takes the form

[TABLE]

where $\rho=\lambda\nu^{-1}$, the upper-dot stands for $d/d\tau$ and $\{\beta_{s}(\tau),\,s\in\mathbb{Z}^{d}_{L}\}$ is another set of standard independent complex Wiener processes. Below we use $\rho$, $\nu$ and $L$ as the parameters of the equation.

In the context of eq. (1.5), the objective of WT is to study its solutions $u$ when

[TABLE]

as well as main characteristics of the solutions such as the energy spectrum

[TABLE]

In [9, 10] and in this paper we study formal decompositions in $\rho$ of solutions to eq. (1.5), regarding $\rho$ as an independent parameter, and those of the energy spectrum $n_{s}$. In Section 1.3, where we present the results of [9] and [10], we assume that $\rho\sim\nu^{-1/2}$ or $\rho\leavevmode\nobreak\ \sim L$ correspondingly, since due to [9] and [10] exactly under these scalings the quadratic in $\rho$ part of the decomposition of the energy spectrum has a non-trivial behaviour under the limit (1.6), taken in the regimes, considered in [9] and [10].

1.2. The result

Let us take the Fourier transform of eq. (1.5):

[TABLE]

$s\in{{\mathbb{Z}}}^{d}_{L}$, where $|s|$ stands for the Euclidean norm of a vector $s$ and

[TABLE]

Note that

[TABLE]

In view of the factor ${\delta^{\prime}}^{12}_{3s}$, in the sum in (1.8) the variable $s_{3}$ is a function of $s_{1},s_{2},s$. That is why below we write the sum as

[TABLE]

We fix $0\leq T\leq+\infty$ and study equation (1.5)=(1.8) with zero initial condition at $\tau=-T$, so

[TABLE]

Let us write solutions of (1.8), (1.11) as formal series in $\rho$: 555As we mentioned above, in paper [9] we choose $\rho\sim\nu^{-1/2}$ to be large. However, the first $d$ coefficients $v_{s}^{(i)}$ of series (1.12) are in fact of order $\lesssim\nu^{i/2}$ and we expect that the others also satisfy this estimate, see in Section 1.4. So there is a hope that the series converges. A similar situation takes place in paper [10], where we assume $\rho\sim L$.

[TABLE]

where $v_{s}^{(i)}=v_{s}^{(i)}(\tau;\nu,L)$. The processes $v_{s}^{(0)}$ satisfy the linear equations

[TABLE]

so these are independent Gaussian processes. The linear in $\rho$ term $v_{s}^{(1)}$ satisfies

[TABLE]

and $v^{(1)}_{s}(-T)=0$. Other terms $v_{s}^{(n)}$ can be found recursively in a similar way.

Our goal is to study correlations of the terms of series (1.12) under the limit (1.6). Repeating the argument of Lemma 2.3 from [9] it is straightforward to show that ${\mathbb{E}}v_{s}^{(m)}(\tau_{1})v_{s^{\prime}}^{(n)}(\tau_{2})=0$ for any $s,s^{\prime}\in\mathbb{Z}^{d}_{L}$ and that ${\mathbb{E}}v_{s}^{(m)}(\tau_{1})\bar{v}_{s^{\prime}}^{(n)}(\tau_{2})=0$ for $s\neq s^{\prime}$. Our main result, given below in Theorem 1.1, establishes upper bounds for the correlations with $s=s^{\prime}$. Its particular case is crucially used in [9], see below in Section 1.3. Our second main result is Theorems 5.7, 5.9, which are steps in the proof of Theorem 1.1. There we develop an instrumental representation of the correlations above in a form of explicit sums, convenient for further analysis. This result is crucially used in [10].

To state Theorem 1.1 we use the function

[TABLE]

(we recall that $0<\nu\leq 1/2$). For an $x\in\mathbb{R}$ we denote $\lceil x\rceil=\min\{n\in\mathbb{Z}:n\geq x\};$ and denote by $C^{\#}(\cdot)$ various positive continuous functions on $\mathbb{R}^{k}$, $k\in\mathbb{N}$, fast decaying at infinity:

[TABLE]

for every $n\in\mathbb{N}$, with a suitable constant $c_{n}>0$.

Theorem 1.1**.**

Assume that $d\geq 2$. Then for any integers $m,n\geq 0$ there exists a function $C^{\#}$ such that for each $s\in\mathbb{Z}^{d}_{L}$ and $\tau_{1},\tau_{2}\geq-T$ we have

[TABLE]

where $N=m+n$. For $d=1$ the assertion above remains true if we modify the r.h.s. of (1.17) by adding the term $C^{\#}(s)L^{-1}$.

In our work [9] estimate (1.17) (with $d\geq 2$) is used in the situation when $\nu^{-2}L^{-2}\leq\nu^{\min(\lceil N/2\rceil,d)}$ – then it implies that ${\mathbb{E}}|v_{s}^{(m)}(\tau)|^{2}\leq C^{\#}(s)\nu^{\min(m,d)}$, uniformly in $L$. The above restriction on the parameters well agrees with a postulate, usually accepted by the WT community, that in (1.6) $L$ should go fast to infinity when $\nu\to 0$.

To prove the theorem we approximate, with accuracy $\nu^{-2}L^{-2}$, the expectation ${\mathbb{E}}v_{s}^{(m)}\bar{v}_{s}^{(n)}$ by a finite sum of integrals, independent from $L$ and parametrized by a suitable class $\mathfrak{F}_{m,n}$ of Feynman diagrams (the cardinality of the set $\mathfrak{F}_{m,n}$ grows factorially with $m+n$). Next we study the behaviour of each integral as $\nu\ll 1$. To this end, in Section 5, for every diagram $\mathfrak{F}\in\mathfrak{F}_{m,n}$ we find a special coordinate system such that in these coordinates the integral, corresponding to $\mathfrak{F}$, has the simple explicit form:

[TABLE]

with explicit functions $\Omega^{\mathfrak{F}}$ and $F^{\mathfrak{F}}_{s}$, see Theorem 5.9. Here the phase function $\Omega^{\mathfrak{F}}$ is a real quadratic form in $z$ (usually degenerate) which linearly depends on $l$. The density function $F^{\mathfrak{F}}_{s}$ is Schwartz in $(s,z)$, while as a function of $l$ it is piecewise smooth and fast decays as $|l|\to\infty$. Then in Sections 6 and 7 we show that

[TABLE]

which implies (1.17). More precisely, there we prove abstract Theorem 6.2 where we get an upper estimate for a class of fast oscillating integrals of the form (1.18).

The diagram decomposition and transformation of coordinates from Sections 5 can be rather straightforwardly generalized to NLS equations with higher order polynomial nonlinearities. Together with abstract Theorem 6.2 this allows to generalize Theorem 1.1 for these equations.

Behaviour of energy spectra of small-amplitude solutions for nonlinear equations when the space-period of the solution grows to infinity has been intensively studied by physicists since the 1960’s. Last decade this problem draw attention of mathematicians starting [18], see [1, 2, 3, 4, 5, 6, 13, 14] and discussions in introduction to [3, 9, 6]. In some of the mentioned works decompositions of energy spectra to series of the form (1.12) where examined, evoking the techniques of Feynman diagrams, inspired by the works [11, 12] (where the diagram techniques were used to analyse expansions of solutions in different, but related situations). Our diagram presentation is based on similar ideas, but is rather different from those in the mentioned works. In particular, as explained above, our approach leads to an explicit presentation of the correlations ${\mathbb{E}}v_{s}^{(m)}\bar{v}_{s}^{(n)}$ which may be treated by asymptotical methods, developed in the present paper and in [9, 10].

In the next two subsections we briefly present the results of [9, 10] and explain the role of the present paper for the research started in [9, 10].

1.3. Kinetic limit

Let $u(\tau,x)$ be a solution of equation (1.5), so that v(\tau)=\mathcal{F}\big{(}u(\tau)\big{)} is a solution of (1.8), and $n_{s}(\tau)$ be its energy spectrum (see (1.7)). One of objective of WT is to study the behaviour of the energy spectrum under the limits (1.6), with a properly scaled $\rho$, see [21, 22, 19]. The order of limits is not quite clear but, as we mentioned above, in WT it is usually accepted that $L$ grows to infinity fast when $\nu$ goes to zero (for a discussion of other possibilities see introductions to [9] and [10] and see below). Accordingly in [9] we assume that

[TABLE]

or that first $L\to\infty$ and then $\nu\to 0$. It is traditional in WT to study the quadratic in $\rho$ truncation of series (1.12), postulating that it well approximates small amplitude solutions. Thus motivated, in [9] we considered the process

[TABLE]

which we called a quasisolution of equation (1.8), (1.11), and examined its energy spectrum $\widetilde{n}=(\widetilde{n}_{s})_{s\in\mathbb{Z}^{d}_{L}}$, $\widetilde{n}_{s}={\mathbb{E}}|\widetilde{v}_{s}|^{2}$,

[TABLE]

where

[TABLE]

Assuming $d\geq 2$, we proved that $n_{s}^{(2)}\thicksim\nu$ as $\nu\to 0$ uniformly in $L$ satisfying (1.19), while a simple computation showed that $n_{s}^{(0)}\sim 1$ and $n_{s}^{(1)}=0.$ This indicates that the right scaling for $\rho$ is such that $\rho^{2}\nu\thicksim 1$ as $\nu\to 0$, and accordingly in [9] we chose $\rho$ to be of the form

[TABLE]

Here $0<\varepsilon\leq 1$ is a fixed small number (independent from $\nu$ and $L$). Using estimate (1.17) under the scaling (1.19), (1.22) we found that $|\rho^{3}n_{s}^{(3)}|,$ $|\rho^{4}n_{s}^{(4)}|\lesssim\varepsilon^{2}$. Accordingly,

[TABLE]

when $\nu\to 0$, since $n_{s}^{(2)}\sim\nu$.

Thus, the parameter $\varepsilon$ measures the properly scaled amplitude of oscillations, described by eq. (1.5), so indeed it should be small for the methodology of WT apply to the solutions. Next we establish that for $\nu$ small in terms of $\varepsilon$ the energy spectrum $\widetilde{n}_{s}(\tau)$ is $\varepsilon^{2}$-close to a function $m(\tau,s)$, which is a unique solution of the damped/driven wave kinetic equation:

[TABLE]

$m(-T)=0.$ Here $K(m)(s)$ is the wave kinetic integral

[TABLE]

where $m_{i}=m(s_{i})$, $m_{s}=m(s)$ with $s_{3}=s_{1}+s_{2}-s$ (cf. (1.9)). The set ${}^{s}\Sigma_{*}$ is the resonant surface (cf. (2.5)-(2.6))

[TABLE]

and $ds_{1}\,ds_{2}\mid_{{}^{s}\Sigma_{*}}$ stands for the volume element on ${}^{s}\Sigma_{*}$, corresponding to the euclidean structure in $\mathbb{R}^{2d}$. This is exactly the wave kinetic integral which appears in physical works on WT to describe the 4–waves interaction, see [22], p. 71 and [19], p. 91.

In particular from what was said it follows that in the regime (1.19), the scaling $\rho\sim\nu^{-1/2}$ is the only scaling of $\rho$ when the energy spectrum admits a non-trivial kinetic limit of order one; see [9] for a more detailed discussion.

In [10] we consider the opposite to (1.19) order of limits, when first $\nu\to 0$ and then $L\to\infty$. As in [9], we study the energy spectrum (1.21) of the quasisolution, but now analysis of the term $n_{s}^{(2)}$ shows that the correct scaling is $\rho=\sqrt{\varepsilon}L$. 666In [10] the notation is slightly different: $\varepsilon$ is written as $\varepsilon^{2}$. Besides when $d=2$, $\rho$ should be multiplied by the factor $(\ln L)^{-1/2}$. In this setting, crucially using Theorem 5.7, we establish a result, similar to the discussed above that in [9], but the limiting WKE (1.23) should be modified. In particular, the WKE in [10] is non-autonomous.

Results of the present paper also are used in our work in progress, where by means of the KAM-technique we are aiming to show that the results in [9, 10] remain true for the energy spectrum of the exact solution for eq. (1.5), under the corresponding limits (and not only for the quasisolution). See subsection What next? in [10, Section 1.3].

1.4. Discussion of results. Higher order truncations

Let us come back to the decomposition (1.12) and accordingly write the energy spectrum $n_{s}$, defined in (1.7), as formal series in $\rho$:

[TABLE]

Here the terms $n_{s}^{0}$, $n_{s}^{1}$ and $n_{s}^{2}$ coincide with $n_{s}^{(0)}$, $n_{s}^{(1)}$ and $n_{s}^{(2)}$ in (1.21), while the terms $n_{s}^{3}$ and $n_{s}^{4}$ are slightly different (but still satisfy the estimate (1.26) below). According to Theorem 1.1,

[TABLE]

where for definiteness we assumed that $d\geq 2$. For $k\leq 2d$ in the exponent in r.h.s. the minimum with $d$ can be removed, so (1.26) takes the form

[TABLE]

In particular, (1.27) holds for $k\leq 4$ since $d\geq 2.$ Using estimate (1.27) with $k\leq 2d$ it is straightforward to show that under the scaling (1.22) the kinetic limit for quasisolution, established in [9] and discussed in the previous section, takes place for the truncation $\sum_{i=0}^{m}\rho^{i}v_{s}^{(i)}$ of series (1.12) not only of order $m=2$ as in (1.20) but of any order $2\leq m\leq d$ (with the same kinetic equation (1.23) and at least with the same accuracy $\varepsilon^{2}$, once $L$ is sufficiently large in terms of $\nu^{-1}$).

If it was true that (1.27) is valid for any $k$, so that $|\rho^{k}n_{s}^{k}|\lesssim\varepsilon^{k/2}$ under the scaling (1.22) with $L\gg\nu^{-1}$, then the kinetic limit would hold for truncations of any order $m\geq 2$. On the contrary, if for some $k$ we have $|n^{k}|_{1}:=L^{-d}\sum_{s}|n^{k}_{s}|\gg\nu^{k/2}$, then this is not the case since the term $\rho^{k}n^{k}$ explodes as $\nu\to 0$.

The question if estimate (1.27) holds for high $k$ is complicated. Namely, estimate (1.26) follows from (1.17), and in Theorem 1.1 we established the latter by showing that each integral $\widetilde{J}_{s}(\mathfrak{F})$ is bounded by the r.h.s. of (1.17). We are able to prove that this estimate is optimal at least for some integrals $\widetilde{J}_{s}(\mathfrak{F})$ in the sense that for some of them the minimum with $d$ in the exponent cannot be removed, so for these terms estimate (1.27) fails. However, in the sum $\sum_{\mathfrak{F}}\widetilde{J}_{s}(\mathfrak{F})$ that approximates ${\mathbb{E}}v_{s}^{(m)}\bar{v}_{s}^{(n)}$ there are some non-trivial cancellations in a few first orders of its decomposition in $\nu$ which could lead to the validity of (1.27). In Section 8.1 we discuss this phenomenon in more detail.

Notation. By $\mathbb{Z}_{+}^{n}$ we denote the set of integer $n$-vectors with non-negative components. For vectors in $\mathbb{R}^{n}$ we denote by $|v|$ their Euclidean norms and by $v\cdot u$ – the Euclidean scalar product. By $C^{\#},C_{1}^{\#},\widetilde{C}_{1}^{\#},\dots$ we denote various positive continuous functions satisfying (1.16) for any $n\in\mathbb{N}$. If we want to indicate that a function $C^{\#}$ depends on a parameter $m$, we write it as $C^{\#}_{m}$. The functions $C^{\#},C_{1}^{\#},\ldots$ and constants $C,C_{1},\ldots$ never depend on the parameters $\nu,L,\rho,\varepsilon$ and the moments of time $T,\tau,\tau_{1},\ldots$ unless otherwise stated. In Section 1.4 of [9] it is shown that for any function $C^{\#}(x,y)$, where $(x,y)\in\mathbb{R}^{d_{1}+d_{2}}$, $d_{1},d_{2}\geq 1$, there exist functions $C_{1}^{\#}(x)$, $C_{2}^{\#}(y)$ such that

[TABLE]

Acknowledgements. AD was funded by a grant from the Russian Science Foundation (Project 20-41-09009), and SK – by Agence Nationale de la Recherche through the grant 17-CE40-0006.

2. Preliminaries and outline of the proof

2.1. Approximate equation

We start by getting rid in the r.h.s. of equation (1.8) of the term $-i\rho L^{-d}|v_{s}|^{2}v_{s}$ since the latter is small but rather inconvenient for analysis. With this term being dropped, equation (1.8), (1.11) reads

[TABLE]

As in (1.12), we decompose its solution $y_{s}$ into a formal series in $\rho$:

[TABLE]

Then $y_{s}^{(0)}=v_{s}^{(0)}$ is a Gaussian process satisfying equation (1.13), $y_{s}^{(1)}(\tau)$ satisfies equation (1.14) without the term $|v_{s}^{(0)}|^{2}v_{s}^{(0)}$, etc.

In Section 8.6 we prove that correlations of the processes $v_{s}^{(i)}$ and $y_{s}^{(i)}$ are $L^{-d}$-close:

Proposition 2.1**.**

For any $m,n\geq 0$ there exists a function $C^{\#}$ such that for each $s\in\mathbb{Z}^{d}_{L}$ and $\tau_{1},\tau_{2}\geq-T$,

[TABLE]

In view of Proposition 2.1 it suffices to establish inequality (1.17) (and its 1d analogy) when the processes $v_{s}^{(m)}$ are replaced by $y_{s}^{(m)}$. Accordingly below we study the latter instead of the former.

2.2. Interaction representation

It is convenient to work in the interaction representation, that is in the $a$-variables defined as

[TABLE]

We set

[TABLE]

(cf. (1.24)). Then equation (2.1), written in the $a$-variables reads

[TABLE]

where $\{\beta_{s}\}$ is another set of standard independent complex Wiener processes. Formal expansion (2.2) takes the form

[TABLE]

where $a_{s}^{(j)}(\tau)=y_{s}^{(j)}(\tau)e^{-i\nu^{-1}\tau|s|^{2}}.$ For a process $a_{s}^{(j)}(\tau)$ we call the integer $j\geq 0$ its degree,

[TABLE]

and call the vector $s\in\mathbb{Z}_{L}^{d}$ its index. Since $a_{s}^{(k)}(\tau)$ differs from $y_{s}^{(k)}(\tau)$ by the factor $e^{-i\nu^{-1}\tau|s|^{2}}$, then $|{\mathbb{E}}a_{s}^{(m)}\bar{a}_{s}^{(n)}|=|{\mathbb{E}}y_{s}^{(m)}\bar{y}_{s}^{(n)}|$. Thus, in view of (2.3), it suffices to establish Theorem 1.1 with the processes $v_{s}^{(m)}$ replaced by $a_{s}^{(m)}$:

Theorem 2.2**.**

Assume that $d\geq 2$. Then for any integers $m,n\geq 0$ there exists a function $C^{\#}$ such that for $s\in\mathbb{Z}^{d}_{L}$ and any $\tau_{1},\tau_{2}\geq-T$,

[TABLE]

where $N=m+n$ and $\chi_{d}^{N}$ is defined in (1.15). If $d=1$, the assertion above holds if we modify the r.h.s. of (2.8) by adding the term $C^{\#}(s)L^{-1}$. Moreover, the correlation ${\mathbb{E}}a_{s}^{(m)}(\tau_{1})\bar{a}_{s}^{(n)}(\tau_{2})$ extends to a Schwartz function of $s\in\mathbb{R}^{d}\supset\mathbb{Z}^{d}_{L}$ which satisfies the same estimate (2.8).

Note that for $N=1$ the l.h.s. of (2.8) vanishes, see in Section 4.2 and Lemma 2.3 in [9].

The processes $a_{s}^{(i)}$ can be computed inductively. Namely, $a_{s}^{(0)}$ satisfies the linear equation

[TABLE]

so

[TABLE]

and $\{a^{(0)}_{s}(\tau),s\in\mathbb{Z}^{d}_{L}\}$, are independent Gaussian processes. The process $a^{(1)}$ satisfies

[TABLE]

so

[TABLE]

is a Wiener chaos of third order (see [16]). For what follows, it is convenient to introduce in (2.6) a fictitious index $s_{4}$ and write the function $\mathcal{Y}_{s}$ from (2.6) as

[TABLE]

where $\delta_{s_{4}}^{s}$ is the Kronecker symbol and we denote

[TABLE]

Then (2.10) takes the form

[TABLE]

Similar, for $m\geq 1$,

[TABLE]

This is a Wiener chaos of order $2m+1$.

In what follows, till the end of Section 5.3, we do not use the explicit form (2.11) of the function $\theta$ (except the next section, where we give an outline of the proof of Theorem 2.2). Instead we assume that the processes $a^{(m)}_{s}$ are given by equation (2.13) where

[TABLE]

In particular, we may choose for $\theta$ the function $\theta(x,t)=\mathbb{I}_{\{0\}}(x)$, where $\mathbb{I}_{\{0\}}(x)$ is the indicator function of the point $x=0$ (in this case $\theta$ does not depend on $t$). This choice of $\theta$ is used in our paper [10], where we use that in some sense a properly scaled function $(s_{1},s_{2},s_{3},s_{4})\mapsto\exp(i\nu^{-1}\tau\omega^{12}_{34})$ with $\tau\neq 0$, regarded as a “generalised function on $(\mathbb{Z}^{d}_{L})^{4}$”, converges to $\mathbb{I}_{\{0\}}(\omega^{12}_{34})$ as $\nu\to 0$.

We will use correlations of the process $a_{s}^{(0)}$. By (2.9), for any $\tau_{1},\tau_{2}\geq-T$ and $s,s^{\prime}\in\mathbb{Z}^{d}_{L}$,

[TABLE]

(see (2.8) in [9] for the calculation).

2.3. Outline of the proof of Theorem 2.2

We divide a quite detailed outline of the proof to four steps, approximately corresponding to Sections 3, 4, 5 and 6+7. In the outline we assume that $\theta$ takes the form (2.11).

