domain: math phys

text:
Andrey Dymov
email: dymov@mi-ras.ru

Sergei Kuksin
email: sergei.kuksin@imj-prg.fr

FORMAL EXPANSIONS IN STOCHASTIC MODEL FOR WAVE TURBULENCE 2: METHOD OF DIAGRAM DECOMPOSITION

). 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. * for some r * ≥ 0. 3 i.e. β s = β 1 s + iβ 2 s , where {β j s , s ∈ Z d L , j = 1, 2} are standard independent real Wiener processes.

4 Often it is assumed that the intensity b(x) of the noise η ω is non-negative, but we do not impose this condition. Still note that if b(x) ≡ 0, then our results become trivial since below we will provide (1.3) with the zero initial conditions.

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 [START_REF] Dymov | Formal expansions in stochastic model for wave turbulence 1: kinetic limit[END_REF][START_REF] Dymov | The large-period limit for equations of discrete turbulence[END_REF]. Results of this work are crucially used in [START_REF] Dymov | Formal expansions in stochastic model for wave turbulence 1: kinetic limit[END_REF][START_REF] Dymov | The large-period limit for equations of discrete turbulence[END_REF] 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 [START_REF] Dymov | Formal expansions in stochastic model for wave turbulence 1: kinetic limit[END_REF][START_REF] Dymov | The large-period limit for equations of discrete turbulence[END_REF] 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 [START_REF] Nazarenko | Wave Turbulence[END_REF][START_REF] Newell | Wave Turbulence[END_REF][START_REF] Zakharov | Kolmogorov Spectra of Turbulence[END_REF], and for an additional discussion of the relevance of our results for WT -to the introduction in [START_REF] Dymov | Formal expansions in stochastic model for wave turbulence 1: kinetic limit[END_REF]. (1.1)

u(x) = L -d/2 s∈Z d L v s e 2πis⋅x , Z d L = L -1 Z d .
Here the vector of Fourier coefficients v = (v s ) s∈Z d L is given by the Fourier transform of u(x),

v = F(u), v s = L -d/2 T d L u(x)e -2πis⋅x dx for s ∈ Z d L .
The Parseval identity reeds ∥u∥

2 = L -d ∑ s∈Z d L |v s | 2 .
We will study solutions u(t, x) whose norms satisfy ∥u(t, ⋅)∥ ∼ 1 as L → ∞.

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

(1.2) ∂ ∂t u + i∆u -iλ |u| 2 -2∥u∥ 2 u = 0, x ∈ T d L ,
where u = u(t, x), ∆ = (2π)

-2 ∑ d j=1 (∂ 2 /∂x 2
j
) and λ ∈ (0, 1] is a small parameter. The modification of the nonlinearity by the term 2iλ∥u∥ 2 u keeps the main features of the standard cubic NLS equation, reducing some non-crucial technicalities, see the introduction to [START_REF] Dymov | Formal expansions in stochastic model for wave turbulence 1: kinetic limit[END_REF] for a detailed discussion.

The objective of WT is to study solutions of (1.2) under the wave turbulence limit L → ∞ and λ → 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 [START_REF] Zakharov | Statistical description of nonlinear wave fields[END_REF], we consider the NLS equation (1.2) dumped by a (hyper) viscosity and driven by a random force:

(1.3) ∂ ∂t u + i∆u -iλ |u| 2 -2∥u∥ 2 u = -νA(u) + ν ∂ ∂t η ω (t, x).
Here ν ∈ (0, 1/2] is a small parameter and A is the dissipative linear operator The real function γ 0 (y) ≥ 1 is smooth and has at most a polynomial growth at infinity, together with all its derivatives. 2 The random noise η ω is given by the Fourier series

η ω (t, x) = L -d/2 s∈Z d L b(s)β ω s (t)e 2πis⋅x ,
where {β s (t), s ∈ Z d L } are standard independent complex Wiener processes 3 and b(x) is a Schwartz function on R d ⊃ Z d L . 4 If the function γ 0 (y) grows at infinity as |y| r * with r * sufficiently big, then eq. (1.3) is well posed. In difference with [START_REF] Dymov | Formal expansions in stochastic model for wave turbulence 1: kinetic limit[END_REF][START_REF] Dymov | The large-period limit for equations of discrete turbulence[END_REF], where we assumed d ≥ 2, in this work we allow d = 1 and impose less restrictions on the function γ 0 .