Step 1. Let us write the variable $a_{s}^{(m)}(\tau_{1})$ in the form (2.13) and then apply the Duhamel formula (2.13) to the variables $a_{s_{i}}^{(m_{i})}(l)$ in its r.h.s. with the degrees $m_{i}\geq 1$ (see (2.7)). Iterating the procedure, we represent $a_{s}^{(m)}(\tau_{1})$ by a sum of terms of the form

[TABLE]

here and below we call terms of this form sums. The integrating zone in (2.16) is a convex polyhedron in $[-T,\tau_{1}]^{m}$ and the summation is taken over the vectors $s_{1},\ldots,s_{4m}\in\mathbb{Z}^{d}_{L}$ which are subject to certain linear relations, following from the factors $\delta^{\prime 12}_{34}$ and $\delta^{s}_{s_{4}}$ in (2.13). The summand in brackets is a product of functions $e^{-\gamma_{s^{\prime}}(l_{k}-l_{r})}$, $\exp(i\nu^{-1}\omega^{s^{\prime}_{1}s^{\prime}_{2}}_{s^{\prime}_{3}s^{\prime}_{4}}l_{q})$ and the processes $[a^{(0)}_{s^{{}^{\prime\prime}}}(l_{j})]^{*}$ where $a^{*}$ is either $a$ or $\bar{a}$, with various indices $1\leq k,r,q,j\leq m$ and $s^{\prime},s^{\prime}_{i},s^{\prime\prime}\in\{s_{1},\ldots,s_{4m}\}$. This product is of degree $2m+1$ with respect to the process $a^{(0)}$ (see (2.12) for the case $m=1$).

To each sum $\widetilde{I}_{s}$ as in (2.16) we associate a diagram $\mathfrak{D}$ as in fig. 1(a), constructed according to a rule, explained below; we then write $\widetilde{I}_{s}=\widetilde{I}_{s}(\mathfrak{D})$. The root of a diagram $\mathfrak{D}$, associated with a sum $\widetilde{I}_{s}$, is the vertex $a^{(m)}_{s}(\tau_{1})$. If $a^{(p)}_{s^{\prime}}(t)$, $p\geq 1$, is any vertex of the diagram $\mathfrak{D}$, then it generates a block of four vertices, three of which correspond to a choice of the three terms $a^{(m_{1})},a^{(m_{2})},\bar{a}^{(m_{3})}$ in the decomposition (2.13) of $a^{(p)}_{s^{\prime}}(t)$. These three are called real vertices, and the vertex $a^{(p)}_{s^{\prime}}$ is called the parent of the block. The fourth vertex of the block, denoted by $\bar{w}_{s^{\prime\prime}}$, is called the conjugated virtual vertex. It corresponds to the factor $\delta^{s^{\prime}}_{s^{\prime\prime}}$ in (2.13), is coupled with the parent $a^{(p)}_{s^{\prime}}(t)$ by an edge and one can think that its role is to couple the block with its parent. Similarly, each vertex $\bar{a}^{(p)}_{s^{\prime}}(t)$ with degree $p\geq 1$ generates a block, where the three real vertices correspond to a choice of the terms $\bar{a}^{(m_{1})},\bar{a}^{(m_{2})},a^{(m_{3})}$ in the formula, conjugated to (2.13), and the fourth vertex $w_{s^{\prime\prime}}$ is the non-conjugated virtual vertex (corresponding to factor $\delta_{s^{\prime\prime}}^{s^{\prime}}$).

We denote by $\mathfrak{D}_{m}$ the set of all diagrams $\mathfrak{D}$ associated with different terms $\widetilde{I}_{s}$ from (2.16), sum of which gives $a_{s}^{(m)}(\tau_{1})$. 777As we will see, structure of the set $\mathfrak{D}_{m}$ depends on $a^{(m)}_{s}(\tau_{1})$ through the index $m$ only. This will become more clear in Section 3, where we will construct $\mathfrak{D}_{m}$ carefully. Then in every diagram $\mathfrak{D}\in\mathfrak{D}_{m}$ there are $m$ blocks, and in each block there are two conjugated and two non-conjugated vertices. By convention, in each block we draw and enumerate first the non-conjugated vertices and then the conjugated. The virtual vertex is always positioned at the second or forth place, depending whether it is conjugated or not, see fig. 1(a).

We denote by $\xi_{1},\ldots,\xi_{2m}$ the indices $s_{j}\in\mathbb{Z}^{d}_{L}$, enumerating the non-conjugated vertices, and by $\sigma_{1},\ldots,\sigma_{2m}$ – those enumerating the conjugated vertices. Then, setting $\xi_{0}=s$, the diagram from fig. 1(a) takes the form as in fig. 1(b). For more examples see fig. 2, 3(a-c), where we write $c$ instead of $a$ and omit indices $\xi$ and $\sigma$.

Now the linear relations, imposed on the multi-indices $\xi_{i},\sigma_{j}$ with $i,j\geq 1$ in a sum (2.16) take the following compact form:

[TABLE]

where the first relation comes from the factor $\delta^{\prime 12}_{34}$ in (2.13) while the second one – from the factor $\delta_{s_{4}}^{s}$. Moreover, we see that in this notation the frequency of a fast rotating exponent, hidden in the term $(\dots)$ in (2.16) and arising from the factor $\theta(\omega^{12}_{34},\nu^{-1}l)=e^{i\nu^{-1}\omega^{12}_{34}l}$ in (2.13), also takes the compact form $i\nu^{-1}\sum_{j=1}^{m}\,\omega^{\xi_{2j-1}\xi_{2j}}_{\sigma_{2j-1}\sigma_{2j}}\,l_{j}$.

We have seen that

[TABLE]

where each sum $\widetilde{I}_{s}(\mathfrak{D})$ has the form (2.16). Similarly, $\bar{a}^{(n)}_{s}(\tau_{2})=\sum_{\bar{\mathfrak{D}}\in\overline{\mathfrak{D}}_{n}}\widetilde{I}_{s}(\bar{\mathfrak{D}})$, where $\overline{\mathfrak{D}}_{n}$ is the set of diagrams with the root $\bar{a}^{(n)}_{\sigma_{0}}(\tau_{2})$, $\sigma_{0}:=s$, constructed by the same inductive rule as diagrams from the set $\mathfrak{D}_{m}$ (see fig. 3(d)). 888Equivalently, the set $\overline{\mathfrak{D}}_{n}$ can be obtained by conjugating diagrams from the set $\mathfrak{D}_{n}$, then exchanging positions of conjugated and non-conjugated vertices in each block (we recall that, by convention, the non-conjugated vertices should always situate before the conjugated vertices), and exchanging the indices $\xi_{j}$ with $\sigma_{j}$, so that again the non-conjugated vertices are enumerated by the indices $\xi_{j}$ while the conjugated  – by the indices $\sigma_{j}$.

Finally, we introduce the set $\mathfrak{D}_{m}\times\overline{\mathfrak{D}}_{n}$ of diagrams $\mathfrak{D}=\mathfrak{D}^{1}\sqcup\bar{\mathfrak{D}}^{2}$, which are obtained by drawing side by side pairs $\mathfrak{D}^{1}$ and $\bar{\mathfrak{D}}^{2}$ with various $\mathfrak{D}^{1}\in\mathfrak{D}_{m}$ and $\bar{\mathfrak{D}}^{2}\in\overline{\mathfrak{D}}_{n}$. Here in the diagrams $\bar{\mathfrak{D}}^{2}$ the vertices (except the root) are enumerated by the indices $\xi_{2m+1},\ldots,\xi_{2N},\sigma_{2m+1},\ldots,\sigma_{2N}$, $N:=m+n$. Since the root of $\bar{\mathfrak{D}}^{2}$ is denoted $\sigma_{0}$, then all vertices of $\mathfrak{D}$ are enumerated by $4N+2$ indices $\xi_{i},\sigma_{j},$ where $0\leq i,j\leq 2N$ and always $\xi_{0}=\sigma_{0}=s$; see fig. 4. Then setting $I_{s}(\mathfrak{D})=\widetilde{I}_{s}(\mathfrak{D}^{1})\widetilde{I}_{s}(\bar{\mathfrak{D}}^{2})$, where $\mathfrak{D}=\mathfrak{D}^{1}\sqcup\bar{\mathfrak{D}}^{2}$, we find

[TABLE]

In fact, we do not write down the sums $\widetilde{I}_{s}$ but directly write the products $I_{s}$, see Lemma 3.1 for their explicit form.

Step 2. To estimate the r.h.s. of (2.18) we study separately the expectation ${\mathbb{E}}I_{s}(\mathfrak{D})$ for each diagram $\mathfrak{D}=\mathfrak{D}^{1}\sqcup\bar{\mathfrak{D}}^{2}\in\mathfrak{D}_{m}\times\overline{\mathfrak{D}}_{n}$. Due to (2.16),

[TABLE]

where the term in the brackets is obtained by taking the product of the terms $(\ldots)$ in the integrands of $\widetilde{I}_{s}(\mathfrak{D}^{1})$ and $\widetilde{I}_{s}(\bar{\mathfrak{D}}^{2})$, corresponding to the leaves of the diagram $\mathfrak{D}$. So, the brackets in (2.19) contain a product of $2N+2$ random variables of the form $[a_{s^{\prime}}^{(0)}(t)]^{*}$, multiplied by a deterministic factor. Since $a_{s^{\prime}}^{(0)}(t)$ are independent complex Gaussian random variables whose correlations are given by (2.15), then by the Wick theorem (see e.g. in [16]) the expectation ${\mathbb{E}}(\ldots)$ is a sum over different Wick-pairings of the non-conjugated variables $a_{\xi_{j}}^{(0)}(l_{k})$ with conjugated $\bar{a}_{\sigma_{r}}^{(0)}(l_{q})$, where

[TABLE]

The summands can be parametrised by Feynman diagrams $\mathfrak{F}$ (see e.g. in [16]), obtained from the diagram $\mathfrak{D}$ by connecting with an edge every pair of the Wick-coupled vertices $a_{\xi_{j}}^{(0)}(l_{k})$ and $\bar{a}_{\sigma_{r}}^{(0)}(l_{q})$, see fig. 5. So to every diagram $\mathfrak{D}$ correspond several Feynman diagrams $\mathfrak{F}$, one diagram for each Wick-pairing. By construction, every Feynman diagram decomposes to pairs of adjacent vertices, where in each pair either both vertices have zero (recall (2.7)) degree, or one vertex is virtual and another one has positive degree.

Therefore, by (2.18),

[TABLE]

where the sum is taken over the set $\mathfrak{F}_{m,n}$ of Feynman diagrams $\mathfrak{F}$ obtained via all possible Wick-couplings in various diagrams $\mathfrak{D}\in\mathfrak{D}_{m}\times\overline{\mathfrak{D}}_{n}$. Each (deterministic) sum $J_{s}(\mathfrak{F})$ has the form

[TABLE]

where the term in the brackets $(\dots)$ is obtained from those in (2.19) by a Wick-pairing associated to the diagram $\mathfrak{F}$. See Lemma 4.1 for an explicit form of the sums $J_{s}(\mathfrak{F})$.

Step 3. The indices $\xi_{i},\,\sigma_{j}$ in (2.21) are subject to the linear relations (2.17), where $1\leq j\leq N$ and in (2.17)(2) the diagram $\mathfrak{D}$ is replaced by some $\mathfrak{F}$ (see (2.20)). By (2.17)(2) (with $\mathfrak{D}$ replaced by $\mathfrak{F}$), the multi-index $\sigma=(\sigma_{1},\ldots,\sigma_{2N})$ is a function of the multi-index $\xi=(\xi_{1},\ldots,\xi_{2N})$, so it remains to study equation (2.17)(1), where we substitute $\sigma=\sigma(\xi)$. Analysing the diagrams $\mathfrak{F}$, forming the class $\mathfrak{F}_{m,n}$, we find an $\mathfrak{F}$-dependent affine parametrization of solutions $\xi$ for this equation by poly-vectors $z=(z_{1},\dots,z_{N})$, $z_{j}\in\mathbb{Z}^{d}_{L}$. Then we write the normalized sum $L^{-Nd}\sum_{\xi,\sigma}$ in (2.21) as L^{-Nd}\sum_{z}\, approximate it by an integral over $\mathbb{R}^{Nd}$ with accuracy $L^{-2}\nu^{-2}$, and accordingly find that

[TABLE]

Here $l=(l_{1},\ldots,l_{N})\in\mathbb{R}^{N}$, the density function $F_{s}^{\mathfrak{F}}$ (which we write down explicitly) is Schwartz in the variables $(s,\,z)$, piecewise smooth and fast decaying in $l$, while the phase function $\Omega^{\mathfrak{F}}$ is quadratic in $z$ and linear in $l$: $\Omega^{\mathfrak{F}}(l,z)=\sum_{1\leq i,j\leq N}\alpha^{\mathfrak{F}}_{ij}\,z_{i}\cdot z_{j}(l_{i}-l_{j}),$ where $z_{i}\cdot z_{j}$ is the scalar product in $\mathbb{R}^{d}$; see Theorem 5.9. The constant skew-symmetric matrix $\alpha^{\mathfrak{F}}=(\alpha^{\mathfrak{F}}_{ij})$, $\alpha_{ij}^{\mathfrak{F}}\in\{-1,0,1\},$ is given by an explicit formula. Usually it is degenerate, but all its rows and columns are non-zero vectors.

Step 4. The last step of the proof is to estimate integral (2.22) when $\nu\ll 1$. To this end, inspired by the stationary phase method, we apply to the latter the integral Parseval’s identity, involving the fast oscillating Gaussian kernel $e^{i\nu^{-1}\Omega^{\mathfrak{F}}}$. The task is made complicated by the degeneracy of the quadratic form $\Omega^{\mathfrak{F}}(l,\cdot)$. To handle it, in Sections 6, 7 and 8.8 we prove abstract theorems which allow to estimate integrals of a more general forms than (2.22).

3. Diagrams $\mathfrak{D}$ and formula for products $a^{(m)}_{s}\bar{a}^{(n)}_{s}$

In this section we construct the set of diagrams $\mathfrak{D}_{m}\times\overline{\mathfrak{D}}_{n}$, associated to the product $a^{(m)}_{s}(\tau_{1})\bar{a}^{(n)}_{s}(\tau_{2})$, and express the latter through the random processes $a^{(0)}_{k}$ and $\bar{a}^{(0)}_{k}$ (see Lemma 3.1). The construction is rather tedious but the main difficulties are notational: once the diagrammatic language is developed and convenient notation are introduced, proof of Lemma 3.1 becomes a simple computation.

3.1. The set of diagrams $\mathfrak{D}_{m}\times\overline{\mathfrak{D}}_{n}$

For integers $m,n\geq 0$ we define the set $\mathfrak{D}_{m}\times\overline{\mathfrak{D}}_{n}$ as a direct product of the sets of diagrams $\mathfrak{D}_{m}$ and $\overline{\mathfrak{D}}_{n}$, associated, correspondingly, with the variables $a_{s}^{(m)}$ and $\bar{a}_{s}^{(n)}$.

We start by constructing the set $\mathfrak{D}_{m}$, construction of $\overline{\mathfrak{D}}_{n}$ is similar. For example, the set $\mathfrak{D}_{1}$ consists of a unique diagram, which can be obtained from the diagram in fig. 3(a) by erasing the isolated vertex $\bar{c}_{0}^{(0)}$. Here each $c^{(p)}_{j}$ is regarded as an argument, which should be substituted by a variable $a_{\xi_{j}}^{(p)}(l)$, while $\bar{c}^{(p)}_{j}$ should be substituted by an $\bar{a}_{\sigma_{j}}^{(p)}(l^{\prime})$. E.g., this substitution, applied to the diagram in fig. 2(c), gives the diagram in fig. 1(b). The set $\mathfrak{D}_{2}$ consists of three diagrams in fig. 2. The set $\mathfrak{D}_{3}$ consists of 12 diagrams, two of which are given in fig. 3(b,c). Construction of the diagrams seems to be quite clear from these examples. Nevertheless, below we explain it in detail.

3.1.1. Agreements and first notation

Each diagram $\mathfrak{D}\in\mathfrak{D}_{m}$ consists of a number of vertices, some of them are coupled by edges. Each vertex is either non-conjugated or conjugated. The non-conjugated vertices are denoted by $c_{j}$ while the conjugated  – by $\bar{c}_{j}$. If we do not want to indicate whether a vertex is conjugated or not, we write it as $\hat{c}_{j}$. Each vertex is characterised by its degree which we often write as an upper index:

[TABLE]

and each vertex is either virtual or real. If a vertex $c_{j},\bar{c}_{j}$ or $\hat{c}_{j}$ is virtual, we may write it as $w_{j},\bar{w}_{j}$ or $\hat{w}_{j}$. Each virtual vertex $\hat{w}_{j}$ has zero degree,

[TABLE]

so we do not write for it the upper index. Real vertices of zero degree are called leaves.

The vertices of a diagram are organised in blocks of four, where the $k$-th block, $k\geq 1$, is $\{c_{2k-1},c_{2k},\bar{c}_{2k-1},\bar{c}_{2k}\}.$ Among these four vertices either the vertex $c_{2k}$ or the vertex $\bar{c}_{2k}$ is virtual, while the other are real. If the vertex $c_{2k}$ is virtual then we denote this block by $B_{k}$,

[TABLE]

and call it the non-conjugated $k$-th block. If the vertex $\bar{c}_{2k}$ is virtual, we denote

[TABLE]

and call it the conjugated $k$-th block. If we do not want to emphasize whether a block is conjugated or not, we denote it by $\hat{B}_{k}$. We define the degree of each block as the sum of degrees of the forming it vertices. So

[TABLE]

(since deg$\,\bar{w}_{2k}=0$), and similarly for $\deg\bar{B}_{k}$. E.g., the diagram from fig. 2(a) has two conjugated blocks $\bar{B}_{1}$ and $\bar{B}_{2}$, where $\deg\bar{B}_{1}=1$ and $\deg\bar{B}_{2}=0.$

3.1.2. Construction of the diagrams

If a vertex $c_{j}$ is coupled with a virtual vertex $\bar{w}_{2k}$ of a block $\bar{B}_{k}$ then we say that $c_{j}$ generates the block $\bar{B}_{k}$ or that $c_{j}$ is a parent of the block $\bar{B}_{k}$. The notion that a vertex $\bar{c}_{j}$ generates a block $B_{k}$ is defined similarly.

The set of diagrams $\mathfrak{D}_{m}$ is constructed by the following inductive procedure. The set $\mathfrak{D}_{0}$ is formed by a unique diagram which consists of a unique (real) vertex $c_{0}^{(0)}$. If $m\geq 1$, the root of each diagram $\mathfrak{D}\in\mathfrak{D}_{m}$ is a real vertex $c_{0}^{(m)}$. Each vertex $\hat{c}_{j}$ of the diagram $\mathfrak{D}$ with $\deg\hat{c}_{j}\geq 1$ (i.e. which is real and is not a leave) generates a block $\hat{B}_{k}$ satisfying

[TABLE]

Non-conjugated vertices $c_{j}$ generate conjugated blocks $\bar{B}_{k}$ while conjugated vertices $\bar{c}_{j}$ generate non-conjugated blocks $B_{k}$. The numeration of the blocks (i.e. the dependence of $k$ as a function of $(\hat{c}_{j})$) is chosen in arbitrary way. E.g., from the left to the right and from the top to the bottom, as in fig. 2,3.

Rule (3.3) does not determine uniquely the degrees of vertices in the block $\hat{B}_{k}$, but fixes only their sum. So, for $m\neq 0,\,1$ the diagram $\mathfrak{D}$ obtained by the inductive procedure above is not unique. We define $\mathfrak{D}_{m}$ as the set of all possible diagrams obtained by this procedure.

The set of diagrams $\overline{\mathfrak{D}}_{n}$ is constructed by a similar procedure. The only difference is that in diagrams $\bar{\mathfrak{D}}\in\overline{\mathfrak{D}}_{n}$ the root vertex is conjugated and has the form $\bar{c}_{0}^{(n)}$, see fig. 3(d). The set of diagrams $\overline{\mathfrak{D}}_{n}$ equals to the set $\mathfrak{D}_{n}$ in which we conjugate each diagram, and in every block of each diagram we exchange positions of conjugated and non-conjugated vertices, so that the non-conjugated vertices are situated first.

Finally, for a pair of diagrams $\mathfrak{D}^{1}\in\mathfrak{D}_{m}$ and $\bar{\mathfrak{D}}^{2}\in\overline{\mathfrak{D}}_{n}$ we consider the diagram $\mathfrak{D}^{1}\sqcup\bar{\mathfrak{D}}^{2}$ (see fig. 3(a) and 4), obtained by drawing the diagrams $\mathfrak{D}^{1}$ and $\bar{\mathfrak{D}}^{2}$ side by side and enumerate the vertices of $\bar{\mathfrak{D}}^{2}$, except the root $\bar{c}_{0}$, not from $1$ to $2n$ but from $2m+1$ to $2m+2n$. Thus, in $\mathfrak{D}^{1}\sqcup\bar{\mathfrak{D}}^{2}$ we denote the two roots as $c_{0}$ and $\bar{c}_{0}$, then enumerate the remaining vertices of the diagram $\mathfrak{D}^{1}$, and finally those of the diagram $\bar{\mathfrak{D}}^{2}$. We define

[TABLE]

3.1.3. Properties of the set $\mathfrak{D}_{m}\times\overline{\mathfrak{D}}_{n}$ and some terminology

Throughout the paper we use the notation

[TABLE]

The following properties of a diagram $\mathfrak{D}\in\mathfrak{D}_{m}\times\overline{\mathfrak{D}}_{n}$ are obvious.

$\bullet$ The number of blocks of $\mathfrak{D}$ equals to $N$.

$\bullet$ The diagram $\mathfrak{D}$ has $4N+2$ vertices, half of them are conjugated while another half are non-conjugated.

$\bullet$ If vertices $\hat{c}_{i}$ and $\hat{c}_{j}$ are coupled by an edge, we call them adjacent and write $\hat{c}_{i}\thicksim\hat{c}_{j}$. Each pair of adjacent vertices in $\mathfrak{D}$ consists of a conjugated and non-conjugated vertex, and has either the form $\{c^{(p)}_{i},\bar{w}_{j}\}$ or $\{\bar{c}^{(p)}_{i},w_{j}\}$, where the degree $p\geq 1$.

$\bullet$ An edge which joins $c_{i}$ with $\bar{c}_{j}$ is denoted $(c_{i},\bar{c}_{j})$ (either $c_{i}$ or $\bar{c}_{j}$ is a virtual vertex). Each diagram $\mathfrak{D}\in\mathfrak{D}_{m}\times\overline{\mathfrak{D}}_{n}$ has $N$ edges.

$\bullet$ Recall that a real vertex $\hat{c}_{j}^{(0)}$ of zero degree is called a leaf. There is no vertex in $\mathfrak{D}$, adjacent to a leaf. The number of conjugated leaves equals to the number of non-conjugated leaves and equals to $N+1$.

3.1.4. Values of vertices $\hat{c}_{j}$

Consider a diagram $\mathfrak{D}\in\mathfrak{D}_{m}\times\overline{\mathfrak{D}}_{n}$. To every vertex $\hat{c}_{i}$ of a block $\hat{B}_{k}$, $1\leq k\leq N$, we assign the same time variable $l_{k}\geq-T$. E.g., in the diagram from fig. 4(b) the times assigned to the vertices $c_{2}$ and $\bar{c}_{6}$ are $l_{1}$ and $l_{3}$, respectively. To the roots $c_{0}$ and $\bar{c}_{0}$ we assign the times $\tau_{1}$ and $\tau_{2}$. The vectors

[TABLE]

are called the vectors of times. Let

[TABLE]

where $\xi_{i},\sigma_{j}\in\mathbb{Z}^{d}_{L}$, so $(\xi,\sigma)\in\mathbb{Z}^{(4N+2)d}_{L}$, and we recall that $4N+2$ is the number of vertices in $\mathfrak{D}$. These will be the indices for summation in a formula for $a_{s}^{(m)}(\tau_{1})\bar{a}_{s}^{(n)}(\tau_{2})$, where $\xi_{j}$’s are indices for the non-conjugated variables $a_{s}$ and $\sigma_{j}$ – for the conjugated variables $\bar{a}_{s}$ in the following sense.

We regard a real vertex $c_{j}^{(p)}$ of the diagram $\mathfrak{D}$ as a position to put there a component $a^{(p)}_{\xi_{j}}(t)$ of the random vector $a^{(p)}(t)$, and a real vertex $\bar{c}_{j}^{(p)}$ – as a position for a component $\bar{a}^{(p)}_{\sigma_{j}}(t^{\prime})$ of the conjugate vector, with suitable $t=t(\bf l)$, $t^{\prime}=t^{\prime}(\bf l)$. More precisely, for $p\geq 0$ we view the real vertices $c_{j}^{(p)}$, $\bar{c}_{j}^{(p)}$ as random functions of the variables $\bf l,\xi,\sigma$, defined by

[TABLE]

where $t$ and $t^{\prime}$ are components of the time-vector $\bf l$ in (3.4), assigned to the vertices $c_{j}^{(p)}$ and $\bar{c}_{j}^{(p)}$. We say that the vertices $c_{j}^{(p)}$ and $\bar{c}_{j}^{(p)}$ take values $a_{\xi_{j}}^{(p)}(t)$ and $\bar{a}_{\sigma_{j}}^{(p)}(t^{\prime})$. For virtual vertices $w_{j}$ and $\bar{w}_{j}$ we say that they take values $w_{\xi_{j}}$ and $\bar{w}_{\sigma_{j}}$.

3.1.5. Restrictions on the multi-indices $(\xi,\sigma)$

To calculate the value of a diagram $\mathfrak{D}\in\mathfrak{D}_{m}\times\overline{\mathfrak{D}}_{n}$ we will make substitutions (3.5), using multi-indices $(\xi,\sigma)\in\mathbb{Z}_{L}^{(4N+2)d}$. At some stages of our constructions it will be necessary to put restrictions on the $(\xi,\sigma)$ and substitute in the diagram only the corresponding multi-indices which we call admissible. The set of all admissible multi-indices $(\xi,\sigma)$ is denoted

[TABLE]

where we take $s,\xi_{i},\sigma_{j}$ in $\mathbb{R}^{d}$ rather than in $\mathbb{Z}^{d}_{L}$ to be able to extend the correlation from Theorem 2.2 to a function on $\mathbb{R}^{d}=\{s\}$. The set $\mathcal{A}_{s}(\mathfrak{D})$ is defined by the following three relations:

$\bullet$ We have

[TABLE]

$\bullet$ For all $1\leq i\leq N$,

[TABLE]

$\bullet$ If the vertices $c_{i}$ and $\bar{c}_{j}$ are adjacent, the corresponding indices are equal:

[TABLE]

We will also use the corresponding discrete subset

[TABLE]

which is exactly the set of multi-indices in which we sum in (2.21). In view of condition (3.6) it may be not empty only if $s\in\mathbb{Z}^{d}_{L}$.

3.1.6. More notation

Let $\mathfrak{D}\in\mathfrak{D}_{m}\times\overline{\mathfrak{D}}_{n}$.

$\bullet$ We denote by $E(\mathfrak{D})$ the set of all edges of a diagram $\mathfrak{D}$, $E(\mathfrak{D})=\{(c_{i},\bar{c}_{j})\}.$

$\bullet$ Let $(\xi,\sigma)\in\mathcal{A}_{s}(\mathfrak{D})$ be an admissible index–vector. Then $\xi_{i}=\sigma_{j}$ if $c_{i}\sim\bar{c}_{j}$. Denoting by $\varphi$ an edge $\varphi=(c_{i},\bar{c}_{j})$, we set

[TABLE]

($\gamma_{s}$’s are defined in (1.4)).

$\bullet$ Recall that in each pair of adjacent vertices in $\mathfrak{D}$ always one vertex is real while another is virtual. Then, by $l^{\varphi}_{r}$ and $l^{\varphi}_{w}$ we denote the times assigned to the real and virtual vertices, connected by the edge $\varphi$. So each $l^{\varphi}_{r,w}$ is a mapping from $E(\mathfrak{D})$ to the set $\{\tau_{1},\tau_{2},l_{1},\dots,l_{N}\}$.

$\bullet$ By $l^{\hat{c}_{i}}$ we denote the time, assigned to a vertex $\hat{c}_{i}$.

$\bullet$ By

[TABLE]

we denote the sets such that $\{c_{i}\}_{i\in L(\mathfrak{D})}$ and $\{\bar{c}_{i}\}_{i\in\bar{L}(\mathfrak{D})}$ form the sets of all non-conjugated leaves and all conjugated leaves, correspondingly. In other words, abusing notation and identifying the vertices with their values, we have $\{c_{i}\}_{i\in L(\mathfrak{D})}=\{a^{(0)}_{\xi_{i}}\}$ and $\{\bar{c}_{i}\}_{i\in\bar{L}(\mathfrak{D})}=\{\bar{a}^{(0)}_{\sigma_{i}}\}$. E.g., for the diagram from fig. 4(b) we have $L(\mathfrak{D})=\{1,2,5,6\}$ and $\bar{L}(\mathfrak{D})=\{1,3,4,5\}$.

3.2. Formula for the product

$a^{(m)}_{s}\bar{a}^{(n)}_{s}$

Let us denote

[TABLE]

where we recall that $\tau_{1},\tau_{2}$ are the times for the roots $c_{0},\bar{c}_{0}$. Recall also that the sets $E(\mathfrak{D})$, $L(\mathfrak{D})$, $\bar{L}(\mathfrak{D})$, the functions $\gamma^{\varphi}=\gamma^{\varphi}(\xi,\sigma)$ and the times $l^{\varphi}_{r,w}$ and $l^{c_{j},\bar{c}_{j}}$ are defined in Section 3.1.6; in difference with the functions $\gamma^{\varphi}$, the time variables $l^{\varphi}_{r},l^{\varphi}_{w},l^{c_{j}},l^{\bar{c}_{j}}$ are independent from $(\xi,\sigma)$. To each diagram $\mathfrak{D}\in\mathfrak{D}_{m}\times\overline{\mathfrak{D}}_{n}$ we associate the (random) function

[TABLE]

which is defined on the set of admissible multi-indices $(\xi,\sigma)\in\mathcal{A}_{s}^{L}(\mathfrak{D})$ (see (3.9)). See below for a discussion of this formula. Here $\mathbb{I}_{\{-T\leq l^{\varphi}_{w}\leq l^{\varphi}_{r}\}}$ denotes the indicator function of the set in which $-T\leq l^{\varphi}_{w}\leq l^{\varphi}_{r}$ and $c_{\mathfrak{D}}$ is the constant

[TABLE]

where $\#w_{\mathfrak{D}}$ is the number of non-conjugated virtual vertices of the diagram $\mathfrak{D}$, $N=m+n$ and $i=\sqrt{-1}$ is the imaginary unit. Set

[TABLE]

we recall that the quadratic form $\omega^{s_{1}s_{2}}_{s_{3}s}$ is defined in (2.5). For vectors $\vec{x}=(x_{1},\ldots,x_{N})$ and $\vec{t}=(t_{1},\ldots,t_{N})$ set also

[TABLE]

where $\theta$ is as in (2.14).

Lemma 3.1**.**

For any integers $m,n\geq 0$ satisfying $N:=m+n\geq 1$, $s\in\mathbb{Z}^{d}_{L}$ and $\tau_{1},\tau_{2}\geq-T$, we have

[TABLE]

where $I_{s}(\mathfrak{D})=I_{s}(\tau;\mathfrak{D})$,

[TABLE]

Components of the formula (3.17) are quite simple. Indeed, the factor $\prod_{j\in L(\mathfrak{D})}a^{(0)}_{\xi_{j}}(l^{c_{j}})\prod_{j\in\bar{L}(\mathfrak{D})}\bar{a}^{(0)}_{\sigma_{j}}(l^{\bar{c}_{j}})$, entering $G_{\mathfrak{D}}(\tau,l,\xi,\sigma)$ (see (3.12)), is just the product of values of all leaves of the diagram $\mathfrak{D}$. The term $l^{\varphi}_{r}-l^{\varphi}_{w}$ is a difference of the times assigned to vertices, coupled by an edge $\varphi$, and the corresponding exponent in $G_{\mathfrak{D}}(\tau,l,\xi,\sigma)$ comes from the iteration of the Duhamel formula (2.13) that generates $\varphi$. It appears together with the indicator function $\mathbb{I}_{\{-T\leq l^{\varphi}_{w}\leq l^{\varphi}_{r}\}}$, which is there due to the fact that in (2.13) the integration is performed not over $\mathbb{R}$ but only over the interval $[-T,\tau]$. The factor $\theta(\omega_{j},\nu^{-1}l_{j})$, entering the function $\Theta(\omega,\nu^{-1}l)$, comes from the iteration of the Duhamel formula, generating the $j$-th block of $\mathfrak{D}$.

The assertion of Lemma 3.1 easily (but clumsily) follows by iterating the formula (2.13). Instead of giving a complete proof we restrict ourselves to considering an example below.

3.2.1. Example

Here we establish formula (3.16) for the case $m=1$ and $n=0$. We get from (2.12) that

[TABLE]

where we use notation (3.14). The set $\mathfrak{D}_{1}\times\overline{\mathfrak{D}}_{0}$ consists of a unique diagram, given in fig. 3(a), which we denote by $\mathfrak{D}^{1,0}$. Now the total number of vertices is $4N+2=6$, and the corresponding set of admissible multi-indices $\mathcal{A}_{s}^{L}(\mathfrak{D}^{1,0})$ has the form

[TABLE]

Then we rewrite the r.h.s. of (3.18) as

[TABLE]

The unique edge of the diagram $\mathfrak{D}^{1,0}$ is $\varphi:=(c_{0}^{(1)},\bar{w}_{2})$, so using the introduced in Section 3.1.6 notation we write $e^{-\gamma_{\sigma_{2}}(\tau_{1}-l_{1})}=e^{-\gamma^{\varphi}(l_{r}^{\varphi}-l_{w}^{\varphi})}$ and $\int_{-T}^{\tau_{1}}\,dl_{1}=\int_{\mathbb{R}}\mathbb{I}_{\{-T\leq l^{\varphi}_{w}\leq l^{\varphi}_{r}\}}\,dl_{1}$. Then, noting that the term $a^{(0)}_{\xi_{1}}a^{(0)}_{\xi_{2}}{\bar{a}_{\sigma_{1}}}^{(0)}\bar{a}^{(0)}_{\sigma_{0}}$ is a product of values of all leaves in the diagram $\mathfrak{D}^{1,0}$, we see that (3.18) takes the form $a^{(1)}_{s}(\tau_{1})\bar{a}^{(0)}_{s}(\tau_{2})=I_{s}(\mathfrak{D}^{1,0})$, where the sum $I_{s}(\mathfrak{D}^{1,0})$ is defined in (3.17). Thus, formula (3.16) holds for $m=1,n=0$.

4. Feynman diagrams and formula for expectations ${\mathbb{E}}a^{(m)}_{s}\bar{a}^{(n)}_{s}$

In this section we compute the expectation ${\mathbb{E}}a^{(m)}_{s}(\tau_{1})\bar{a}^{(n)}_{s}(\tau_{2})$. Due to (3.16), it suffices to find ${\mathbb{E}}I_{s}(\mathfrak{D})$ for every diagram $\mathfrak{D}\in\mathfrak{D}_{m}\times\overline{\mathfrak{D}}_{n}$. Note that the randomness enters $I_{s}(\mathfrak{D})$ only via the product of values of leaves $a_{\xi_{i}}^{(0)}$, $\bar{a}_{\sigma_{j}}^{(0)}$ in (3.12), which are Gaussian random variables. Since their correlations are given by (2.15), due to the Wick theorem ${\mathbb{E}}I_{s}(\mathfrak{D})$ is given by a sum over all Wick pairings of values $a_{\xi_{i}}^{(0)}(l^{c_{i}})$ of non-conjugated leaves in $\mathfrak{D}$ with values $\bar{a}_{\sigma_{j}}^{(0)}(l^{\bar{c}_{j}})$ of conjugated leaves. We parametrize this sum by Feynman diagrams $\mathfrak{F}$ (see [16] for the notion), which are obtained from the diagram $\mathfrak{D}$ by connecting the Wick paired leaves by edges, see fig. 5.

Next we give a detailed definition of the Feynman diagrams and discuss related notions.

4.1. Feynman diagrams

Let us fix $m,n\geq 0$ satisfying $N=m+n\geq 1$, and consider a diagram $\mathfrak{D}\in\mathfrak{D}_{m}\times\overline{\mathfrak{D}}_{n}$. Its set of leaves consists of $N+1$ conjugated leaves $\bar{c}_{i}^{(0)}$ and $N+1$ non-conjugated leaves $c_{j}^{(0)}$. Consider any partition of this set to $N+1$ non-intersecting pairs of the form $(c_{i}^{(0)},\bar{c}_{j}^{(0)})$, where the leaves $c_{i}^{(0)}$ and $\bar{c}_{j}^{(0)}$ do not belong to the same block of $\mathfrak{D}$ (if such partition exists). We denote by $\mathfrak{F}$ a diagram obtained from $\mathfrak{D}$ by connecting the leaves in each pair by an edge (see fig. 5(a)) and call it a Feynman diagram. We never couple leaves belonging to the same block since for any $j$-th block the variables $a_{\xi_{\alpha}}^{(0)}$, $\bar{a}_{\sigma_{\beta}}^{(0)}$, assigned to the vertices in this block as in (3.5), satisfy ${\mathbb{E}}a^{(0)}_{\xi_{\alpha}}\bar{a}^{(0)}_{\sigma_{\beta}}=0$ for $\alpha,\beta\in\{2j-1,2j\}$ and any admissible mutli-index $(\xi,\sigma)\in\mathcal{A}_{s}^{L}(\mathfrak{D})$, since $\xi_{\alpha}\neq\sigma_{\beta}$ in view of (1.10) and (3.7).

Let $\mathfrak{F}(\mathfrak{D})$ be the set of all Feynman diagrams, which may be obtained from $\mathfrak{D}$ using different partitions of the set of leaves to pairs. We define

[TABLE]

This is the set of all Feynman diagrams corresponding to the product $a^{(m)}_{s}\bar{a}^{(n)}_{s}$. By construction, the set of vertices of any Feynman diagram $\mathfrak{F}\in\mathfrak{F}_{m,n}$ is partitioned into pairs of adjacent vertices $(c_{i},\bar{c}_{j})$. Each pair has either the form $(c^{(p)}_{i},\bar{w}_{j})$ or $(w_{i},\bar{c}^{(p)}_{j})$, where $p\geq 1$, or the form $(c_{i}^{(0)},\bar{c}_{j}^{(0)})$. Moreover, adjacent vertices never belong to the same block. To any $\mathfrak{D}\in\mathfrak{D}_{m}\times\overline{\mathfrak{D}}_{n}$ the construction above corresponds many diagrams $\mathfrak{F}\in\mathfrak{F}_{m,n}$, but the inverse mapping

[TABLE]

is well defined: the diagram $\mathfrak{D}_{\mathfrak{F}}$ is obtained from $\mathfrak{F}$ by erasing the edges that couple its leaves.

In fact, by demanding that in Feynman diagrams coupled leaves never belong to the same block, we have excluded only a part of vanishing Wick pairings. To exclude the remaining part, in Section 5.2 we will replace the set $\mathfrak{F}_{m,n}$ of Feynman diagrams by a smaller set $\mathfrak{F}^{\,true}_{m,n}\subset\mathfrak{F}_{m,n}$. For the Wick pairings corresponding to diagrams $\mathfrak{F}\notin\mathfrak{F}^{\,true}_{m,n}$ the assumption $\{s_{1},s_{2}\}\neq\{s_{3},s\}$, following from (1.9), again is violated due to (3.7), but in a less trivial way (see Proposition 5.6).

4.1.1. Set of admissible multi-indices $\mathcal{A}_{s}^{\prime}(\mathfrak{F})$

Let us orient edges of a Feynman diagram $\mathfrak{F}\in\mathfrak{F}_{m,n}$ in the direction from conjugated vertices to non-conjugated, as in fig. 5(b), and consider a permutation $f_{\mathfrak{F}}$ of the set $\{0,\dots,2N\}$ such that

[TABLE]

see fig. 5(b). For a vector $\xi\in\mathbb{R}^{(2N+1)d}$ we denote

[TABLE]

Let $\mathcal{A}^{\prime}_{s}(\mathfrak{F})$ be a set of all multi-indices $\xi\in\mathbb{R}^{(2N+1)d}$ satisfying the following relations: for $\sigma=\xi_{f_{\mathfrak{F}}}$,

• •

We have

[TABLE]

• •

For any $1\leq i\leq N$,

[TABLE]

In other words, we require (4.5) and (4.6) to be satisfied by the multi-indices $\xi,\sigma$, where $\sigma$ is obtained from $\xi$ accordingly to the edges of diagram $\mathfrak{F}$: $\sigma_{j}=\xi_{i}$ if a vertex $\bar{c}_{j}$ is coupled with the vertex $c_{i}$. In particular, if $\xi\in\mathcal{A}_{s}^{\prime}(\mathfrak{F})$ then $(\xi,\xi_{f_{\mathfrak{F}}})\in\mathcal{A}_{s}(\mathfrak{D}_{\mathfrak{F}})$, where $\mathcal{A}_{s}(\mathfrak{D})$ is defined in Section 3.1.5 and the diagram $\mathfrak{D}_{\mathfrak{F}}$ – in (4.2). Indeed, (3.8) holds by the definition (4.3) of the permutation $f_{\mathfrak{F}}$ while (3.6) and (3.7) are obvious.

We also consider the discrete set $\mathcal{A}_{s}^{\prime L}(\mathfrak{F})=\mathcal{A}^{\prime}_{s}(\mathfrak{F})\cap\mathbb{Z}^{(2N+1)d}_{L}$.

Our motivation behind these definitions is as follows. As it is explained in the beginning of the section, each term ${\mathbb{E}}I(\mathfrak{D})$ can be written as a sum over the Feynman diagrams $\mathfrak{F}$,

[TABLE]

Each summand $J_{s}(\mathfrak{F})$ has the form (3.17) where $G_{\mathfrak{D}}$ is replaced by a function obtained by the corresponding to $\mathfrak{F}$ Wick pairing of the values $a_{\xi_{i}}^{(0)}$, $\bar{a}_{\sigma_{j}}^{(0)}$ of leaves. In this formula for $J_{s}(\mathfrak{F})$ we should take summation only over those multi-indices $(\xi,\sigma)\in\mathcal{A}_{s}^{L}(\mathfrak{D})$ for which $\xi_{i}=\sigma_{j}$ once the vertices $c_{i}$ and $\bar{c}_{j}$ are adjacent in $\mathfrak{F}$, that is over vectors of the form $(\xi,\xi_{f_{\mathfrak{F}}})$, where $\xi\in\mathcal{A}_{s}^{\prime L}(\mathfrak{F})$.

4.2. Formula for the expectation ${\mathbb{E}}a^{(m)}_{s}\bar{a}^{(n)}_{s}$

Below we introduce some more notation. Let $\mathfrak{F}\in\mathfrak{F}_{m,n}$ be a Feynman diagram.

$\bullet$ We denote by $E(\mathfrak{F})$ the set of edges of the diagram $\mathfrak{F}$, then $E(\mathfrak{D}_{\mathfrak{F}})\subset E(\mathfrak{F})$ (see (4.2)). We also set

[TABLE]

$\bullet$ For an edge $\vartheta\in E(\mathfrak{F})$ which couples some vertices $c_{i}$ and $\bar{c}_{j}$, we set $c_{\vartheta}:=c_{i}$, $\bar{c}_{\vartheta}:=\bar{c}_{j}$, and denote by $l^{c_{\vartheta}}$ and $l^{\bar{c}_{\vartheta}}$ the times assigned to the vertices $c_{\vartheta}$ and $\bar{c}_{\vartheta}$. We also set

[TABLE]

where $\gamma_{s}$ is defined in (1.4) and $b(s)$ enters the definition of the random force $\eta^{\omega}$.