2 For example A = (1 -∆) r

We will study the equation on time intervals of order ν -1 , so it is convenient to pass from t to the slow time τ = νt. Then eq. (1.3) where ρ = λν -1 , the upper-dot stands for d/dτ and {β s (τ ), s ∈ Z d L } is another set of standard independent complex Wiener processes. Below we use ρ, ν 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 (1.6) L → ∞ and ν → 0, as well as main characteristics of the solutions such as the energy spectrum

(1.7) n s (τ ) ∶= E|v s (τ )| 2 , s ∈ Z d L
, where v = F(u). In [START_REF] Dymov | Formal expansions in stochastic model for wave turbulence 1: kinetic limit[END_REF][START_REF] Dymov | The large-period limit for equations of discrete turbulence[END_REF] and in this paper we study formal decompositions in ρ of solutions to eq. (1.5), regarding ρ as an independent parameter, and those of the energy spectrum n s . In Section 1.3, where we present the results of [START_REF] Dymov | Formal expansions in stochastic model for wave turbulence 1: kinetic limit[END_REF] and [START_REF] Dymov | The large-period limit for equations of discrete turbulence[END_REF], we assume that ρ ∼ ν -1/2 or ρ ∼ L correspondingly, since due to [START_REF] Dymov | Formal expansions in stochastic model for wave turbulence 1: kinetic limit[END_REF] and [START_REF] Dymov | The large-period limit for equations of discrete turbulence[END_REF] exactly under these scalings the quadratic in ρ part of the decomposition of the energy spectrum has a non-trivial behaviour under the limit (1.6), taken in the regimes, considered in [START_REF] Dymov | Formal expansions in stochastic model for wave turbulence 1: kinetic limit[END_REF] and [START_REF] Dymov | The large-period limit for equations of discrete turbulence[END_REF].

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

(1.8) vs -iν -1 |s| 2 v s + γ s v s = iρL -d s 1 ,s 2 ,s 3 ∈Z d L δ ′12 3s v s 1 v s 2 vs 3 -|v s | 2 v s + b(s) βs , s ∈ Z d L
, where |s| stands for the Euclidean norm of a vector s and

(1.9)

δ ′12 3s = δ ′ s 1 s 2 s 3 s = 1, if s 1 + s 2 = s 3 + s and {s 1 , s 2 } ≠ {s 3 , s} , 0, otherwise.
Note that

(1.10) if δ ′12 3s = 1, then {s 1 , s 2 } ∩ {s 3 , s} / = ∅.
In view of the factor δ ′ 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

s 1 ,s 2 ,s 3 ∈Z d L δ ′ 12 3s =∶ s 1 s 2 δ ′ 12
3s .

We fix 0 ≤ T ≤ +∞ and study equation (1.5)=(1.8) with zero initial condition at τ = -T , so (1.11) v s (-T ) = 0 for all s ∈ Z d L .

Let us write solutions of (1.8), (1.11) as formal series in ρ: 5 (1.12)

v s = ∞ i=0 ρ i v (i) s , v (i)
s (-T ) = 0, where v

(i) s = v (i)
s (τ ; ν, L). The processes v (0)

s satisfy the linear equations

(1.13) v(0) s (τ ) -iν -1 |s| 2 v (0) s (τ ) + γ s v (0) s (τ ) = b(s) βs (τ ), v (0)
s (-T ) = 0, so these are independent Gaussian processes. The linear in ρ term v

(1) s satisfies (1.14) v(1) s -iν -1 |s| 2 v (1) s + γ s v (1)
s = iL -d s 1 ,s 2 ,s 3 δ ′12 3s v (0) s 1 v (0) s 2 v(0) s 3 -|v (0) s | 2 v (0) s , and v (1)
s (-T ) = 0. Other terms v (n) s 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 [START_REF] Dymov | Formal expansions in stochastic model for wave turbulence 1: kinetic limit[END_REF] it is straightforward to show that Ev s ′ (τ 2 ) = 0 for s ≠ s ′ . Our main result, given below in Theorem 1.1, establishes upper bounds for the correlations with s = s ′ . Its particular case is crucially used in [START_REF] Dymov | Formal expansions in stochastic model for wave turbulence 1: kinetic limit[END_REF], 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 [START_REF] Dymov | The large-period limit for equations of discrete turbulence[END_REF].

To state Theorem 1.1 we use the function

(1.15) χ N d (ν) = -ln ν if (N, d) = (3, 2) or (2 , 1) 1 otherwise
(we recall that 0 < ν ≤ 1/2). For an x ∈ R we denote ⌈x⌉ = min{n ∈ Z ∶ n ≥ x}; and denote by C # (⋅) various positive continuous functions on R k , k ∈ N, fast decaying at infinity:

(1.16) C # (x) ≤ c n (1 + |x|) -n ∀x ∈ R k ,
for every n ∈ N, with a suitable constant c n > 0.

Theorem 1.1. Assume that d ≥ 2. Then for any integers m, n ≥ 0 there exists a function C # such that for each s ∈ Z d L and τ 1 , τ 2 ≥ -T we have

(1.17) |Ev (m) s (τ 1 )v (n) s (τ 2 )| ≤ C # (s) ν -2 L -2
+ ν min(⌈N /2⌉,d) χ N d (ν) , 5 As we mentioned above, in paper [START_REF] Dymov | Formal expansions in stochastic model for wave turbulence 1: kinetic limit[END_REF] we choose ρ ∼ ν -1/2 to be large. However, the first d coefficients v (i) s of series (1.12) are in fact of order ≲ ν 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 [START_REF] Dymov | The large-period limit for equations of discrete turbulence[END_REF], where we assume ρ ∼ L.

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 [START_REF] Dymov | Formal expansions in stochastic model for wave turbulence 1: kinetic limit[END_REF] estimate (1.17) (with d ≥ 2) is used in the situation when ν -2 L -2 ≤ ν min(⌈N /2⌉,d) -then it implies that E|v (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 ν → 0.

s (τ )| 2 ≤ C # (s)ν min(m,
To prove the theorem we approximate, with accuracy ν -2 L -2 , the expectation Ev

(m) s v(n)
s by a finite sum of integrals, independent from L and parametrized by a suitable class F m,n of Feynman diagrams (the cardinality of the set F m,n grows factorially with m + n). Next we study the behaviour of each integral as ν ≪ 1. To this end, in Section 5, for every diagram F we find a special coordinate system such that in these coordinates the integral, corresponding to F, has the simple explicit form: (1.18) Js (τ 1 , τ 2 ;

F) = R N dl R dN dz F F s (τ 1 , τ 2 ; l, z)e iν -1 Ω F (l,z)
with explicit functions Ω F and F F s , see Theorem 5.9. Here the phase function Ω

F is a real quadratic form in z (usually degenerate) which linearly depends on l. The density function F F s is Schwartz in (s, z), while as a function of l it is piecewise smooth and fast decays as |l| → ∞. Then in Sections 6 and 7 we show that

| Js (F)| ≤ C # (s)ν min(⌈N /2⌉,d) χ N d (ν)
∀F, 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 [START_REF] Lukkarinen | Weakly nonlinear Schrödinger equation with random initial data[END_REF], see [1-6, 14, 15] and discussions in introduction to [START_REF] Collot | On the derivation of the homogeneous kinetic wave equation[END_REF][START_REF] Deng | Full derivation of the wave kinetic equation[END_REF][START_REF] Dymov | Formal expansions in stochastic model for wave turbulence 1: kinetic limit[END_REF]. 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 [START_REF] Erdös | Linear Boltzmann equation as the weak coupling limit of a random Schrödinger equation[END_REF][START_REF] Erdös | Derivation of the cubic non-linear Schrödinger equation from quantum dynamics of many-body systems[END_REF] (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 Ev (m) s v(n) s which may be treated by asymptotical methods, developed in the present paper and in [START_REF] Dymov | Formal expansions in stochastic model for wave turbulence 1: kinetic limit[END_REF][START_REF] Dymov | The large-period limit for equations of discrete turbulence[END_REF].