$\bullet$ If $\varphi\in E_{D}(\mathfrak{F})$, we define the times $l_{r}^{\varphi}$ and $l_{w}^{\varphi}$ as in Section 3.1.6.

The main result of this section is Lemma 4.1, stated below. Recalling (3.11) and (3.12), (2.15), to each Feynman diagram $\mathfrak{F}$ we associate the density function $c_{\mathfrak{F}}F^{\mathfrak{F}}(\tau,l,\xi)$, where $c_{\mathfrak{F}}=c_{\mathfrak{D}_{\mathfrak{F}}}$ (see (3.13)) and $F^{\mathfrak{F}}(\tau,l,\xi)$ is the real function

[TABLE]

where $\gamma^{\varphi}=\gamma^{\varphi}(\xi)$, $\gamma^{\psi}=\gamma^{\psi}(\xi)$, $b^{\psi}=b^{\psi}(\xi)$. We also associate to $\mathfrak{F}$ the quadratic forms

[TABLE]

see (3.14) and (4.4).

Lemma 4.1**.**

For any integers $m,n\geq 0$ satisfying $N=m+n\geq 1$, for $s\in\mathbb{Z}^{d}_{L}$ and $\tau_{1},\tau_{2}\geq-T$, we have

[TABLE]

where $J_{s}(\mathfrak{F})=J_{s}(\tau;\mathfrak{F})$

[TABLE]

The density $F^{\mathfrak{F}}$, given by (4.7), is real, smooth in $\xi\in\mathbb{R}^{(2N+1)d}$, piecewise smooth in $l\in\mathbb{R}^{N}$ and such that for any vector $\kappa\in\mathbb{Z}_{+}^{d(2N+1)}$ we have:

[TABLE]

for any $\xi\in\mathcal{A}_{s}^{\prime}(\mathfrak{F})$, $l\in\mathbb{R}^{N}$ and $\tau_{1},\tau_{2}\geq-T$, where $\delta=\delta_{N}>0$.

Proof of (4.9) can be obtained by taking the expectation of both sides of (3.16) and is essentially explained in previous discussion of this section. Estimate (4.11) follows from the explicit form (4.7) of the density $F^{\mathfrak{F}}$. Nevertheless, we give an accurate proof in Appendix 8.2. Below we do not use the explicit form of $F^{\mathfrak{F}}$ but only the estimate (4.11).

Let us note that

[TABLE]

Indeed, in this case $\mathfrak{F}_{m,n}=\emptyset$, since every partition of the set of leaves into pairs couples vertices from the same block. Accordingly, we will often assume that $m+n\geq 2$.

5. Change of coordinates and final formula for expectations ${\mathbb{E}}a^{(m)}_{s}\bar{a}^{(n)}_{s}$

Lemma 4.1 gives an explicit formula for the expectation ${\mathbb{E}}a^{(m)}_{s}\bar{a}^{(n)}_{s}$, however, the structure of the set of multi-indices $\mathcal{A}_{s}^{\prime L}(\mathfrak{F})$, over which we take a summation in (4.10), is too complicated for further analysis. In this section we find new coordinates in which this set and the quadratic forms (4.8) take convenient form.

By (4.6) the vectors $\xi\in\mathcal{A}_{s}^{\prime L}$ satisfy the system of equations

[TABLE]

We start with constructing coordinates $x=(x_{0},\dots,x_{2N})$, where $x_{0}=\xi_{0}$ and for every $1\leq k\leq N$ we set either $x_{2k-1}=\xi_{2k-1}-\sigma_{2k-1}$ and $x_{2k}=\xi_{2k}-\sigma_{2k}$ or $x_{2k-1}=\xi_{2k-1}-\sigma_{2k}$ and $x_{2k}=\xi_{2k}-\sigma_{2k-1}$, so that equations (5.1) take trivial form $x_{2k}=-x_{2k-1}$, $\forall k$. We will make a choice (different for different $k$ and dependent on the diagram $\mathfrak{F}$) between the two possibilities above in such a way that the transformation $\xi\mapsto x$ is invertible. Then, components $z_{k}:=x_{2k-1}$, $1\leq k\leq N$, form coordinates on the set $\mathcal{A}_{s}^{\prime L}$ while components $x_{2k}$ are functions of $z=(z_{k})_{1\leq k\leq N}$.

To construct this transformation $\xi\mapsto x$ we add to the diagram $\mathfrak{F}$ dashed edges which couple vertices inside each block in such a way that the obtained diagram becomes a Hamilton cycle, see fig. 6(b) (we show that it is always possible to do). For every $k$ we choose between the two possibilities above in accordance with the dashed edges. It turns out that in the coordinates $z$ the function $\Theta(\omega^{\mathfrak{F}},\nu^{-1}l)$ with $\theta(x,t)=e^{ixt}$ takes a simple form (5.21)-(5.22). This leads to an explicit formula for the sum $J_{s}(\mathfrak{F})$ and next in Sections 6, 7 allows to analyse its asymptotic behaviour as $\nu\to 0$, $L\to\infty$. Next we explain construction of the transformation $\xi\mapsto x$ in detail.

5.1. Cycles $\mathfrak{C}_{\mathfrak{F}},\hat{\mathfrak{C}}_{\mathfrak{F}}$ and permutation $\pi_{\mathfrak{F}}$

Let us take a Feynman diagram $\mathfrak{F}\in\mathfrak{F}_{m,n}$ with $N=m+n\geq 2$. By construction, if two vertices of $\mathfrak{F}$ belong to the same block then they are not adjacent. Now inside each block $\hat{B}_{k}$, $k\geq 1$, (see (3.1)-(3.2)) we join by four dashed edges conjugated vertices with non-conjugated in all possible ways, see fig. 6(a). We also join by a dashed edge the two roots $c_{0}$, $\bar{c}_{0}$ and denote the resulting graph by $\mathfrak{U}_{\mathfrak{F}}$.

Lemma 5.1**.**

For each Feynman diagram $\mathfrak{F}\in\mathfrak{F}_{m,n}$ the graph $\mathfrak{U}_{\mathfrak{F}}$ has a Hamilton cycle in which solid and dashed edges alternate.

Proof of Lemma 5.1, given in Appendix 8.3, follows from the fact that the diagram, made by the blocks of $\mathfrak{F}$, is connected. In general, the Hamilton cycle in Lemma 5.1 is not unique, so we fix one for each diagram $\mathfrak{U}_{\mathfrak{F}}$ and denote it as $\mathfrak{C}_{\mathfrak{F}}$. Examples of the Hamilton cycles $\mathfrak{C}_{\mathfrak{F}}$ are given in fig. 6(b) and fig. 9. It is straightforward to see that, by construction of the diagrams $\mathfrak{U}_{\mathfrak{F}}$, the cycles $\mathfrak{C}_{\mathfrak{F}}$ contain all solid edges of $\mathfrak{U}_{\mathfrak{F}}$ (which are the edges of $\mathfrak{F}$), and a half of the dashed edges: for each block exactly two of four dashed edges that couple vertices inside the block enter the cycle $\mathfrak{C}_{\mathfrak{F}}$. On the cycle $\mathfrak{C}_{\mathfrak{F}}$ the dashes and solid edges alternate.

We orient the dashed edges of $\mathfrak{C}_{\mathfrak{F}}$ in the direction from non-conjugated vertices to conjugated and recall that the solid edges are oriented in the opposite direction, from conjugated vertices to non-conjugated. Then, $\mathfrak{C}_{\mathfrak{F}}$ becomes an oriented cycle, see fig. 6(b).

Let us denote by $\hat{\mathfrak{C}}_{\mathfrak{F}}$ the reduced oriented cycle $\mathfrak{C}_{\mathfrak{F}}$, that is the cycle $\mathfrak{C}_{\mathfrak{F}}$ in which we keep only non-conjugated vertices and ”forget” all conjugated ones, see fig. 7. The cycle $\hat{\mathfrak{C}}_{\mathfrak{F}}$ defines a cyclic permutation $\pi_{\mathfrak{F}}$ of the set of parameters for the non-conjugated vertices $\{0,\dots,2N\}$ as

[TABLE]

The graph $\mathfrak{U}_{\mathfrak{F}}$ and the Hamilton cycle $\mathfrak{C}_{\mathfrak{F}}$ play a central role in our construction. Nevertheless, below we use only the ”derivative” objects: the reduced cycle $\hat{\mathfrak{C}}_{\mathfrak{F}}$ and the permutation $\pi_{\mathfrak{F}}$.

5.2. The change of coordinates

For a fixed Feynman diagram $\mathfrak{F}\in\mathfrak{F}_{m,n}$ with $N=m+n\geq 1$, we set

[TABLE]

and define the $(2N+1)$–vector $x=(x_{j})_{0\leq j\leq 2N}$, $x_{j}\in\mathbb{R}^{d}$. Since $\pi_{\mathfrak{F}}$ is a cyclic permutation of the set $\{0,\ldots,2N\}$, then the transformation $\xi\mapsto x$ is a bijection of $(\mathbb{R}^{d})^{2N+1}$. Indeed, it is invertible since iterating (5.2) we get

[TABLE]

where $k=k(j)$ denotes minimal positive integer satisfying $\pi^{k+1}_{\mathfrak{F}}(j)=0$. In other words, since $\xi_{0}=x_{0}$,

[TABLE]

where by $[c_{a},c_{b}]$, $a\neq b$, we denote a set (an arc) of vertices of the cycle $\hat{\mathfrak{C}}_{\mathfrak{F}}$ that are situated between the vertices $c_{a}$ and $c_{b}$ according to the orientation of $\hat{\mathfrak{C}}_{\mathfrak{F}}$, including $c_{a}$ and $c_{b}$. E.g., in fig. 7 $[c_{4},c_{3}]=\{c_{4},c_{1},c_{0},c_{3}\}.$ For $a=b$ we set $[c_{a},c_{b}]=\{c_{a}\}$.

Next we write the set of admissible multi-indices $\mathcal{A}_{s}^{\prime}(\mathfrak{F})\subset\mathbb{R}^{d(2N-1)}$ in the coordinates $x$. An important role below is played by the $N\times N$ incidence matrix $\alpha^{\mathfrak{F}}=(\alpha_{ij}^{\mathfrak{F}})_{1\leq i,j\leq N}$ with components 999Formula (5.4) admits the following interpretation. An element $\alpha_{ij}^{\mathfrak{F}}$ equals $\pm 1$ if the arcs $[c_{2i-1},c_{2i}]$ and $[c_{2j-1},c_{2j}]$ are linked, and equals [math] otherwise. The sign ”$+$” or ”$-$” is determined by the order of linking; see fig. 7. So the incidence matrix $\alpha^{\mathfrak{F}}$ describes the linking of the arcs $[c_{2j-1},c_{2j}]$ on the reduced cycle $\hat{\mathfrak{C}}_{\mathfrak{F}}$, or equivalently on the Hamiltonian cycle $\mathfrak{C}_{\mathfrak{F}}$ in the diagram $\mathfrak{F}$.

[TABLE]

It is straightforward to see that the matrix $\alpha^{\mathfrak{F}}$ is skew-symmetric, so it is S-regular in the following sense:

Definition 5.2**.**

A square matrix $\alpha=\{\alpha_{ij}\}$ is called S-regular if it is skew-symmetric and all its elements $\alpha_{ij}$ are such that $\alpha_{ij}\in\{-1,0,1\}.$ If in addition all rows and columns of $\alpha$ are non-zero vectors, then it is called SS-regular.

Given below Propositions 5.3 and 5.5 are in the hart of the proof of the main result of this work – Theorem 1.1. Previous constructions in Sections 3-5.1 essentially were made in order to create the notation, needed to state and prove these results.

Proposition 5.3**.**

For any diagram $\mathfrak{F}\in\mathfrak{F}_{m,n}$ and any $s\in\mathbb{R}^{d}$, the set $\mathcal{A}_{s}^{\prime}(\mathfrak{F})$ consists of all vectors $\xi=\xi(x)$, where $\xi(x)$ is given by (5.3) and $x\in(\mathbb{R}^{d})^{2N+1}$ satisfies the following relations, for every $1\leq k\leq N$:

[TABLE]

Moreover, for vectors $x$ satisfying (5.5)(b) we have

[TABLE]

Proof. In this proof we omit the lower index $\mathfrak{F}$ in the notation $\pi_{\mathfrak{F}}$ and $f_{\mathfrak{F}}$. We start by noting that for every $k\geq 1$

[TABLE]

Indeed, the permutation $\pi$ encodes the change of indices of non-conjugated vertices of the cycle $\mathfrak{C}_{\mathfrak{F}}$ under the double shift, the first along dashed edges and the second along solid edges. The first shift preserves the set of indices $\{2k-1,2k\}$ while the second is encoded by the permutation $f$.

Now let us recall that the set $\mathcal{A}_{s}^{\prime}(\mathfrak{F})$ consists of all multi-indices $\xi$ satisfying (4.5) and (4.6) with $\sigma=\xi_{f}$. In view of what is told above, we have $\{\xi_{f(2k-1)},\xi_{f(2k)}\}=\{\xi_{\pi(2k-1)},\xi_{\pi(2k)}\}$, so $\delta^{\prime\xi_{2k-1}\xi_{2k}}_{\xi_{f(2k-1)}\xi_{f(2k)}}=\delta^{\prime\xi_{2k-1}\xi_{2k}}_{\xi_{\pi(2k-1)}\xi_{\pi(2k)}}$. Then equation (4.6) is equivalent to $\delta^{\prime\xi_{2k-1}\xi_{2k}}_{\xi_{\pi(2k-1)}\xi_{\pi(2k)}}=1$, that is to the equation

[TABLE]

jointly with

[TABLE]

(see (1.10)). Equation (5.9) is equivalent to (5.5)(b). The equality $\xi_{0}=s$ from (4.5) is equivalent to (5.5)(a). The equality $\sigma_{0}=s$, in view of (5.3), takes the form

[TABLE]

Since $f(0)=\pi(0)$, then $[c_{f(0)},c_{0}]=[c_{\pi(0)},c_{0}]$, and this interval contains all vertices of the cycle $\hat{\mathfrak{C}}_{\mathfrak{F}}$. Then, using (5.5)(b), we see that the r.h.s. of (5.11) equals to $x_{0}$, so (5.11) again takes the form (5.5)(a).

Now it remains to show that (5.10) is equivalent to (5.6). In view of (5.9), relation (5.10) is equivalent to

[TABLE]

The first inequality above is equivalent to (5.6)(a), while the second is equivalent to (5.6)(b) if (5.7) is established. So it remains to verify (5.7). Due to (5.3),

[TABLE]

Let us first assume that $c_{\pi(2k)}\in[c_{2k-1},c_{0}]$ and $c_{\pi(2k)}\neq c_{2k-1}$. Then the identity above takes the form

[TABLE]

where we have used that the vertex situating in $\hat{\mathfrak{C}}_{\mathfrak{F}}$ before $c_{\pi(2k)}$ is $c_{2k}$. In view of the assumption above, $c_{0}\notin[c_{2k-1},c_{2k}]$. Then, due to cancellations provided by (5.5)(b), we get (5.7).

The case $c_{\pi(2k)}\notin[c_{2k-1},c_{0}]$ can be considered similarly. If $c_{\pi(2k)}=c_{2k-1}$, that is $\pi(2k)=2k-1$, then the both sides of equation (5.7) vanish. Indeed, $\alpha^{\mathfrak{F}}_{ki}=0$ $\forall i$ since the arc $[c_{2k-1},c_{2k}]$ contains all vertices $c_{j}\in\hat{\mathfrak{C}}_{\mathfrak{F}}$. ∎

Let us denote

[TABLE]

Due to Proposition 5.3, vector $z$ forms coordinates on the set $\mathcal{A}_{s}^{\prime}(\mathfrak{F})$ and, by (5.5)(b), $x_{i}=(-1)^{i+1}z_{\lceil i/2\rceil}$. Then, using (5.5)(a), we see that for $\xi\in\mathcal{A}_{s}^{\prime}(\mathfrak{F})$ relation (5.3) takes the form

[TABLE]

where $[c_{j},c_{0}):=[c_{j},c_{0}]\setminus\{c_{0}\}$, and $\xi_{0}=s$. Here we emphasize that the linear mapping $z\mapsto\xi$ depends on $\mathfrak{F}$ (through the interval $[c_{j},c_{0})$) and that $\xi$ is an affine vector-function of the parameter $s\in\mathbb{R}^{d}$. Thus, in the $z$-coordinates

[TABLE]

where, by (5.6),

[TABLE]

Remark 5.4*.*

Since the choice of the Hamilton cycle $\mathfrak{C}_{\mathfrak{F}}$ in general is not unique, the obtained parametrization $z\mapsto\xi$ is not unique as well. However, one can show that if $z^{\prime}\mapsto\xi$ is another parametrization, obtained by the procedure above, and $\alpha^{\mathfrak{F}\,\prime}$ is the associated incidence matrix, then for each $j$ we have either $z_{j}^{\prime}(\xi)=z_{j}(\xi)$ or $z_{j}^{\prime}(\xi)=\sum_{i=1}^{N}\alpha_{ji}^{\mathfrak{F}}z_{i}$. In the latter case we also have the symmetric relation $z_{j}(\xi)=\sum_{i=1}^{N}\alpha_{ji}^{\mathfrak{F}\,\prime}z^{\prime}_{i}$.

Next we write the quadratic forms $\omega_{j}^{\mathfrak{F}}(\xi)$, defined in (4.8), in the $z$-coordinates.

Proposition 5.5**.**

Functions $\omega_{j}^{\mathfrak{F}}(\xi)$, restricted to the set $\mathcal{A}_{s}^{\prime}(\mathfrak{F})\ni\xi$ and written in the $z$-coordinates, take the form

[TABLE]

where the S-regular incidence matrix $\alpha^{\mathfrak{F}}=(\alpha_{ij}^{\mathfrak{F}})_{1\leq i,j\leq N}$ is defined in (5.4).

Proof. In this proof we again skip the lower index $\mathfrak{F}$ in the notation $\pi_{\mathfrak{F}}$. Due to (5.8) and their definition (3.14),(4.8), the functions $\omega_{j}^{\mathfrak{F}}(\xi)$ have the form

[TABLE]

Using (5.5)(b), we obtain

[TABLE]

Thus, by (5.7), $\omega_{j}^{\mathfrak{F}}(z)=2z_{j}\cdot\sum_{i=1}^{N}\alpha_{ji}^{\mathfrak{F}}z_{i}.$ ∎

Finally, we note that instead of considering the set of Feynman diagrams $\mathfrak{F}_{m,n}$ it suffices to study only its subset. Denote by

[TABLE]

the subset of diagrams $\mathfrak{F}$ for which the corresponding incidence matrix $\alpha^{\mathfrak{F}}$ is not only S-regular, but is SS-regular (see Definition 5.2). In view of (5.15) we have:

Proposition 5.6**.**

If $\mathfrak{F}\notin\mathfrak{F}^{\,true}_{m,n}$, then $\mathcal{Z}(\mathfrak{F})=\emptyset$. If $\mathfrak{F}\in\mathfrak{F}^{\,true}_{m,n}$, then $\mathcal{Z}(\mathfrak{F})$ is the complement to the union of a finite system of linear subspaces of positive codimension. In particular, it is is an open dense subset of $\mathbb{R}^{dN}$.

Due to (5.14), Proposition 5.6 implies that the integrals $J_{s}(\mathfrak{F})$ from (4.10) with $\mathfrak{F}\notin\mathfrak{F}^{\,true}_{m,n}$ vanish.

5.3. Final formula for expectations ${\mathbb{E}}a^{(m)}_{s}\bar{a}^{(n)}_{s}$

Let $F_{s}^{\mathfrak{F}}(\tau,l,z)$ be the function $F^{\mathfrak{F}}(\tau,l,\xi)$, defined in (4.7), restricted to the set $\mathcal{A}_{s}^{\prime}(\mathfrak{F})$ and written in the $z$-coordinates,

[TABLE]

The function $F^{\mathfrak{F}}_{s}$ depends on $s\in\mathbb{R}^{d}$ through the transformation $z\mapsto\xi$, see (5.13). Changing in the sums $J_{s}(\mathfrak{F})$ the coordinates $\xi$ to $z$ and using (1.28) we see that Lemma 4.1 joined with (5.14) and Propositions 5.6, 5.5 implies the following result (we recall (2.14)):

Theorem 5.7**.**

For any integers $m,n\geq 0$ satisfying $N=m+n\geq 1$, $s\in\mathbb{Z}^{d}_{L}$ and $\tau_{1},\tau_{2}\geq-T$, we have

[TABLE]

where $\mathfrak{F}^{\,true}_{m,n}$ is a finite set of diagrams, defined in Section 5.2, the constants $c_{\mathfrak{F}}\in\{\pm 1,\pm i\}$ are defined in Section 4.2 and

[TABLE]

Here $\Theta(\omega^{\mathfrak{F}},\nu^{-1}l)=\prod_{j=1}^{N}\theta(\omega_{j}^{\mathfrak{F}},\nu^{-1}l_{j})$, while components $\omega_{j}^{\mathfrak{F}}$ of the vector $\omega^{\mathfrak{F}}$ are given by (5.16). The set $\mathcal{Z}(\mathfrak{F})$ is defined in (5.15) and the incidence matrix $\alpha^{\mathfrak{F}}$ from (5.15), (5.16) is SS-regular. The density function $F^{\mathfrak{F}}_{s}$ is real and satisfies

[TABLE]

with a suitable $\delta=\delta_{N}>0$, for any vectors $\mu\in\mathbb{Z}^{d}_{+}$, $\kappa\in\mathbb{Z}_{+}^{dN}$, and any $s\in\mathbb{R}^{d}$, $z\in\mathbb{R}^{Nd}$.