In the next two subsections we briefly present the results of [START_REF] Dymov | Formal expansions in stochastic model for wave turbulence 1: kinetic limit[END_REF][START_REF] Dymov | The large-period limit for equations of discrete turbulence[END_REF] and explain the role of the present paper for the research started in [START_REF] Dymov | Formal expansions in stochastic model for wave turbulence 1: kinetic limit[END_REF][START_REF] Dymov | The large-period limit for equations of discrete turbulence[END_REF]. 1.3. Kinetic limit. Let u(τ, x) be a solution of equation (1.5), so that v(τ ) = F u(τ ) is a solution of (1.8), and n s (τ ) 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 ρ, see [START_REF] Nazarenko | Wave Turbulence[END_REF][START_REF] Zakharov | Statistical description of nonlinear wave fields[END_REF][START_REF] Zakharov | Kolmogorov Spectra of Turbulence[END_REF]. 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 ν goes to zero (for a discussion of other possibilities see introductions to [START_REF] Dymov | Formal expansions in stochastic model for wave turbulence 1: kinetic limit[END_REF] and [START_REF] Dymov | The large-period limit for equations of discrete turbulence[END_REF] and see below). Accordingly in [START_REF] Dymov | Formal expansions in stochastic model for wave turbulence 1: kinetic limit[END_REF] we assume that

(1.19) L ≥ ν -2-δ for some δ > 0,
or that first L → ∞ and then ν → 0. It is traditional in WT to study the quadratic in ρ truncation of series (1.12), postulating that it well approximates small amplitude solutions. Thus motivated, in [START_REF] Dymov | Formal expansions in stochastic model for wave turbulence 1: kinetic limit[END_REF] we considered the process

(1.20) ṽs (τ ) ∶= v (0) s (τ ) + ρv (1)
s (τ ) + ρ 2 v (2) s (τ ), s ∈ Z d L
, which we called a quasisolution of equation (1.8), (1.11), and examined its

energy spectrum ñ = (ñ s ) s∈Z d L , ñs = E|ṽ s | 2 , (1.21) ñs (τ ) = n (0) s (τ ) + ρn (1)
s (τ ) + ρ 2 n (2) s (τ ) + ρ 3 n (3) s (τ ) + ρ 4 n (4)
s (τ ), where n

(k) s = 0≤k 1 ,k 2 ≤2∶ k 1 +k 2 =k Ev (k 1 ) s v(k 2 ) s .
Assuming d ≥ 2, we proved that n

s ∼ ν as ν → 0 uniformly in L satisfying (1.19), while a simple computation showed that n (0) s ∼ 1 and n (1) s = 0. This indicates that the right scaling for ρ is such that ρ 2 ν ∼ 1 as ν → 0, and accordingly in [START_REF] Dymov | Formal expansions in stochastic model for wave turbulence 1: kinetic limit[END_REF] we chose ρ to be of the form

(1.22) ρ = εν -1/2 , so that ρ 2 ν = ε.
Here 0 < ε ≤ 1 is a fixed small number (independent from ν and L). Using estimate (1.17) under the scaling (1.19), (1.22) we found that |ρ

3 n (3) s |, |ρ 4 n (4) s | ≲ ε 2 . Accordingly, ñs = n (0) s + εñ (2) s (ε), where ñ(2) s = ν -1 n (2) s + O(ε) ∼ 1 when ν → 0, since n (2)
s ∼ ν. Thus, the parameter ε 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 ν small in terms of ε the energy spectrum ñs (τ ) is ε 2 -close to a function m(τ, s), which is a unique solution of the damped/driven wave kinetic equation:

(1.23) ṁ(τ, s) = -2γ s m(τ, s) + 2b(s) 2 + εK(m(τ, ⋅))(s), s ∈ R d ; m(-T ) = 0. Here K(m)(s) is the wave kinetic integral 2π s Σ * ds 1 ds 2 |s Σ * m 1 m 2 m 3 m s |s 1 -s| 2 + |s 2 -s| 2 1 m s + 1 m 3 - 1 m 1 - 1 m 2 ,
where

m i = m(s i ), m s = m(s) with s 3 = s 1 + s 2 -s (cf. (1.9)). The set s Σ * is the resonant surface (cf. (2.5)-(2.6)) (1.24) s Σ * = {(s 1 , s 2 ) ∈ R 2d ∶ |s 1 | 2 +|s 2 | 2 = |s 3 | 2 +|s| 2 , s 3 = s 1 +s 2 -s},
and ds 1 ds 2 |s Σ * stands for the volume element on s Σ * , corresponding to the euclidean structure in R 2d . This is exactly the wave kinetic integral which appears in physical works on WT to describe the 4-waves interaction, see [START_REF] Zakharov | Kolmogorov Spectra of Turbulence[END_REF], p. 71 and [START_REF] Nazarenko | Wave Turbulence[END_REF], p. 91.

In particular from what was said it follows that in the regime (1.19), the scaling ρ ∼ ν -1/2 is the only scaling of ρ when the energy spectrum admits a non-trivial kinetic limit of order one; see [START_REF] Dymov | Formal expansions in stochastic model for wave turbulence 1: kinetic limit[END_REF] for a more detailed discussion.

In [START_REF] Dymov | The large-period limit for equations of discrete turbulence[END_REF] we consider the opposite to (1.19) order of limits, when first ν → 0 and then L → ∞. As in [START_REF] Dymov | Formal expansions in stochastic model for wave turbulence 1: kinetic limit[END_REF], we study the energy spectrum (1.21) of the quasisolution, but now analysis of the term n [START_REF] Buckmaster | Onset of the wave turbulence description of the longtime behaviour of the nonlinear Schrödinger equation[END_REF] s shows that the correct scaling is ρ = εL. 6 In this setting, crucially using Theorem 5.7, we establish a result, similar to the discussed above that in [START_REF] Dymov | Formal expansions in stochastic model for wave turbulence 1: kinetic limit[END_REF], but the limiting WKE (1.23) should be modified. In particular, the WKE in [START_REF] Dymov | The large-period limit for equations of discrete turbulence[END_REF] 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 [START_REF] Dymov | Formal expansions in stochastic model for wave turbulence 1: kinetic limit[END_REF][START_REF] Dymov | The large-period limit for equations of discrete turbulence[END_REF] 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 [START_REF] Dymov | The large-period limit for equations of discrete turbulence[END_REF]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 ρ:

(1.25) n s = ∞ k=0 ρ k n k s , n k s (τ ) = k 1 +k 2 =k Ev (k 1 ) s (τ )v (k 2 ) s (τ ).
Here the terms n

0 s , n 1 s and n 2 s coincide with n (0) s , n (1)
s and n

s in (1.21), while the terms n 3 s and n 4 s are slightly different (but still satisfy the estimate (1.26) 6 In [START_REF] Dymov | The large-period limit for equations of discrete turbulence[END_REF] the notation is slightly different: ε is written as ε 2 . Besides when d = 2, ρ should be multiplied by the factor (ln L)

|n k s | ≤ C # (s) (ν -2 L -2 + ν min(⌈k/2⌉,d) χ k d (ν) ≤ C # 1 (s) ν -2 L -2 + ν min(k/2,d) ,
where for definiteness we assumed that d ≥ 2. For k ≤ 2d in the exponent in r.h.s. the minimum with d can be removed, so (1.26) takes the form

(1.27) |n k s | ≤ C # (s) ν -2 L -2 + ν k/2 .
In particular, (1.27) holds for k ≤ 4 since d ≥ 2. Using estimate (1.27) with k ≤ 2d it is straightforward to show that under the scaling (1.22) the kinetic limit for quasisolution, established in [START_REF] Dymov | Formal expansions in stochastic model for wave turbulence 1: kinetic limit[END_REF] and discussed in the previous section, takes place for the truncation

∑ m i=0 ρ i v (i)
s of series (1.12) not only of order m = 2 as in (1.20) but of any order 2 ≤ m ≤ d (with the same kinetic equation (1.23) and at least with the same accuracy ε 2 , once L is sufficiently large in terms of ν -1 ).