Since the density $F_{s}^{\mathfrak{F}}$ in (5.19) is an affine function of $s\in\mathbb{R}^{d}$ and satisfies (5.20), the sum $J_{s}(\mathfrak{F})$ is a Schwartz function of $s\in\mathbb{R}^{d}$. Thus, the correlations ${\mathbb{E}}a^{(m)}_{s}(\tau_{1})\bar{a}^{(n)}_{s}(\tau_{2})$ extends to a Schwartz function of $s\in\mathbb{R}^{d}$ via the equality (5.18), as is affirmed in Theorem 2.2. In view of this result, below we always assume that $s\in\mathbb{R}^{d}$.

Subsequent analysis of the sums $J_{s}(\mathfrak{F})$ is based on the lemma below, where the function $\Theta(\omega^{\mathfrak{F}}(z),\nu^{-1}l)$ is defined as in (3.13) with $\theta(y,t)=e^{ity}$ (cf. (2.11)).

Lemma 5.8**.**

If $\theta(y,t)=e^{ity}$, then

[TABLE]

where the phase function $\Omega^{\mathfrak{F}}$ is given by

[TABLE]

Proof. Since $\theta(x,t)=e^{itx}$, the function $\Theta$ is given by (5.21) with $\Omega^{\mathfrak{F}}(l,z)=\sum_{j=1}^{N}l_{j}\omega^{\mathfrak{F}}_{j}(z).$ From (5.16) it follows that $\Omega^{\mathfrak{F}}=2\sum_{1\leq i,j\leq N}l_{j}\alpha_{ji}^{\mathfrak{F}}\,z_{j}\cdot z_{i}$. Now (5.22) follows from the skew symmetry of the matrix $\alpha^{\mathfrak{F}}$. ∎

5.4. Continuous approximation

From now on till Section 8.6 we always assume that

[TABLE]

Using that by Proposition 5.6  $\mathcal{Z}(\mathfrak{F})$ with $\mathfrak{F}\in\mathfrak{F}^{\,true}_{m,n}$ is an open dense subset of $\mathbb{R}^{dN}$, we approximate the sum $L^{-Nd}\sum_{z\in\mathcal{Z}(\mathfrak{F})\cap\mathbb{Z}^{Nd}_{L}}$ from (5.19) by an integral over $\mathbb{R}^{dN}$ by applying Theorem 3.1 from [9]. For $d\geq 2$ we get

[TABLE]

where we used (5.21). When $d=1$ the r.h.s. of the inequality above should be modified by adding the term $C^{\#}(s)L^{-1}$, see Remark 3.2 in [9].

Together with (5.18) this estimate implies the main result of the section:

Theorem 5.9**.**

Assume that $d\geq 2$ and the function $\theta$ is as above. Then for any integers $m,n\geq 0$ satisfying $N=m+n\geq 1$, any $s\in\mathbb{R}^{d}$ and $\tau_{1},\tau_{2}\geq-T$, we have

[TABLE]

where

[TABLE]

The constant $c_{\mathfrak{F}}=c_{\mathfrak{D}_{\mathfrak{F}}}$ is defined in (3.13) and the diagram $\mathfrak{D}_{\mathfrak{F}}$ – in (4.2). The real valued density $F_{s}^{\mathfrak{F}}$ is given by (5.17) together with (4.7) and satisfies the estimate (5.20). The phase function $\Omega^{\mathfrak{F}}$ is given by (5.22) where the incidence matrix $\alpha^{\mathfrak{F}}$ is SS-regular.

If $d=1$ then the r.h.s. of (5.23) should be modified by adding to it the term $C^{\#}(s)L^{-1}$.

Theorem 5.9 provides an explicit formula for a correlation ${\mathbb{E}}a_{s}^{(m)}\bar{a}_{s}^{(n)}$ with arbitrary $m$ and $n$, up to an error term of the size $O(L^{-2}\nu^{-2})$. In Section 8.4 we consider an example when $m=n=2$.

Since the functions $F^{\mathfrak{F}}$ and $\Omega^{\mathfrak{F}}$ are given by explicit formulas (5.17) and (5.22), the integral over the variable $l$ in (5.24) can be found explicitly. In Section 8.7 we give the result of this computation, relevant for further study of the energy spectrum of the quasisolutions.

6. Main estimate

6.1. Estimation of integrals $\widetilde{J}_{s}(\mathfrak{F})$

Next we estimate the integrals $\widetilde{J}_{s}(\mathfrak{F})$ from Theorem 5.9.

Theorem 6.1**.**

For any integers $m,n\geq 0$, satisfying $N=m+n\geq 2$, any $\mathfrak{F}\in\mathfrak{F}^{\,true}_{m,n}$ and $s\in\mathbb{R}^{d}$,

[TABLE]

uniformly in $\tau_{1},\tau_{2}\geq-T$, wherethe function $\chi_{d}^{N}$ is defined in (1.15).

Together with Theorem 5.9, estimate (6.1) implies the desired inequality (2.8) and concludes the proof of Theorems 2.2 and 1.1. 101010We recall that $\mathfrak{F}_{m,n}^{\,true}=\emptyset$ for $N=1$, see (4.12), so we study only the case $N\geq 2$.

We deduce (6.1) from an abstract theorem below where we estimate integrals of the following more general form. Let

[TABLE]

where $M\geq 2$, $K\geq 1$, $d\geq 1$, $l=(l_{1},\ldots,l_{K})$ and $z$ is the polyvector $(z_{1},\ldots,z_{M})$ with $z_{i}=(z_{i}^{1},\ldots,z_{i}^{d})\in\mathbb{R}^{d}$. The phase function $Q(l,z)$ is assumed to be linear in $l$ and quadratic in $z$,

[TABLE]

Here $q_{ij}=q_{ji}=(q_{ij}^{k})_{1\leq k\leq K}$ are $K$-vectors, so for $l\in\mathbb{R}^{K}$, Q(l)=\big{(}q_{ij}\cdot l\big{)}_{1\leq i,j\leq M} is a real symmetric $M\times M$–matrix. Denoting by $Q_{k}$, $1\leq k\leq K$, the real symmetric matrices

[TABLE]

we write $Q(l)$ as $\sum_{k}Q_{k}l_{k}$. We denote by $R\leq K$ the rank of the system of vectors $q_{ij}\in\mathbb{R}^{K}$, $1\leq i,j\leq M.$ It equals to the rank of the system of matrices $\{Q_{k},1\leq k\leq K\}$ in the space of $M\times M$ matrices. Indeed, consider the row

[TABLE]

interpreting it as a matrix with columns $Q_{j}$ (regarded as vectors in $\mathbb{R}^{M^{2}}$). Then the two ranks we talk about are the column - and raw-ranks of this matrix, so they coincide.

Note that the phase function $\Omega^{\mathfrak{F}}$ in (5.22) may be written as $Q(l,z)$.

The complex density function $F_{s}(l,z)$ in (6.2) is assumed to be bounded and measurable in $l\in\mathbb{R}^{K},z\in\mathbb{R}^{dN}$, smooth in $z$, depend on the parameter $s\in\mathbb{R}^{d}$, and satisfy

[TABLE]

for any $\kappa\in\mathbb{Z}_{+}^{dM}$ and some $l_{0}\in\mathbb{R}^{K}$. Each function $h_{\kappa}$ is assumed to be monotone decreasing, bounded and such that the function $h_{\kappa}(|l|)$ is integrable over $\mathbb{R}^{K}$. Then

[TABLE]

for appropriate constants $C_{\kappa}>0$. For $k\in\mathbb{N}$ we introduce the functions

[TABLE]

Theorem 6.2**.**

Assume that $\operatorname{tr}Q_{k}=0$ for each $k$. Then, under assumption (6.3)

[TABLE]

The function $C^{\#}$ depends on $F_{s}$ only through the functions $C_{\kappa}^{\#}$ and the constants $C_{\kappa}$ in (6.3), (6.4). It also depends on the tensor $(q^{k}_{ij})$.

Theorem 6.2 is proven in the next subsection. In Appendix 8.8 we note that it implies an asymptotic estimate for parameter-depending integrals of some quotients with asymptotically degenerating divisors, including those which are obtained from the integrals $\widetilde{J}_{s}(\mathfrak{F})$ by explicit integration over $\mathbb{R}_{l}^{N}$. That result is not needed for the present paper, however we believe that it is of independent interest. It is related to [17].

In view of (5.20), the integrals $\widetilde{J}_{s}(\mathfrak{F})$ satisfy the assumptions of Theorem 6.2 with the functions $C^{\#}_{\kappa}$ and the constants $C_{\kappa}$ independent from $\tau_{1},\tau_{2}$. Here, in view of (5.22), we have $q_{ij}\cdot l=\alpha_{ij}^{\mathfrak{F}}(l_{i}-l_{j})$ and it can be shown that the rank $R$ satisfies $R\geq\lceil N/2\rceil$ (this follows solely from the fact that the matrix $\alpha^{\mathfrak{F}}$ is skew-symmetric and does not have zero rows and columns since it is SS-regular). So Theorem 6.2 implies estimate (6.1), where the function $\chi_{d}^{N}$ is replaced by $\psi_{d}^{R}$. In the case $R=d=\lceil N/2\rceil$ (which holds true for certain matrices $\alpha^{\mathfrak{F}}$) this estimate is slightly weaker than (6.1). Indeed, it gives

[TABLE]

so compare to (6.1) we get an additional factor $\ln\nu^{-1}$, unless $N=2,3$. This is insufficient for the purposes of [9]. To overcome this difficulty, in Theorem 7.1 we prove that if the tensor $(q_{ij}^{k})$ has a block-diagonal form, the estimate (6.6) improves and implies (6.1). This is mostly a technical issue, and ideologically (6.1) follows from (6.6). So we postpone Theorem 7.1 till the next Section 7 and first establish estimate (6.6).

6.2. Proof of Theorem 6.2

In this proof the positive constants $C,C_{1}$, etc. and the functions $C^{\#}$ depend on $F_{s}$ as in Theorem 6.2. To simplify presentation we assume that in (6.3)   $l_{0}=0$, but emphasize that the estimates we obtain are uniform in $l_{0}$.

Let $\lambda_{1}(l),\ldots,\lambda_{M}(l)$ be (real) continuous in $l$ eigenvalues of the symmetric matrix $Q(l)$, enumerated in decreasing order:

[TABLE]

The corresponding normalized eigenvectors are vector–functions of $l$, forming an orthonormal basis in $\mathbb{R}^{M}$ for every $l$, and analytic outside an analytic subset of $\mathbb{R}^{K}_{l}$ of positive codimension (while as functions of $l\in\mathbb{R}^{k}$ they even are not continuous). Denoting by $w(l,z)=w=(w_{1},\ldots,w_{M})$, $w_{i}\in\mathbb{R}^{d}$, the polyvector $z=(z_{1},\ldots,z_{M})$, written in this basis, we write function $Q$ as

[TABLE]

Since the transformation $z\mapsto w$ is orthogonal for every $l$, then $dz=dw$ and

[TABLE]

Here $H_{s}(l,w)=F_{s}(l,z(l,w))$, so that the function $H_{s}(l,w)$ again satisfies the estimate (6.3). The following observation plays the key role in the proof.

Lemma 6.3**.**

Let $A=(a_{ij})$ be a real symmetric $M\times M$-matrix, $M\geq 2$, such that $\operatorname{tr}A=0$, and let $\lambda_{1},\ldots,\lambda_{M}$ be its eigenvalues. Then

[TABLE]

Proof. Let $\|A\|_{HS}$ be the Hilbert-Schmidt norm of the matrix $A$. From one hand, $\|A\|_{HS}^{2}$ equals to the l.h.s. of (6.9). From another hand,

[TABLE]

It remains to note that \Big{(}\sum_{i=1}^{M}\lambda_{i}\Big{)}^{2}=(\operatorname{tr}A)^{2}=0. ∎

Applying Lemma 6.3 to the matrix $Q(l)$, we find

[TABLE]

Let us consider the sets

[TABLE]

Below for a set $V\subset\mathbb{R}^{K}_{l}$ we denote by $\langle J_{s},V\rangle$ the integral (6.8) with the domain of integration $\mathbb{R}_{l}^{K}$ replaced by $V$. Recall that $R$ is the rank of the system of $K$–vectors $(q_{ij})$.

Lemma 6.4**.**

Under assumption (6.3) we have $|\langle J_{s},\mathcal{N}_{\nu}^{c}\rangle|\leq C^{\#}(s)\nu^{R}.$

Proof of Lemma 6.4 is given at the end of this section.

It remains to establish estimate (6.6) for the integral $\langle J_{s},\mathcal{N}_{\nu}\rangle$. In view of (6.7),

[TABLE]

So for $l\in\mathcal{N}_{\nu}$ we have $|\lambda_{1}(l)\lambda_{2}(l)|\geq M^{-2}\nu^{2}$. It is known (see [15, 7]) that the (generalized) Fourier transform of the function

[TABLE]

is

[TABLE]

where $\zeta$ is a complex constant of unit norm; here we define the Fourier transform of a regular function $u(x)$ as $\int e^{-ix\cdot\xi}u(x)\,dx.$ Inspired by the stationary phase method [15, 7], we apply the Parseval’s identity to the partial integral over $w_{1},w_{2}$ and get

[TABLE]

for any $w_{\geq 3}=(w_{3},\ldots,w_{M})$, where $\hat{H}_{s}^{1,2}$ stands for the Fourier transform of function $H_{s}$ with respect to the variables $w_{1},w_{2}$. 111111 See Section 12.2 in [9] for justification of this formula without using the generalized Fourier transform. Since $H_{s}$ satisfies assumption (6.3) (where $l_{0}=0$), the r.h.s. above is bounded in absolute value by

[TABLE]

where the function $h\geq 0$ satisfies (6.4). So

[TABLE]

Then, due to (6.10), we have

[TABLE]

Lemma 6.5**.**

Assume that the function $h$ satisfies assumption (6.4). Then $I^{\nu}\leq C\nu^{\min(R,d)}\psi^{R}_{d}(\nu)$.

Lemma 6.5 together with Lemma 6.4 concludes the proof of the theorem. The lemmas are established below. ∎

Proof of Lemma 6.4. Since function $H_{s}$ satisfies (6.3) with $l_{0}=0$, then

[TABLE]

Let us denote by $v_{1},\ldots,v_{R}$ a maximal collection of linearly independent vectors from the set $(q_{ij})_{1\leq i,j\leq M}$. In the integral $\langle J_{s},\mathcal{N}_{\nu}^{c}\rangle$ we make a linear (non-degenerate) change of variable from $l$ to $t=(t_{1},\ldots,t_{K})$ in such a way that $t_{k}=v_{k}\cdot l$ for $k\leq R$. Let $t^{<}=(t_{1},\ldots,t_{R})$ and $t^{>}=(t_{R+1},\ldots,t_{K})$. Due to the identity \mathcal{K}(l)=\frac{1}{2}\sum\big{(}q_{ij}\cdot l\big{)}^{2},

[TABLE]

So denoting by $\mathcal{N}_{\nu,t}^{c}$ the set $\mathcal{N}_{\nu}^{c}$ written in the $t$-coordinates, we obtain $\mathcal{N}_{\nu,t}^{c}\subset\{t:\,|t^{<}|<\sqrt{2}\nu\}.$ Since the function $h_{0}$ is decreasing, inequalities $|l|\geq C|t|$ and $|t|\geq|t^{>}|$ imply that $h_{0}(|l|)\leq h_{0}(C|t|)\leq h_{0}(C|t^{>}|)$. Then

[TABLE]

The last integral is bounded by $C\int_{0}^{\infty}h_{0}(r)r^{K-R-1}\,dr<C_{1}$, in view of assumption (6.4). ∎

Proof of Lemma 6.5. We make the change of coordinates $l\mapsto t$ as in the proof of Lemma 6.4 and use the notation $t^{<},\,t^{>},$ introduced there. Denoting by $\mathcal{N}_{\nu,t}$ the set $\mathcal{N}_{\nu}$ written in the $t$-coordinates, in view of (6.11) we find $\mathcal{N}_{\nu,t}\subset\{t:\,|t^{<}|\geq C^{\prime}\nu\}$. Then, using that $h(|l|)\leq h(C_{1}|t|)$, by (6.11) we get

[TABLE]

Since the function $h(|t|)$ is integrable, the r.h.s. above is bounded by $C\nu^{d}$, if we take the integral only over the set where $|t^{<}|\geq 1$. Then, it remains to get the desired estimate for the integral over the set where $C\nu\leq|t^{<}|\leq 1$. Since the function $h$ is decreasing, we have $h(C_{1}|t|)\leq h(C_{1}|t^{>}|)$, so this integral is bounded by

[TABLE]

where we have integrated out the variable $t^{>}$. Passing to the spherical coordinates, we see that the latter integral is bounded by

[TABLE]

The r.h.s. of the last inequality can be rewritten as $C_{4}\nu^{\min(R,d)}\psi_{d}^{R}(\nu).$ ∎

7. Refinement of estimate (6.6) and proof of Theorem 6.1

In this section we establish Theorem 7.1, in which we improve estimate (6.6), and deduce from it Theorem 6.1.

7.1. Refinement of (6.6)

We consider integral (6.2), where to simplify the formulation of result (and since it suffices for our purpose) we assume $M=K$ and denote $N:=M=K\geq 2$. We consider a decomposition of the set $\{1,\ldots N\}$ into $p\geq 1$ non-empty disjoint subsets

[TABLE]

and denote by $l^{k}$ the vector $l^{k}=(l_{i})_{i\in\mathcal{I}_{k}}$.

We assume that the phase function $Q$ admits a block-decomposition

[TABLE]

where $q_{ij}=q_{ji}\in\mathbb{R}^{|\mathcal{I}_{k}|}$ for $i,j\in\mathcal{I}_{k}$. Concerning the density $F_{s}$ we assume that it satisfies the estimate

[TABLE]

for any $\kappa$ and some vectors $l^{i}_{0}\in\mathbb{R}^{|\mathcal{I}_{i}|}$. The functions $h^{i}_{\kappa}$ are assumed to satisfy the same assumption that we imposed on the functions $h_{\kappa}$ in (6.3), where in (6.4) we replace $K$ by $|\mathcal{I}_{i}|$. Denote by $R_{k}$ the rank of the system of $|\mathcal{I}_{k}|$-vectors $(q_{ij})_{i,j\in\mathcal{I}_{k}}$.

Theorem 7.1**.**

Assume that $\sum_{i\in\mathcal{I}_{k}}q_{ii}=0\in\mathbb{R}^{|\mathcal{I}_{k}|}$ for any $1\leq k\leq p$. Then, under assumption (7.2) we have

[TABLE]

where the function $C^{\#}$ depends on $F_{s}$ as in Theorem 6.2.

Proof. We argue by induction. In the case $p=1$ the assertion follows from Theorem 6.2. Assume that theorem is proven for the case of $p-1$ subsets $\mathcal{I}_{k}$ in (7.1) and let us establish it for the case of $p$ subsets. We recall the notation $l^{p}=(l_{i})_{i\in\mathcal{I}_{p}}$ and set $l^{<}=(l_{i})_{i\notin\mathcal{I}_{p}}$. Similarly, we denote $z^{p}=(z_{i})_{i\in\mathcal{I}_{p}}$ and $z^{<}=(z_{i})_{i\notin\mathcal{I}_{p}}$. Let $N_{p}:=|\mathcal{I}_{p}|$ and

[TABLE]

where we recall that the function $U_{p}(l,z)$ depends only on $l^{p}$ and $z^{p}$. Then,

[TABLE]

for any $\kappa$. Literally repeating the proof of Theorem 6.2, we get

[TABLE]

where $g(\nu)=\nu^{\min(R_{p},d)}\psi^{R_{p}}_{d}(\nu)$ and the functions $h^{i}$ satisfy the same assumptions that the functions $h_{\kappa}^{i}$ from (7.2). Thus, we see that the function $G_{s}(l^{<},z^{<})=(g(\nu))^{-1}I_{s}(l^{<},z^{<})$ satisfies assumption (7.2) with $p=p-1$. Note that

[TABLE]

where $U_{k}(l,z)$ depend only on $l^{<},z^{<}$. Then, by the induction hypothesis,

[TABLE]

∎

7.2. Proof of Theorem 6.1

Recall that the phase function $\Omega^{\mathfrak{F}}$ from the definition (5.24) of the integral $\widetilde{J}_{s}(\mathfrak{F})$ is given by (5.22) and that for $\mathfrak{F}\in\mathfrak{F}^{\,true}_{m,n}$ the matrix $\alpha^{\mathfrak{F}}$ is SS-regular (see Denition 5.2). Consider a partition (7.1) with $p\geq 1$ such that the matrices $\alpha_{k}=(\alpha_{ij}^{\mathfrak{F}})_{i,j\in\mathcal{I}_{k}}$ are irreducible and $\alpha_{ij}^{\mathfrak{F}}=0$ once $i\in\mathcal{I}_{k}$ and $j\in\mathcal{I}_{r}$ with $k\neq r$. Then the integral $\widetilde{J}_{s}(\mathfrak{F})$ satisfies assumptions of Theorem 7.1 with $q_{ij}\cdot l=\alpha_{ij}^{\mathfrak{F}}(l_{i}-l_{j})$.

Lemma 7.2**.**

Let $A=(a_{ij})$ be an irreducible $n\times n$-matrix, $n\geq 2$, and the vectors $r_{ij}\in\mathbb{R}^{n}$ be given by $r_{ij}=a_{ij}(e_{i}-e_{j})$, where $(e_{i})_{1\leq i\leq n}$ is a basis in $\mathbb{R}^{n}$. Then rank $R$ of the system of $n$-vectors $(r_{ij})_{1\leq i,j\leq n}$ equals $n-1$.

Proof of Lemma 7.2 is a simple exercise from linear algebra and we postpone it to Appendix 8.5. Set $N_{k}=|\mathcal{I}_{k}|$. Since the $N_{k}\times N_{k}$-matrices $\alpha_{k}$ are skew-symmetric and non-zero, we have $N_{k}\geq 2$. Applying Lemma 7.2 to the system of vectors $(q_{ij})_{i,j\in\mathcal{I}_{k}}$, we see that its rank $R_{k}$ equals to $N_{k}-1$. Then by Theorem 7.1 we get the estimate

[TABLE]

Let us show that (7.3) implies the desired estimate (6.1). Assume first that $N_{i}-1<d$ for every $1\leq i\leq p$. Then, in view of (7.3) and the identity $\sum_{i=1}^{p}N_{i}=N$, we have

[TABLE]

Since $N_{i}\geq 2$ for any $i$, the number of blocks $p$ satisfies $p\leq\lfloor N/2\rfloor$, where $\lfloor\cdot\rfloor$ denotes the (lower) entire part. Then $N-p\geq\lceil N/2\rceil$, so we get $\nu^{N-p}\leq\nu^{\lceil N/2\rceil}$ which implies (6.1).

Now we assume that

[TABLE]

so $\nu^{\min(N_{j}-1,d)}=\nu^{d}$. Since for every $i$ we have $\nu^{\min(N_{i}-1,d)}\leq\nu$, by (7.3) we find

[TABLE]

In the case $p\geq 2$ this implies (6.1). Now it remains to consider the case when (7.4) holds with $p=1$. Then $N-1\geq d$. If $N-1>d$ then (7.3) with $p=1$ implies (6.1). Otherwise $N-1=d$, so

[TABLE]

If $N-1>\lceil N/2\rceil$ then $\nu^{N-1}\ln\nu^{-1}<C\nu^{\lceil N/2\rceil}$ and we are done. The opposite situation is possible only if $N\leq 3$. Since $d=N-1$, it takes place if $d=2$, $N=3$ or $d=1,$ $N=2$. In these cases we find $N-1=\lceil N/2\rceil=d$, which is 2 or $1$ correspondingly, so again (7.5) implies (6.1). ∎

8. Addenda

8.1. Optimality of the estimate in Theorem 2.2

Throughout this section we assume that $N=m+n\geq 2$ and $d\geq 2$. Accordingly to Theorem 5.9, we have

[TABLE]

where we recall that the integrals $\widetilde{J}_{s}(\mathfrak{F})$ are given by (5.24) and $\mathfrak{F}^{\,true}_{m,n}$ is a certain subset of the set of Feynman diagrams $\mathfrak{F}$ (which are defined in Section 4), associated with the product $a^{(m)}_{s}\bar{a}^{(n)}_{s}$. Each integral $\widetilde{J}_{s}(\mathfrak{F})$ satisfies the estimate (6.1), and this implies the estimate in Theorem 2.2. Since $\nu\ln\nu^{-1}\leq C$ these estimates admit the following slightly weaker forms:

[TABLE]

In this appendix we discuss the question if it is possible to get rid of the minimum with $d$ in the estimates above, that is, if it is true that

[TABLE]

for any $N\geq 2$. We discussed the motivation for this question in Section 1.4.

Let $\mathfrak{F}^{2}_{m,n}\subset\mathfrak{F}^{\,true}_{m,n}$ be a subset of Feynman diagrams $\mathfrak{F}$ for which there is $q=q(\mathfrak{F})$ such that the matrix $\alpha^{\mathfrak{F}}$ from (5.22) satisfies $\alpha_{ij}^{\mathfrak{F}}=0$ if both $i\neq q$ and $j\neq q$ (so, all elements of the matrix $\alpha^{\mathfrak{F}}$ outside the $q$-th line and column are zero). It can be shown that $\mathfrak{F}^{2}_{m,n}\neq\emptyset$ if $m,n\geq 1$.

Proposition 8.1**.**

If $m,n\geq 1$ and $N>2d$, then for any Feynman diagram $\mathfrak{F}\in\mathfrak{F}^{2}_{m,n}$ we have $C_{1}^{\#}(s)\nu^{d}\leq|\widetilde{J}_{s}(\mathfrak{F})|\leq C_{2}^{\#}(s)\nu^{d}$.

Proposition 8.1 shows that the first estimate from (8.3) is false for $\mathfrak{F}\in\mathfrak{F}^{2}_{m,n}$ when $N>2d$, so in general we have only (8.2). Then, if the large integrals $\widetilde{J}_{s}(\mathfrak{F})$ of the size $\nu^{d}$ did not cancel each other in the sum from (8.1), the second estimate from (8.3) would also fail. Moreover, if they did not cancel in the sums from (1.25) that give $n_{s}^{k}$, then as we have discussed in Section 1.4, this would imply that the energy spectrum corresponding to a truncation $\sum_{i=0}^{m}\rho^{i}v_{s}^{(i)}$ with $m>d$ of the decomposition (1.12) cannot be approximated by a solution of the WKE (1.23). However, we find some strong cancellations in the sum from (8.1):

Proposition 8.2**.**

We have \big{|}\sum_{\mathfrak{F}\in\mathfrak{F}^{2}_{m,n}}\widetilde{J}_{s}(\mathfrak{F})\big{|}\leq C^{\#}(s)\nu^{N-1} for $N\geq 2$.

Since $\nu^{N-1}\leq\nu^{N/2}$, this estimate does not prevent the second estimate from (8.3) to be true. We expect that for $N\gg 2d$ the integrals $\widetilde{J}_{s}(\mathfrak{F})$ with $\mathfrak{F}\in\mathfrak{F}^{2}_{m,n}$ are the biggest (see below) but still there are many other diagrams $\mathfrak{F}\notin\mathfrak{F}^{2}_{m,n}$ for which the corresponding integrals also are large (bigger than $\nu^{N/2}$). Nevertheless, we expect that similar cancellations take place for them as well, but do not know if they go till the order $\nu^{N/2}$. So, we state the following

Problem 8.3**.**

Prove that for any $d\geq 2$ and $m,n$ satisfying $N=m+n\geq 2$,

[TABLE]

If (8.4) is true, we get the second estimate from (8.3). In particular, under the ”kinetic” scaling $\rho=\sqrt{\varepsilon}\nu^{-1/2}$ the energy spectrum corresponding to a truncation $\sum_{i=0}^{m}\rho^{i}v_{s}^{(i)}$ of series (1.12) with any $m\geq 2$ can be approximated by a solution of the WKE (1.23) at least with the accuracy $\varepsilon^{2}$, if $L^{-2}\nu^{-2}\leq\nu^{m}$.

Discussion of proofs of Propositions 8.1, 8.2 and of Problem 8.3. Below we do not give proofs but present only their rough ideas. Let us consider the matrix \alpha^{\mathfrak{F}}_{l}=\big{(}\alpha_{ij}^{\mathfrak{F}}(l_{i}-l_{j})\big{)}_{1\leq i,j\leq N}, which depends on $l\in\mathbb{R}^{N}$ (see (5.22)). Since it is symmetric, it has $N$ real eigenvalues, which we enumerate in the decreasing order $|\lambda_{1}(l)|\geq|\lambda_{2}(l)|\geq\ldots\geq|\lambda_{N}(l)|\geq 0$, for any $l\in\mathbb{R}^{N}$. It is possible to show that the set $\mathfrak{F}^{2}_{m,n}$ can be equivalently defined as follows. For $k\geq 1$ we denote by $\mathfrak{F}^{k}_{m,n}$ a set of Feynman diagrams $\mathfrak{F}$ for which the eigenvalues $\lambda_{1}(l),\ldots,\lambda_{k}(l)$ are not identically zero while $\lambda_{i}(l)\equiv 0$ for any $i\geq k+1$. Since the matrix $\alpha_{l}^{\mathfrak{F}}$ is non-zero but has zero trace, we have $\mathfrak{F}^{1}_{m,n}=\emptyset$, so $\mathfrak{F}^{\,true}_{m,n}=\cup_{k=2}^{N}\mathfrak{F}^{k}_{m,n}$.

Let us take a diagram $\mathfrak{F}\in\mathfrak{F}^{2}_{m,n}$ and consider the integral over $dz$ from the definition (5.24) of $\widetilde{J}_{s}(\mathfrak{F})$. Applying to it the Parseval identity in the eigendirections corresponding to the eigenvalues $\lambda_{1}$ and $\lambda_{2}$, and using that for $\mathfrak{F}\in\mathfrak{F}^{2}_{m,n}$ we have $\lambda_{3}=\ldots=\lambda_{N}=0$, we find that $\displaystyle{\widetilde{J}_{s}(\mathfrak{F})\sim\nu^{d}\int_{\mathbb{R}^{N}}\frac{h_{s}^{\mathfrak{F}}(l)\,dl}{|\lambda_{1}(l)\lambda_{2}(l)|^{d/2}}}$ if $N>2d$, for an appropriate integrable function $h_{s}^{\mathfrak{F}}$, which is fast decaying in $s$. Due to Lemma 6.3, we have $\lambda_{1}\lambda_{2}=-\frac{1}{2}\sum\limits_{1\leq i,j\leq N}(\alpha_{ij}^{\mathfrak{F}})^{2}(l_{i}-l_{j})^{2}$. Using this formula, we show that the latter integral converges, which proves Proposition 8.1.

Below by a graph of a Feynman diagram we mean the diagram, where we do not specify which vertices are virtual, which have positive degree and which are leaves, so the graph is just a collection of vertices coupled by edges. To establish Proposition 8.2, we note that, up to a renumbering of vertices, all diagrams $\mathfrak{F}\in\mathfrak{F}^{2}_{m,n}$ have the same graph which we denote by $\mathfrak{G}^{2}$. To recover the set of diagrams $\mathfrak{F}^{2}_{m,n}$ from this graph, it suffices to look over all possible placings of virtual vertices (once positions of the virtual vertices are known, degrees of the other vertices are recovered uniquely).

The graph $\mathfrak{G}^{2}$ is such that in a $k$-th block with any $k\neq 1,\,m+1$, the virtual vertex can be placed in two different positions, $c_{2k}$ or $\bar{c}_{2k}$. Now, using the stationary phase method [7, 15], we develop each integral $\widetilde{J}_{s}(\mathfrak{F})$ in $\nu$. Change of position of a virtual vertex inside of a block, roughly speaking, changes only the sign in first $p$ orders of this decomposition. This is where the cancellations come from. The number $p$ here equals to the number of blocks in which position of the virtual vertex is not determined uniquely, that is $p=N-2$. After some work, we deduce from this Proposition 8.2.

Let us now discuss Problem 8.3. Take a diagram $\mathfrak{F}_{0}\notin\mathfrak{F}^{2}_{m,n}$ and consider its graph $\mathfrak{G}_{\mathfrak{F}_{0}}$. We expect that in the sum

[TABLE]

there are similar cancellations. However, the graph $\mathfrak{G}^{2}$ is ”very symmetric” (we are able to draw it explicitly), while for a general diagram $\mathfrak{F}_{0}$ there are less symmetries in the graph $\mathfrak{G}_{\mathfrak{F}_{0}}$. That is why we expect that in the sum (8.5) the number $p$ of orders, in which the cancellations take place, is smaller than in the sum from Proposition 8.2, so we expect to get less ”additional degrees” of $\nu$ than in Proposition 8.2. Then the estimate (8.4) can be true only if the integrals $\widetilde{J}_{s}(\mathfrak{F})$ are smaller than those from Proposition 8.2, once $N>2d$. Namely, if there exists $r>d$ sufficiently large such that for the diagrams $\mathfrak{F}$ satisfying $\mathfrak{G}_{\mathfrak{F}}=\mathfrak{G}_{\mathfrak{F}_{0}}$ we have $|\widetilde{J}_{s}(\mathfrak{F})|\leq C^{\#}(s)\nu^{r}.$

It is plausible that this is indeed the case and below we explain the reason. Let $k>2$ be such that $\mathfrak{F}_{0}\in\mathfrak{F}^{k}_{m,n}$. Then it is possible to show that $\mathfrak{F}\in\mathfrak{F}^{k}_{m,n}$ as well for any diagram $\mathfrak{F}$ satisfying $\mathfrak{G}_{\mathfrak{F}}=\mathfrak{G}_{\mathfrak{F}_{0}}$. Then, arguing as in the proof of Proposition 8.1 and applying the stationary phase method in the eigendirections corresponding to the eigenvalues $\lambda_{1},\ldots,\lambda_{k}$, we get $\displaystyle{\widetilde{J}_{s}(\mathfrak{F})\sim\nu^{kd/2}\int_{\mathbb{R}^{N}}\frac{h_{s}^{\mathfrak{F}}(l)\,dl}{|\lambda_{1}(l)\cdots\lambda_{k}(l)|^{d/2}}}$, if the latter integral converges. Perhaps, the factor $\nu^{kd/2}$ could lead to the desired estimate. However, we are not able to establish the convergence since we do not know a good estimate from below for the product $\lambda_{1}(l)\cdots\lambda_{k}(l)$ in the case $k>2$, in difference with the situation when $k=2$ considered in Proposition 8.1.

8.2. Proof of Lemma 4.1

We compute the expectation of the both sides of (3.16). Let us take any diagram $\mathfrak{D}\in\mathfrak{D}_{m}\times\overline{\mathfrak{D}}_{n}$ and fix $(\xi,\sigma)\in\mathcal{A}_{s}^{L}(\mathfrak{D})$. Recall that ${\mathbb{E}}a^{(0)}_{s^{\prime}}a^{(0)}_{s^{\prime\prime}}={\mathbb{E}}\bar{a}^{(0)}_{s^{\prime}}\bar{a}^{(0)}_{s^{\prime\prime}}=0$, for any $s^{\prime},s^{\prime\prime}$. Then, by the Wick theorem we find

[TABLE]

where the summation is taken over all bijections $f:\,\bar{L}(\mathfrak{D})\mapsto L(\mathfrak{D})$ of the set $\bar{L}(\mathfrak{D})$ indexing conjugated leaves to the set $L(\mathfrak{D})$ indexing non-conjugated leaves. Let us fix any such bijection $f$ and couple by an edge in the diagram $\mathfrak{D}$ the leaf $c_{f(j)}$ with the leaf $\bar{c}_{j}$ for every $j\in\bar{L}(\mathfrak{D})$ (that is, the leaves which take values $a^{(0)}_{\xi_{f(j)}}$and $\bar{a}^{(0)}_{\sigma_{j}}$); denote the obtained diagram by $\mathfrak{F}_{f}$. The summation over the bijections $f$ in (8.6) can be replaced by the summation over the diagrams $\mathfrak{F}_{f}$.

The only difference between the set $\mathfrak{F}(\mathfrak{D})$ of Feynman diagrams obtained from the diagram $\mathfrak{D}$ with the set of diagrams $\mathfrak{F}_{f}$ is that, by definition, in any $\mathfrak{F}\in\mathfrak{F}(\mathfrak{D})$ there are no leaves from the same block which are adjacent. Thus, for any $\mathfrak{F}_{f}\notin\mathfrak{F}(\mathfrak{D})$ there are adjacent leaves $c_{f(j)}^{(0)}\sim\bar{c}_{j}^{(0)}$ that belong to the same block. Since by (3.7) we have $\xi_{f(j)}\neq\sigma_{j}$ for $(\xi,\sigma)\in\mathcal{A}_{s}^{L}(\mathfrak{D})$, then ${\mathbb{E}}a^{(0)}_{\xi_{f(j)}}\bar{a}^{(0)}_{\sigma_{j}}=0$ by (2.15), so the product from the r.h.s. of (8.6), corresponding to the bijection $f$, vanishes. Therefore, it suffices to take the summation only over the set of Feynman diagrams $\mathfrak{F}(\mathfrak{D})$. Then, denoting $\xi_{\psi}:=\xi_{i}$ and $\sigma_{\psi}:=\sigma_{j}$ if vertices $c_{i}$ and $\bar{c}_{j}$ are coupled by an edge $\psi$, we rewrite (8.6) as

[TABLE]

by (2.15), where we recall that the set $E_{L}(\mathfrak{F})$, the times $l^{c_{\psi}},l^{\bar{c}_{\psi}}$ and the functions $b^{\psi}$, $\gamma^{\psi}$ are defined in Section 4.2. Since $E(\mathfrak{D})=E_{D}(\mathfrak{F})$ for $\mathfrak{F}\in\mathfrak{F}(\mathfrak{D})$, we get

[TABLE]

Then, taking the expectation of the both sides of (3.17) and using that, by definition of $c_{\mathfrak{F}}$, we have $c_{\mathfrak{F}}=c_{\mathfrak{D}}$ for $\mathfrak{F}\in\mathfrak{F}(\mathfrak{D})$, we find

[TABLE]

Next we note that $(\xi,\sigma)\in\mathcal{A}_{s}^{L}(\mathfrak{D})$ and $\delta_{\xi_{\psi}}^{\sigma_{\psi}}=1$ for every edge $\psi\in E_{L}(\mathfrak{F})$ if and only if $\xi\in\mathcal{A}_{s}^{\prime L}(\mathfrak{F})$ and $\sigma=\xi_{f_{\mathfrak{F}}}$, where we recall that $\xi_{f_{\mathfrak{F}}}$ is defined in (4.4). Then in the sum above we can replace the summation $\sum_{(\xi,\sigma)\in\mathcal{A}_{s}^{L}(\mathfrak{D})}$ by $\sum_{\xi\in\mathcal{A}_{s}^{\prime L}(\mathfrak{F})}$ and substitute $\sigma=\xi_{f_{\mathfrak{F}}}$. Then $\omega(\xi,\sigma)$ replaces by $\omega^{\mathfrak{F}}(\xi)$ (see (4.8)) and we get ${\mathbb{E}}I_{s}(\mathfrak{D})=\sum_{\mathfrak{F}\in\mathfrak{F}(\mathfrak{D})}c_{\mathfrak{F}}J_{s}(\mathfrak{F})$. Now the desired formula (4.9) follows from the identity $\sum_{\mathfrak{D}\in\mathfrak{D}_{m}\times\overline{\mathfrak{D}}_{n}}\sum_{\mathfrak{F}\in\mathfrak{F}(\mathfrak{D})}=\sum_{\mathfrak{F}\in\mathfrak{F}_{m,n}},$ which is due to (4.1).

It remains to derive the estimate (4.11). Recall that $\gamma_{s}\geq 1$ for any $s\in\mathbb{R}^{d}$. Let $\widetilde{F}^{\mathfrak{F}}$ be a function defined as $F^{\mathfrak{F}}$, where in the product over $\varphi\in E_{D}(\mathfrak{F})$ the exponent $\gamma^{\varphi}$ is replaced by $\gamma^{\varphi}-1/2$, that is

[TABLE]

Since the product over $\varphi$ above does not depend on $\xi$, it suffices to show that

[TABLE]

for $\xi\in\mathcal{A}^{\prime}_{s}(\mathfrak{F})$ and

[TABLE]

where $\delta=\delta_{N}>0$.

We start with the estimate (8.7). Since $b$ is a Schwartz function and, clearly, $\prod_{\psi\in E_{L}(\mathfrak{F})}(b^{\psi}(\xi))^{2}=\prod_{i\in L(\mathfrak{F})}(b(\xi_{i}))^{2}$, we find

[TABLE]

where $P_{\kappa}$ is a polynomial. Thus, it remains to show that C_{\kappa}^{\#}\big{(}(\xi_{i})_{i\in L(\mathfrak{F})}\big{)}\leq C_{1\kappa}^{\#}(\xi) for any $\xi\in\mathcal{A}_{s}^{\prime}(\mathfrak{F})$, or equivalently

[TABLE]

for any $j\notin L(\mathfrak{F})$. Inequality (8.9) easily follows from assumption (4.6). Indeed, since for $j\notin L(\mathfrak{F})$ a vertex $c^{(p)}_{j}$ has positive degree $p\geq 1$, it is coupled in $\mathfrak{F}$ with a virtual vertex $\bar{w}_{2k}$. Then, for $\xi\in\mathcal{A}_{s}^{\prime}(\mathfrak{F})$ we have $f_{\mathfrak{F}}(2k)=j$, so (4.6) implies $\xi_{2k-1}+\xi_{2k}=\xi_{f_{\mathfrak{F}}(2k-1)}+\xi_{j}$. Then, among the indices $\xi_{2k-1},\xi_{2k}$ and $\xi_{f_{\mathfrak{F}}(2k-1)}$ there is at least one, say $\xi_{2k-1}$, satisfying $|\xi_{2k-1}|\geq|\xi_{j}|/3$. If $2k-1\in L(\mathfrak{F})$ then we get (8.9), so we are done. Otherwise, the vertex $c^{(p_{1})}_{2k-1}$ is not a leaf and we continue the procedure in the same way until we arrive at a leaf. If it is $\xi_{2k}$ or $\xi_{f_{\mathfrak{F}}(2k-1)}$ who dominates $|\xi_{j}|/3$ then we argue in the same way.

Now we establish the estimate (8.8). We recall that $E_{D}(\mathfrak{F})$ is the set of edges of the diagram $\mathfrak{F}$ which couple its virtual vertices with vertices of positive degree. Fix any $1\leq k\leq m$. There exist edges $\varphi_{1},\ldots,\varphi_{p}\in E_{D}(\mathfrak{F})$ which form a path from the block $\hat{B}_{k}$ of the diagram $\mathfrak{F}$ to the root $a^{(m)}_{s}$. That is, the edge $\varphi_{1}$ couples the virtual vertex of the block $\hat{B}_{k}$ with a vertex of a block $\hat{B}_{j_{1}}$, the edge $\varphi_{2}$ couples the virtual vertex of the block $\hat{B}_{j_{1}}$ with a vertex from a block $\hat{B}_{j_{2}}$, etc. Finally, the edge $\varphi_{p}$ couples the virtual vertex of the block $\hat{B}_{j_{p-1}}$ with the root $c^{(m)}$ (here, essentially, $\hat{B}_{j_{p-1}}=B_{1}$). Accordingly to the indicator function in (8.8), the times $l_{k},l_{j_{1}},\ldots,l_{j_{p-1}}$ assigned to the blocks $\hat{B}_{k},\hat{B}_{j_{1}},\ldots,\hat{B}_{j_{p-1}}$ satisfy $-T\leq l_{k}\leq l_{j_{1}}\leq\ldots\leq l_{j_{p-1}}\leq\tau_{1}$. Then, bounding by one those exponents from (8.8) which correspond to edges $\varphi\notin\{\varphi_{1},\ldots,\varphi_{p}\}$, we get

[TABLE]

Since (8.10) holds for any $1\leq k\leq m$, we get that its l.h.s. is bounded by $e^{-\frac{1}{2m}\sum_{i=1}^{m}|\tau_{1}-l_{i}|}$.

Now we fix any $m+1\leq k\leq N$ and find a path from the block $\hat{B}_{k}$ to the root $\bar{c}^{(n)}$. Arguing as above we get

[TABLE]

Combining the two obtained estimates we get that (8.8) holds with $\delta=\frac{1}{4N}$.

We conclude the proof of the lemma by noting that the absolute convergence of the sum $J_{s}(\mathfrak{F})$ follows from the estimate (4.11). ∎

8.3. Proof of Lemma 5.1

Let us consider a graph $\mathfrak{C}^{\prime}_{\mathfrak{F}}$ which is obtained from $\mathfrak{U}_{\mathfrak{F}}$ by erasing in each block two (of four) dashed edges. We choose the edges we preserve arbitrarily, but in such a way that in the graph $\mathfrak{C}^{\prime}_{\mathfrak{F}}$ each vertex is incident to exactly two edges, one of which is solid and another one is dashed. We will modify the choice of the preserved dashed edges in certain blocks of $\mathfrak{C}^{\prime}_{\mathfrak{F}}$ in such a way that $\mathfrak{C}^{\prime}_{\mathfrak{F}}$ will become a cycle. Since in the graph $\mathfrak{C}^{\prime}_{\mathfrak{F}}$ every vertex is incident to two edges, $\mathfrak{C}^{\prime}_{\mathfrak{F}}$ decouples to a union of $k\geq 1$ disjoint cycles $\mathfrak{C}_{i}$, $1\leq i\leq k$, in which solid and dashed edges alternate. If $k=1$ then we are done. Assume that $k>1.$ We claim that in this case there exists a block $\hat{B}_{r}=\{c_{2r-1},c_{2r},\bar{c}_{2r-1},\bar{c}_{2r}\}$ such that some but not all of its vertices belongs to cycle $\mathfrak{C}_{1}$.

Indeed, assume that the claim is not true, so that each block either entirely belongs to the cycle $\mathfrak{C}_{1}$, or its intersection with $\mathfrak{C}_{1}$ is empty. Take a block $\hat{B}_{p}$ belonging to $\mathfrak{C}_{1}$. Then any vertex $\hat{c}_{j}\notin\hat{B}_{p}$ which is adjacent with a vertex from the block $\hat{B}_{p}$ (by construction of $\mathfrak{C}^{\prime}_{\mathfrak{F}}$ such vertex exists), belongs to the cycle $\mathfrak{C}_{1}$ and, consequently, all other vertices from the block to which belongs $\hat{c}_{j}$ belong to $\mathfrak{C}_{1}$ as well. Arguing in this way, by induction we obtain that the cycle $\mathfrak{C}_{1}$ contains all blocks of the diagram $\mathfrak{C}^{\prime}_{\mathfrak{F}}$, as well as the vertices $c_{0}$ and $\bar{c}_{0}$. Thus, $\mathfrak{C}_{1}=\mathfrak{C}^{\prime}_{\mathfrak{F}}$ which contradicts to the assumption $k>1$, so the claim is true.

Let $\hat{B}_{r}$ be a block as in the claim. By construction of the diagram $\mathfrak{C}^{\prime}_{\mathfrak{F}}$, its dashed edges either provide the couplings $c_{2r-1}\thicksim\bar{c}_{2r-1}$ and $c_{2r}\thicksim\bar{c}_{2r}$ or the couplings $c_{2r-1}\thicksim\bar{c}_{2r}$ and $c_{2r}\thicksim\bar{c}_{2r-1}$. For definiteness, we assume that $c_{2r-1}\thicksim\bar{c}_{2r-1}\in\mathfrak{C}_{1}$ and $c_{2r}\thicksim\bar{c}_{2r}\in\mathfrak{C}_{2}$, as in fig. 8(a). Let us erase the (dashed) edges that couple the vertices $c_{2r-1}$ with $\bar{c}_{2r-1}$ and $c_{2r}$ with $\bar{c}_{2r}$, and instead couple the vertices $c_{2r-1}$ with $\bar{c}_{2r}$ and $c_{2r}$ with $\bar{c}_{2r-1}$. Then from the diagram $\mathfrak{C}^{\prime}_{\mathfrak{F}}$ we obtain another diagram $\mathfrak{C}^{\prime\prime}_{\mathfrak{F}}$, constructed from $\mathfrak{U}_{\mathfrak{F}}$ in the same way as the diagram $\mathfrak{C}^{\prime}_{\mathfrak{F}}$, in which there is one cycle $\mathfrak{C}_{12}$ instead of the two disjoint cycles $\mathfrak{C}_{1}$ and $\mathfrak{C}_{2}$ in $\mathfrak{C}^{\prime}_{\mathfrak{F}}$, see fig. 8(b). So the total number of cycles in the diagram $\mathfrak{C}^{\prime\prime}_{\mathfrak{F}}$ is $k-1$, where we recall that by our assumption in $\mathfrak{C}^{\prime}_{\mathfrak{F}}$ there are $k>1$ cycles. Then, arguing by induction, we arrive at a diagram $\mathfrak{C}_{\mathfrak{F}}$ which is a cycle itself.

8.4. Example: computation of ${\mathbb{E}}a_{s}^{(2)}\bar{a}_{s}^{(2)}$

In this section we apply techniques, developed in Sections 3-5, to analyse the expectation ${\mathbb{E}}a_{s}^{(2)}\bar{a}_{s}^{(2)}$. Namely, we fix one of Feynman diagrams $\mathfrak{F}\in\mathfrak{F}^{\,true}_{2,2}$, – say, that from fig. 9, – and find explicitly the corresponding integral $\widetilde{J}_{s}(\mathfrak{F})$ (in total there are about 25 different Feynman diagrams in the set $\mathfrak{F}^{\,true}_{2,2}$).

The cycle $\mathfrak{C}_{\mathfrak{F}}$ for the considered diagram $\mathfrak{F}$ can be chosen as in fig 9, so that the associated cycle $\hat{\mathfrak{C}}_{\mathfrak{F}}$ has the form

[TABLE]

Accordingly to (5.22), the phase function $\Omega^{\mathfrak{F}}$ has the form

[TABLE]

Assuming for simplicity $T=\infty$, by (4.7) the function $F^{\mathfrak{F}}$ reads

[TABLE]

where we denote $B(s)=(b(s))^{2}/\gamma_{s}$. Here the first two lines correspond to the product over the set $E_{D}(\mathfrak{F})$ in the formula (4.7) and the last to lines  – to the product over $E_{L}(\mathfrak{F})$. Accordingly to (5.17), to write down the function $F_{s}^{\mathfrak{F}}$ from (5.24) it remains to write the functions $\xi_{i}(z)$, which are given by (5.13):

[TABLE]

In Appendix 8.7, assuming for simplicity $\tau_{1}=\tau_{2}$ and $T=\infty$, we explicitly compute the integral over the time variable $dl$ in (5.24) and discuss the motivation for this. Below we present the result of this computation for the example we are considering (here we do not restrict ourselves to the case $\tau_{1}=\tau_{2}$ but assume $T=\infty$).

We have $\omega_{1}^{\mathfrak{F}}(z)=2z_{1}\cdot(z_{2}+z_{3})$, $\omega_{2}^{\mathfrak{F}}(z)=2z_{2}\cdot(-z_{1}-z_{3}),$ $\omega_{3}^{\mathfrak{F}}(z)=2z_{3}\cdot(-z_{1}+z_{2}-z_{4})$ and $\omega_{4}^{\mathfrak{F}}(z)=2z_{4}\cdot z_{3}$. Computing the integral over $dl$ we get

[TABLE]

where for $j=1,2,3$ we take ”$+$” in the term $\pm\gamma_{s}$, for $j=4,5,6$ we take ”$-$”, $\xi=\xi(z)$ and

[TABLE]

Here the quadratic forms $D^{j}_{k}$ are given by sums of $\omega^{\mathfrak{F}}_{k}$ while the functions $\Gamma_{k}^{j}\geq 1$ – by sums of $\gamma_{\xi_{i}}$. Namely, omitting the upper index $\mathfrak{F}$,

[TABLE]

since $\sum_{j=1}^{4}\omega_{j}=0$. Similarly,

[TABLE]

The functions $\Gamma_{k}^{j}$ read

[TABLE]

8.5. Proof of Lemma 7.2

Since the matrix $A$ is irreducible, there exists $k$ and indices $i_{1},\ldots,i_{k}$ satisfying $i_{j}\neq i_{j+1}$ $\forall j$, such that $a_{i_{1}i_{2}}a_{i_{2}i_{3}}\cdots a_{i_{k-1}i_{k}}\neq 0$ and the sequence $i_{1},\ldots,i_{k}$ contains every number from $1$ to $n$. We construct a set $B$ of linearly independent vectors $r_{ij}$ by the following inductive procedure. For the base of induction we consider the indices $i_{1},i_{2}$ and set $B=\{r_{i_{1}i_{2}}\}$. For the inductive step, we assume that the indices $i_{1},\ldots,i_{m-1}$ are already considered, and we consider the index $i_{m}$. If the set $B$ does not contain any vector $r_{uv}$ for which $u=i_{m}$ or $v=i_{m}$ then we add the vector $r_{i_{m-1}i_{m}}$ to $B$ and pass to the next step. Otherwise we just pass to the next step. We conclude the procedure when all indices $i_{1},\dots,i_{k}$ are considered.

Since the sequence $i_{1},\ldots,i_{k}$ contains every $1\leq j\leq n$, after the end of the procedure above we have $|B|=n-1$. Due to the form of the vectors $r_{ij}$, vectors from the set $B$ are linearly independent, so the rank $R$ satisfies $R\geq n-1$. Thus, it remains to show that $R<n$. If $R=n$ then the basis vectors $e_{m}$ can be uniquely expressed through the vectors $r_{ij}$. However, this is not possible since the vectors $r_{ij}$ are invariant with respect to translations of the set $(e_{i})$: if we replace $e_{i}$ by $e_{i}+e$ for every $i$ and a vector $e$ then the vectors $r_{ij}$ remain unchanged. ∎

8.6. Approximate equation

In this section we adopt the method of Feynman diagrams, developed for equation (2.6), to equation (1.8), and prove Proposition 2.1. In fact, for the proof we do not need to develop the theory in so detailed way (cf. proof of Corollary 2.2 in [9]), but we also use the results of this section in our work [10].

8.6.1. Extended Feynman diagrams and correlations ${\mathbb{E}}\mathfrak{a}_{s}^{(m)}\bar{\mathfrak{a}}_{s}^{(n)}$

Similarly to Section 2.2 we write equation (1.8) in the interaction representation $\mathfrak{a}_{s}(\tau)=v_{s}(\tau)e^{-i\nu^{-1}\tau|s|^{2}}\,:$

[TABLE]

where $\delta^{=12}_{\ 3s}=1$ if $s_{1}=s_{2}=s_{3}=s$ and $\delta^{=12}_{\ 3s}=0$ otherwise. Note that

[TABLE]

Expansion (1.12), written in the $\mathfrak{a}$-coordinates, takes the form

[TABLE]

where $\mathfrak{a}_{s}^{(0)}=a_{s}^{(0)}$ and, similarly to (2.13),

[TABLE]

Here and below in this section $\theta$ is as in (2.14).

Literally repeating the proof of Lemma 3.1, we see that the product $\mathfrak{a}_{s}^{(m)}(\tau_{1})\bar{\mathfrak{a}}_{s}^{(n)}(\tau_{2})$ is given by formula (3.16) where in the definition (3.17) of the sum $I_{s}(\mathfrak{D})$ the set $\mathcal{A}_{s}^{L}(\mathfrak{D})$ is replaced by the set $\mathcal{B}_{s}^{L}(\mathfrak{D})$, defined by relations (3.6), (3.8) and

[TABLE]

(we recall (8.11)), cf. (3.7). Moreover, the function $G_{\mathfrak{D}}$ from (3.17) is replaced by the function $(-1)^{r(\xi,\sigma)}G_{\mathfrak{D}}(\tau,l,\xi,\sigma),$ where $r(\xi,\sigma)$ is the number of indices $1\leq i\leq N$ for which $\delta^{=\xi_{2i-1}\xi_{2i}}_{\ \sigma_{2i-1}\sigma_{2i}}=1$.

Next we compute the expectation ${\mathbb{E}}\mathfrak{a}_{s}^{(m)}\bar{\mathfrak{a}}_{s}^{(n)}$. In difference with the computation of ${\mathbb{E}}I_{s}(\mathfrak{D})$ in Section 4, now we should take into account Wick pairings which couple leaves belonging to the same block of the diagram $\mathfrak{D}$. Indeed, the situation when $\xi_{i}=\sigma_{j}$ for $(\xi,\sigma)\in\mathcal{B}_{s}^{L}(\mathfrak{D})$ and $i,j$ such that the vertices $c_{i}$ and $\bar{c}_{j}$ belong to the same block $\hat{B}_{k}$ (that is $i,j\in\{2k-1,2k\}$) now is not impossible: it occurs if $\delta^{=\xi_{2k-1}\xi_{2k}}_{\ \sigma_{2k-1}\sigma_{2k}}=1$. Arguing as when proving Lemma 4.1, we get

[TABLE]

where the set $\mathfrak{F}^{+}_{m,n}$ of extended Feynman diagrams is defined in the same way as the set $\mathfrak{F}_{m,n}$ of Feynman diagrams from Section 4 except that in the diagrams $\mathfrak{F}\in\mathfrak{F}^{+}_{m,n}$ we allow leaves belonging to the same block to be coupled. The terms $\mathcal{J}_{s}(\mathfrak{F})$ are defined similarly to (4.10):

[TABLE]

Here the set of admissible multi-indices $\mathcal{B}^{\prime L}_{s}(\mathfrak{F})\subset(\mathbb{Z}^{d}_{L})^{2N+1}$ is given by the relation $\xi_{0}=\sigma_{0}=s$ together with (8.13), cf. (4.5), (4.6). From now on we set

[TABLE]

where the permutation $f_{\mathfrak{F}}$ is defined by (4.3) in Section 4. The function $\mathcal{F}^{\mathfrak{F}}$ is given by

[TABLE]

(see (4.7)), while the vector $\omega^{\mathfrak{F}}(\xi)=(\omega^{\mathfrak{F}}(\xi)_{j})$ is defined as in Section 4.

Our next goal is to resolve relations (8.13) with $\sigma=\xi_{f_{\mathfrak{F}}}$, entering the definition of the set $\mathcal{B}^{\prime L}_{s}(\mathfrak{F})$. In Section 5 we dealt with an analogous problem, but now there is an additional difficulty: the transformation $\xi\mapsto x$ used in Section 5 is not well defined if in the diagram $\mathfrak{F}$ there are coupled vertices belonging to the same block. That is if $\mathfrak{F}\in\mathfrak{F}^{+}_{m,n}\setminus\mathfrak{F}_{m,n}$.

Below we call irregular those blocks of $\mathfrak{F}$ which contain a pair of coupled vertices. By definition, an $\mathfrak{F}$ belongs to $\mathfrak{F}^{+}_{m,n}\setminus\mathfrak{F}_{m,n}$ if and only if it has at least one irregular block. To overcome the difficulty mentioned above, we will regularize such diagrams by erasing the irregular blocks thus reducing our study to the case $\mathfrak{F}\in\mathfrak{F}_{m,n}$, possibly, with smaller $m,n$.

Before explaining the procedure let us note that if $\xi\in\mathcal{B}^{\prime L}_{s}(\mathfrak{F})$ then for any $k$ such that the block $\hat{B}_{k}$ is irregular, we have

[TABLE]

We regularize a diagram $\mathfrak{F}\in\mathfrak{F}^{+}_{m,n}\setminus\mathfrak{F}_{m,n}$ in several steps. In the first one we take an irregular block $\hat{B}_{k}$ with the least number $k$. It contains a pair of coupled leaves, say for definiteness these are $c_{2k-1}\sim\bar{c}_{2k-1}$. Also assume for definiteness that the block $\hat{B}_{k}$ is conjugated, i.e. $\hat{B}_{k}=\bar{B}_{k}$. Then, apart from the leaves $c_{2k-1}$ and $\bar{c}_{2k-1}$, it contains a virtual vertex $\bar{w}_{2k}$ which is coupled with a parent $c_{k^{\prime}}$ of the block and a real vertex $c_{2k}$ coupled with a vertex $\bar{c}_{k^{\prime\prime}}$ that does not belong to the block $\bar{B}_{k}$. We erase in $\mathfrak{F}$ the block $\bar{B}_{k}$, couple directly the parent $c_{k^{\prime}}$ with the vertex $\bar{c}_{k^{\prime\prime}}$ and decrease by $1$ the degree of the parent $c_{k^{\prime}}$, degree of the parent of the block to which belongs $c_{k^{\prime}}$, and so on until we arrive at the root vertex $c_{0}$ or $\bar{c}_{0}$, see fig. 10. Let us denote the resulting diagram by $\mathfrak{F}_{1}$. If we re-numerated the vertices and blocks of $\mathfrak{F}_{1}$ to remove the gaps in the numerations 121212 For that we should decrease by one the numbers $j$ of all blocks $\hat{B}_{j}$ with $j>k$ and renumber the vertices belonging to these blocks accordingly. (which we do not do!), then we would have that $\mathfrak{F}_{1}\in\mathfrak{F}_{m^{\prime},n^{\prime}}^{+}$ with $m^{\prime}+n^{\prime}=m+n-1$. So, slightly abusing notation, we write $\mathfrak{F}_{1}\in\mathfrak{F}_{m^{\prime},n^{\prime}}^{+}$. The diagram $\mathfrak{F}_{1}$ has one block less than the initial diagram $\mathfrak{F}$ and in comparison with the latter has a new edge $(c_{k^{\prime}},\bar{c}_{k^{\prime\prime}})$. Note that if the vertices $c_{k^{\prime}}$ and $\bar{c}_{k^{\prime\prime}}$ as above belong to the same block, then the latter cannot be irregular in the diagram $\mathfrak{F}$ but is necessarily irregular in $\mathfrak{F}_{1}$ (see fig. 10(c)). At the same time, all other blocks of $\mathfrak{F}_{1}$, irregular or not, are equally irregular or not in $\mathfrak{F}$.

Since the vertices $c_{k^{\prime}},\bar{c}_{k^{\prime\prime}}$ are coupled in $\mathfrak{F}$ to those in the block $\bar{B}_{k}$, then in view of (8.16) we get $\xi_{k^{\prime}}=\sigma_{k^{\prime\prime}}$ for $\xi\in\mathcal{B}^{\prime L}_{s}(\mathfrak{F})$. Then the multi-index $\xi^{1}:=(\xi_{i})_{i:\,c_{i}\in\mathfrak{F}_{1}}$, associated with the diagram $\mathfrak{F}_{1}$, satisfies $\xi^{1}\in\mathcal{B}^{\prime L}_{s}(\mathfrak{F}_{1})$. In particular, (8.16) holds for the indices associated with any irregular block of $\mathfrak{F}_{1}$.

In the second step we apply the same procedure to the diagram $\mathfrak{F}_{1}$ and obtain a diagram $\mathfrak{F}_{2}\in\mathfrak{F}_{m^{\prime\prime},n^{\prime\prime}}^{+}$ with $m^{\prime\prime}+n^{\prime\prime}=m+n-2$ (we again slightly abuse the notation). Moreover, we get $\xi^{2}\in\mathcal{B}^{\prime L}_{s}(\mathfrak{F}_{2})$, where $\xi^{2}=(\xi_{i})_{i:\,c_{i}\in\mathfrak{F}_{2}}$. In the third step we apply the procedure to the diagram $\mathfrak{F}_{2}$, etc. We continue in the same way until we get a final diagram $\mathfrak{F}_{p}$, which we denote $\mathfrak{F}_{\mathcal{R}}$, that does not have irregular blocks. That is $\mathfrak{F}_{\mathcal{R}}\in\mathfrak{F}_{\widetilde{m},\widetilde{n}}$, where $\widetilde{m}+\widetilde{n}=m+n-p$.131313If $\widetilde{m}=\widetilde{n}=0$ then the diagram $\mathfrak{F}_{\mathcal{R}}$ is trivial, see fig. 10(d). Let us repeat that in $\mathfrak{F}_{\mathcal{R}}$ we keep the original numeration of vertices and blocks, inherited from $\mathfrak{F}$.

Let us denote by $\mathcal{R}\subset\{0,\dots,2N\}$ the set, made by the numbers of vertices in $\mathfrak{F}_{\mathcal{R}}$, and by $\mathcal{R}_{B}\subset\{1,\dots,N\}$ – the set, made by the numbers of blocks in $\mathfrak{F}_{\mathcal{R}}$. So

[TABLE]

E.g. for the diagram $\mathfrak{F}$ from fig. 10(a), $\mathcal{R}_{B}=\emptyset$ and $\mathcal{R}=\{0\}$. Note that the construction of $\mathcal{R}_{B}$ does not exclude that for some blocks $\hat{B}_{k}$ with $k\in\mathcal{R}_{B}$ holds (8.16).

We will say that vertex $c_{i}$, $i\in\mathcal{R}$, confines a block $\hat{B}_{k}$ of the diagram $\mathfrak{F}$, erased during the regularization procedure (so $k\notin\mathcal{R}_{B}$) if there exists an edge $(c_{i},\bar{c}_{j})$ of $\mathfrak{F}_{\mathcal{R}}$, not contained in $\mathfrak{F}$, which appeared in $\mathfrak{F}_{\mathcal{R}}$ as a result of erasing a set of blocks which includes $\hat{B}_{k}$. E.g., the vertex $c_{0}$ confines all blocks of the diagram from fig. 10(a).

Let us denote $\xi^{\mathcal{R}}:=(\xi_{i})_{i\in\mathcal{R}}$. From what is told above, by induction we get

Lemma 8.4**.**

(i) Every $\xi\in\mathcal{B}^{\prime L}_{s}(\mathfrak{F})$ satisfies (8.16) for $k\notin\mathcal{R}_{B}$.

*(ii)  \mathcal{B}^{\prime L}_{s}(\mathfrak{F})=\big{\{}\xi:\,\xi^{\mathcal{R}}\in\mathcal{B}^{\prime L}_{s}(\mathfrak{F}_{\mathcal{R}})\quad\mbox{and}\quad(\xi)_{j\notin\mathcal{R}}=\Xi(\xi^{\mathcal{R}})\big{\}},

where $\Xi(\xi^{\mathcal{R}})$ is the multi-index such that for $i\in\mathcal{R}$ and $j\notin\mathcal{R}$, $\Xi(\xi^{\mathcal{R}})_{j}=\xi_{i}$ if the vertex $c_{i}$ confines the block which contains $c_{j}$.*

In view of Lemma 8.4(ii) in order to understand the set $\mathcal{B}^{\prime L}_{s}(\mathfrak{F})$ we should first understand $\mathcal{B}^{\prime L}_{s}(\mathfrak{F}_{\mathcal{R}})$. To this end we apply to the diagram $\mathfrak{F}_{\mathcal{R}}$ the argument developed in Section 5. Namely, we construct the reduced cycle $\hat{\mathfrak{C}}_{\mathfrak{F}_{\mathcal{R}}}$, the associated permutation $\pi_{\mathfrak{F}_{\mathcal{R}}}$ and the S-regular incidence matrix $\alpha^{\mathfrak{F}_{\mathcal{R}}}$ of size $\#\mathcal{R}_{B}$. Then we make the transformation $\xi^{\mathcal{R}}\mapsto x^{\mathcal{R}}:=(x_{i})_{i\in\mathcal{R}}$, defined by formulas (5.2) and (5.3), and write $\xi^{\mathcal{R}}=\xi^{\mathcal{R}}(x^{\mathcal{R}})$.

Similarly to Proposition 5.3 we get

Proposition 8.5**.**

(i) The set $\mathcal{B}^{\prime L}_{s}(\mathfrak{F}_{\mathcal{R}})$ consists of all vectors $\xi^{\mathcal{R}}=\xi^{\mathcal{R}}(x^{\mathcal{R}})$, where components $x_{k}\in\mathbb{Z}^{d}_{L}$ of $x^{\mathcal{R}}$ satisfy the relations

[TABLE]

and $x_{2k-1}=0$ if and only if $\sum_{i\in\mathcal{R}_{B}}\alpha_{ki}^{\mathfrak{F}_{\mathcal{R}}}x_{2i-1}=0$.

(ii)  If $\xi^{\mathcal{R}}=\xi^{\mathcal{R}}(x^{\mathcal{R}})$ then $x_{2k-1}=0$ if and only of $\delta^{=\xi_{2k-1}\xi_{2k}}_{\ \sigma_{2k-1}\sigma_{2k}}=1$, where $\sigma=\xi_{f_{\mathfrak{F}_{\mathcal{R}}}}$.

Proof. Let us decompose

[TABLE]

where the union is taken over all subsets $\mathcal{K}$ of $\mathcal{R}_{B}$ (including the empty one and $\mathcal{R}_{B}$) and the set $\mathcal{B}^{\prime L}_{s,\mathcal{K}}(\mathfrak{F}_{\mathcal{R}})$ consists of those vectors $\xi^{\mathcal{R}}$ for which

[TABLE]

(we recall (8.11)). Fix any $\mathcal{K}\subset\mathcal{R}_{B}$ and take $k\in\mathcal{K}$ (if $\mathcal{K}\neq\emptyset$). Since we have (5.8), then the assumption $\delta^{=\xi_{2k-1}\xi_{2k}}_{\ \sigma_{2k-1}\sigma_{2k}}=1$ imposed on $\xi^{\mathcal{R}}\in\mathcal{B}^{\prime L}_{s,\mathcal{K}}(\mathfrak{F}_{\mathcal{R}})$ is equivalent to $\xi_{2k-1}=\xi_{2k}=\xi_{\pi(2k-1)}=\xi_{\pi(2k)}$. Or, to $x_{2k-1}=x_{2k}=0$ together with

[TABLE]

In particular, we get (8.17)(b) for $k\in\mathcal{K}$. This also proves the implication $\Leftarrow$ in item (ii) of the proposition.

Take now $k\notin\mathcal{K}$ (if $\mathcal{K}\neq\mathcal{R}_{B}$). Then, as in Proposition 5.3, relation $\delta^{\prime\xi_{2k-1}\xi_{2k}}_{\ \sigma_{2k-1}\sigma_{2k}}=1$ imposed on $\xi^{\mathcal{R}}\in\mathcal{B}^{\prime L}_{s,\mathcal{K}}(\mathfrak{F}_{\mathcal{R}})$ is equivalent to (5.9) together with (5.10), or to (8.17)(b) with $x_{2k-1}\neq 0$ and

[TABLE]

(see (5.12)). In particular, we get (8.17)(b) for $k\notin\mathcal{K}$ and an implication $\Rightarrow$ in item (ii) of the proposition.

As in the proof of Proposition 5.3, using (8.17)(b), we get that the relation $\xi_{0}=\sigma_{0}=s$, imposed on $\xi^{\mathcal{R}}\in\mathcal{B}^{\prime L}_{s,\mathcal{K}}(\mathfrak{F}_{\mathcal{R}})$, is equivalent to (8.17)(a). We also get that

[TABLE]

see (5.7). Then, in view of (8.18), (8.19), we obtain the last assertion in item (i) of the proposition. ∎

Similarly to Section 5, for $j\in\mathcal{R}_{B}$ we denote $z_{j}:=x_{2j-1}$ and set $z:=(z_{j})_{j\in\mathcal{R}_{B}}$. Due to Proposition 8.5(i), vector $z$ forms coordinates on the set $\mathcal{B}^{\prime L}_{s}(\mathfrak{F}_{\mathcal{R}})$. Namely, let

[TABLE]

cf. (5.15). Then

[TABLE]

where the function $\xi^{\mathcal{R}}(z)$ is defined via the $s$-dependent transformation $z\mapsto x^{\mathcal{R}}\mapsto\xi^{\mathcal{R}}$, given by a natural modification of formula (5.13).

According to Lemma 8.4(ii) the vector $z$ forms coordinates on the set $\mathcal{B}^{\prime L}_{s}(\mathfrak{F})$ as well. The functions $\omega_{j}^{\mathfrak{F}}(\xi)$ from (8.14), restricted to the set $\mathcal{B}_{s}^{\prime L}(\mathfrak{F})\ni\xi$ and written in the $z$-coordinates, take the form

[TABLE]

Indeed, the first equality follows by literally repeating the proof of Lemma 5.5. The second identity immediately follows from Lemma 8.4(i).

Now, using Lemma 8.4(ii) and (8.21), we write $\xi\in\mathcal{B}^{\prime L}_{s}(\mathfrak{F})$ as $\xi=\xi(z)$ in the integral (8.14), and similarly to Theorem 5.7 we get

Proposition 8.6**.**

For any integers $m,n\geq 0$ satisfying $N=m+n\geq 1$, $s\in\mathbb{Z}^{d}_{L}$ and $\tau_{1},\tau_{2}\geq-T$, we have

[TABLE]

where

[TABLE]

Here $\Theta(\omega^{\mathfrak{F}},\nu^{-1}l)=\prod_{j=1}^{N}\theta(\omega_{j}^{\mathfrak{F}},\nu^{-1}l_{j})$, while components $\omega_{j}^{\mathfrak{F}}$ of the vector $\omega^{\mathfrak{F}}$ are given by (8.22). The incidence matrix $\alpha^{\mathfrak{F}_{\mathcal{R}}}$ from (8.20), (8.22) is S-regular. The real density function $\mathcal{F}_{s}^{\mathfrak{F}}(\tau,l,z):=\mathcal{F}^{\mathfrak{F}}(\tau,l,\xi(z))$ is given by (8.15) and satisfies

[TABLE]

with a suitable $\delta=\delta_{N}>0$, for any vectors $\mu\in\mathbb{Z}^{d}_{+}$, $\kappa\in\mathbb{Z}_{+}^{d\#\mathcal{R}_{B}}$, and any $s\in\mathbb{R}^{d}$, $z\in\mathbb{R}^{d\#\mathcal{R}_{B}}$.

Let us note that the matrix $\alpha^{\mathfrak{F}_{\mathcal{R}}}$ may have zero lines and rows, and in difference with the situation in Section 5 this does not imply that $\mathcal{Z}^{+}(\mathfrak{F}_{\mathcal{R}})=\emptyset$ (cf. Proposition 5.6). That is why we do not introduce an analogue of the subset of diagrams $\mathfrak{F}^{\,true}_{m,n}$.

8.6.2. Proof of Proposition 2.1

To establish the proposition it suffices to estimate the difference ${\mathbb{E}}\mathfrak{a}_{s}^{(m)}(\tau_{1})\bar{\mathfrak{a}}_{s}^{(n)}(\tau_{2})-{\mathbb{E}}a_{s}^{(m)}(\tau_{1})\bar{a}_{s}^{(n)}(\tau_{2})$ since its modulus equals to the l.h.s. of (2.3). According to Theorem 5.7 and Proposition 8.6,

[TABLE]

Here we have used that, according to Proposition 5.6, $J_{s}(\mathfrak{F})=0$ for $\mathfrak{F}\in\mathfrak{F}_{m,n}\setminus\mathfrak{F}^{\,true}_{m,n}$, so we can replace $\mathfrak{F}^{\,true}_{m,n}$ by $\mathfrak{F}_{m,n}$ in the formulation of Theorem 5.7.

Let us first simplify the second sum. Since $\mathfrak{F}\in\mathfrak{F}_{m,n}$, we have $\mathfrak{F}_{\mathcal{R}}=\mathfrak{F}$, $\mathcal{R}_{B}=\{1,\ldots,N\}$, $\#\mathcal{R}_{B}=N$ and $\xi^{\mathcal{R}}=\xi$. Then, in the integral $\mathcal{J}_{s}(\mathfrak{F})$ the summation it taken over $z\in\mathcal{Z}^{+}(\mathfrak{F})\subset\mathbb{Z}^{dN}_{L}$, while in $J_{s}(\mathfrak{F})$  – over $z\in\mathcal{Z}(\mathfrak{F})\cap\mathbb{Z}^{dN}_{L}$. Note that $\mathcal{Z}(\mathfrak{F})\cap\mathbb{Z}^{dN}_{L}\subset\mathcal{Z}^{+}(\mathfrak{F})$; more specifically,

[TABLE]

see (5.15). We then write

[TABLE]

where in the integral $\mathcal{J}_{s}^{1}(\mathfrak{F})$ the set $\mathcal{Z}^{+}(\mathfrak{F})$ is replaced by $\mathcal{Z}(\mathfrak{F})\cap\mathbb{Z}^{dN}_{L}$, and in $\mathcal{J}_{s}^{2}(\mathfrak{F})$ – by $\mathcal{Z}^{+}(\mathfrak{F})\setminus\mathcal{Z}(\mathfrak{F})$.

We claim that $\mathcal{J}^{1}_{s}(\mathfrak{F})=J_{s}(\mathfrak{F})$. Indeed, the function $\xi(z)$, entering the integrand of the integral $\mathcal{J}^{1}_{s}(\mathfrak{F})$, coincides with the function $\xi(z)$, entering the integrand of $J_{s}(\mathfrak{F})$. Thus, these integrals differ only in the density functions $\mathcal{F}_{s}^{\mathfrak{F}}$ and $F_{s}^{\mathfrak{F}}$, where $\mathcal{F}_{s}^{\mathfrak{F}}=(-1)^{r(z)}F_{s}^{\mathfrak{F}}$ and $r(z)$ stands for the number of indices $1\leq i\leq N$ for which $\delta^{=\xi_{2k-1}\xi_{2k}}_{\ \sigma_{2k-1}\sigma_{2k}}=1$, $\xi=\xi(z)$. But $r(z)=0$ for $z\in\mathcal{Z}(\mathfrak{F})$, due to Proposition 8.5(ii), since $x_{2k-1}=z_{k}\neq 0$ for every $k$.

We have seen that the second sum from (8.27) takes the form $\sum_{\mathfrak{F}\in\mathfrak{F}_{m,n}}c_{\mathfrak{F}}\mathcal{J}_{s}^{2}(\mathfrak{F})$, where

[TABLE]

since the set $\mathcal{Z}^{+}(\mathfrak{F})\setminus\mathcal{Z}(\mathfrak{F})$ consists of $z\in\mathcal{Z}^{+}(\mathfrak{F})$ satisfying $z_{j}=0$ for some $j$. Thus, we get

Proposition 8.7**.**

For any $m,n\geq 0$ satisfying $m+n\geq 1$, any $s\in\mathbb{Z}^{d}_{L}$ and $\tau_{1},\tau_{2}\geq-T$,

[TABLE]

where the integrals $\mathcal{J}_{s}(\mathfrak{F})$ and $\mathcal{J}^{2}_{s}(\mathfrak{F})$ are given by (8.24) and (8.26) correspondingly.

We estimate the two sums from (8.27) separately. We first consider the case $\mathfrak{F}\in\mathfrak{F}^{+}_{m,n}\setminus\mathfrak{F}_{m,n}.$ According to (8.24) and (8.25),

[TABLE]

since the diagram $\mathfrak{F}$ has at least one irregular block, that is $\#\mathcal{R}_{B}<N$. Thus, the first sum from (8.27) is bounded by $C^{\#}(s)L^{-d}$.

Now we consider the case $\mathfrak{F}\in\mathfrak{F}_{m,n}$. According to (8.25),

[TABLE]

Thus, the second sum from (8.27) is bounded by $C^{\#}(s)L^{-d}$ as well. This concludes the proof of Proposition 2.1.

8.6.3. Example

In our work [10] we apply the theory, developed in this section, to the situation when the function $\theta(x,t)$ is not given by (2.11) (as in Theorem 5.9), but is the indicator function $\mathbb{I}_{\{0\}}(x)$ of the point $x=0$ (so it is independent from $t$). Below we briefly discuss this case. Again our goal is to estimate the difference ${\mathbb{E}}\mathfrak{a}_{s}^{(m)}(\tau_{1})\bar{\mathfrak{a}}_{s}^{(n)}(\tau_{2})-{\mathbb{E}}a_{s}^{(m)}(\tau_{1})\bar{a}_{s}^{(n)}(\tau_{2})$ when $\theta(x,t)=\mathbb{I}_{\{0\}}(x)$.

As before, we estimate the two sums from (8.27) separately. Let first $\mathfrak{F}\in\mathfrak{F}^{+}_{m,n}\setminus\mathfrak{F}_{m,n}$. According to (8.25),

[TABLE]

Assume now $\mathfrak{F}\in\mathfrak{F}_{m,n}$. Then,

[TABLE]

8.7. Integrals $\widetilde{J}_{s}(\mathfrak{F})$ in which the time variable $l$ is integrated out

In this appendix we integrate out the time variable $l$ in the integrals $\widetilde{J}_{s}(\mathfrak{F})$, which are defined in (5.24). We believe that it is relevant for further study of the energy spectrum (1.25) of quasisolutions since for the moment of writing we know asymptotical as $\nu\to 0$ behaviour of the integrals $\widetilde{J}_{s}(\mathfrak{F})$ only in case $N=2$, and to find it we have to write the integrals in such integrated in $l$ form. See in [9], where we found this asymptotic and, using a particular case of Theorem 5.9, deduced from it the asymptotical as $\nu\to 0,\,L\to\infty$ behaviour of the moment ${\mathbb{E}}|a_{s}^{(1)}|^{2}$. Note, however, that in case of general $N$, even to get an upper bound for the integrals $\widetilde{J}_{s}(\mathfrak{F})$ in Theorem 6.2 we write them in the form (5.24) and we do not know how to get the upper bound using their integrated in $l$ form.

For simplicity of computation we assume $\tau_{1}=\tau_{2}=:t$ and $T=\infty$. We denote by $\mathcal{S}^{N}$ the set of all permutations $q$ of the set $\{1,\ldots,N\}$ and let

[TABLE]

and the sets $\mathcal{L}_{q}^{N}$ with different $q$ can intersect only over sets of zero Lebesgue measure. Then, in view of (4.7),

[TABLE]

where

[TABLE]

where $b^{\psi}(z)=b^{\psi}(\xi(z))$, $\gamma^{\psi}(z)=\gamma^{\psi}(\xi(z))$ and $\gamma^{\varphi}(z)=\gamma^{\varphi}(\xi(z))$. It remains to compute the integrals $I_{s}^{q}$, and for this purpose we introduce the following notation. For $0\leq i<j\leq N$ we set

[TABLE]

where the sum is taken over all edges $\vartheta\in E(\mathfrak{F})$ of the diagram $\mathfrak{F}$ which couple a vertex from the block $\hat{B}_{i}$ with a vertex from the block $\hat{B}_{j}$. Here by $\hat{B}_{0}$ we denote the set of roots $\{c_{0},\bar{c}_{0}\}$.

Let $\mathcal{S}^{N}_{\mathfrak{F}}\subset\mathcal{S}^{N}$ be the set of permutations $q$ for which the relation $l\in\mathcal{L}^{N}_{q}$ implies

[TABLE]

cf. the indicator function in (8.31).

Below we denote $q(0):=0$.

Lemma 8.8**.**

Assume $\tau_{1}=\tau_{2}=t$ and $T=\infty$. Then for any $m,n\geq 0$ satisfying $N:=m+n\geq 2$ and $\mathfrak{F}\in\mathfrak{F}_{m,n}$ the integral $\widetilde{J}_{s}(\mathfrak{F})$ is given by (8.30), where

[TABLE]

if $q\in\mathcal{S}^{N}_{\mathfrak{F}}$ and $I_{s}^{q}(\mathfrak{F};z)\equiv 0$ otherwise. Moreover, $\sum_{0\leq r<k\leq j\leq N}\gamma^{\mathfrak{F}}_{q(r)q(j)}\geq 1$ for any $2\leq k\leq N$, so the denominator in (8.32) is separated from zero.

The lemma can be proven by induction.

Note that outside the intersection of quadrics

[TABLE]

we have $I_{s}^{q}(\mathfrak{F};z)\sim\nu^{N-1}$, while on $Q$ we have $I_{s}^{q}(\mathfrak{F};z)\sim 1$. So the integrand in (8.30) asymptotically degenerates on $Q$ as $\nu\to 0$, and the integral in (8.30) is asymptotically singular.

8.8. Estimation of asymptotically singular

integrals of quotients.

In Lemma 8.8 we integrated out the time variable $l$ in integrals $\widetilde{J}_{s}(\mathfrak{F})$, defined in (5.24), and represented $\widetilde{J}_{s}(\mathfrak{F})$ as sums of integrals of quotients with large quadratic forms in divisors, see (8.30),(8.32). In Theorem 6.2 we got an upper bound for a family of integrals of quotients that have more general form than the integrals $\widetilde{J}_{s}(\mathfrak{F})$. As a corollary, in this appendix we obtain an upper bound for a natural family of integrals of more general form than those in (8.30), (8.32).

As in Section 6.1, we consider the family of quadratic forms

[TABLE]

where $z$ is the polyvector $z=(z_{1},\dots,z_{M}),z_{j}\in\mathbb{R}^{d}$, and $Q_{k}=(q_{ij}^{k})_{1\leq i,j\leq M}$ are real symmetric matrices. As before, we denote by $R$ the rank of the system of matrices $Q_{1},\dots,Q_{K}$ in the space of $M\times M$-matrices, equal to the rank of the system of vectors

[TABLE]

Let

[TABLE]

where $0<\nu\leq 1/2$, $G$ is a Schwartz function, the functions $\Gamma_{k}$ are smooth, have at most polynomial growth at infinity together with their partial derivatives of all orders, and satisfy $\Gamma_{k}(z)\geq C_{\Gamma}$ for some constant $C_{\Gamma}>0$ and any $k$ and $z$.

Proposition 8.9**.**

Assume that $\operatorname{tr}Q_{k}=0$ for every $1\leq k\leq K$. Then

[TABLE]

where the function $\psi^{R}_{d}$ is defined in (6.5).

See [8] for a generalization of this result. In [17] the asymptotic behaviour as $\nu\to 0$ of the integral $J^{\nu}$ was found in the case when $d\geq 2$ is a pair number, $K=2$, $\Gamma_{1}=\Gamma_{2}$ and $Q_{1}=-Q_{2}$, where the quadratic form $Q_{1}$ was assumed to be non-degenerate and to have the index of the form $(m,m)$, for some $m$. This result plays a crucial role in [9] when analysing the asymptotic behaviour of the term $n^{(2)}$ from the decomposition (1.21). In particular, it was shown that $J^{\nu}\sim\nu$ which is in accordance with the estimate (8.34). The family of integrals (8.33) is significantly more complicated than that in [17]. We get for them only the upper bound and for the moment do not now how to examine their asymptotic behaviour.

Proof of Proposition 8.9. Since $\Gamma_{k}(z)>0$ for any $z$ and $k$, we have

[TABLE]

Denote $l=(l_{1},\ldots,l_{K})$,

[TABLE]

Then, using representation (8.35) in (8.33) for every $k$, we get

[TABLE]

Since

[TABLE]

then the integral $J^{\nu}$ has the form (6.2), where the density $F$ is independent from the parameter $s$ (one can think that in (6.2) we fix some $s=s_{0}$ and do not follow the dependence of estimates in $s_{0}$). It is straightforward to see that assumptions on the functions $G$ and $\Gamma_{k}$ imply that the function $F$ satisfies (6.3) with $l_{0}=0$ and $h_{\kappa}(x)=C_{\kappa}e^{-\delta C_{\Gamma}x}$, where the constant $\delta>0$ is sufficiently small and $C_{\kappa}>0$ are sufficiently large.

As $\operatorname{tr}Q_{k}=0$ for any $k$, then the assumptions of Theorem 6.2 are fulfilled and integral $J^{\nu}$ satisfies estimate (6.6) (with $C^{\#}(s)$ replaced by $C$). ∎