If it was true that (1.27) is valid for any k, so that |ρ

k n k s | ≲ ε k/2 under
the scaling (1.22) with L ≫ ν -1 , then the kinetic limit would hold for truncations of any order m ≥ 2. On the contrary, if for some k we have

|n k | 1 ∶= L -d ∑ s |n k s | ≫ ν k/2
, then this is not the case since the term ρ k n k explodes as ν → 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 Js (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 Js (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 ∑ F Js (F) that approximates Ev (m) s v(n) s there are some non-trivial cancellations in a few first orders of its decomposition in ν which could lead to the validity of (1.27). In Section 8.1 we discuss this phenomenon in more detail.

The arXiv version of this paper [10], which is longer than the present one, contains some additional results and examples (not needed for the proofs of our main statements). Also we give in [10] a simple but rather long and tedious proof of Lemma 4.1 below which we omit in the present version of the paper for the sake of brevity.

= iρL -d s 1 ,s 2 ,s 3 ∈Z d L δ ′12 3s y s 1 y s 2 ȳs 3 + b(s) βs , y s (-T ) = 0, s ∈ Z d L .
(2.1)

As in (1.12), we decompose its solution y s into a formal series in ρ:

(2.2) y s = ∞ i=0 ρ i y (i) s , y (i)
s (-T ) = 0.

Then {y (0)

s = v (0)
s } are independent Gaussian processes, satisfying eq. (1.13), y

s (τ 1 )v (n) s (τ 2 ) -Ey (m) s (τ 1 )ȳ (n) s (τ 2 )| ≤ L -d C # (s).
Proof of the proposition is straightforward, so we give it only in the arXiv version of this paper [10]. There, for the needs of [START_REF] Dymov | The large-period limit for equations of discrete turbulence[END_REF], we adopt the method of Feynman diagrams developed for eq. (2.1) to eq. (1.8), and get a stronger result than (2.3). Alternatively, (2.3) immediately follows from Proposition 2.1 in [START_REF] Dymov | Formal expansions in stochastic model for wave turbulence 1: kinetic limit[END_REF] and the Cauchy inequality.

In view of Proposition 2.1 it suffices to establish inequality (1.17) (and its 1d analogy) when the processes v s . Accordingly below we study the latter instead of the former.

Interaction representation.

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

(2.4) a s (τ ) = y s (τ )e -iν -1 τ |s| 2 , s ∈ Z d L . We set (2.5) ω 12 3s = ω s 1 s 2 s 3 s = |s 1 | 2 + |s 2 | 2 -|s 3 | 2 -|s| 2 ,
(cf. (1.24)). Then eq. (2.1), written in the a-variables reads ȧs + γ s a s = iρY s (a, ν

-1 τ ) + b(s) βs , a s (-T ) = 0, Y s (a, t) = L -d s 1 ,s 2 ,s 3 ∈Z d L δ ′12 3s a s 1 a s 2 ās 3 e itω 12 3s , s ∈ Z d L , (2.6)
where {β s } is another set of standard independent complex Wiener processes. Formal expansion (2.2) takes the form

a s = ∞ j=0 ρ j a (j) s , a (j)
s (-T ) = 0, where a (j)

s (τ ) = y (j) s (τ )e -iν -1 τ |s| 2 .
For a process a (j)

s (τ ) we call the integer j ≥ 0 its degree,

(2.7) deg a (j) s (τ ) = deg ā(j) s (τ ) = j,
s satisfies the linear equation

ȧ(0) s (τ ) + γ s a (0) s (τ ) = b(s) βs (τ ), a (0)
s (-T ) = 0 , so (2.9) a (0)

s (τ ) = b(s) τ -T e -γ s (τ -l) dβ s (l),
and {a

s (τ ) = iY s (a (0) (τ ), ν -1 τ ), a (1)
s (-T ) = 0, so a (1)

s (τ ) = i τ -T e -γ s (τ -l) Y s (a (0) (l), ν -1 l) dl , (2.10)
is a Wiener chaos of third order (see [START_REF] Janson | Gaussian Hilbert Spaces[END_REF]). For what follows, it is convenient to introduce in (2.6)

s (τ ) = i τ -T L -d s 1 ,...,s 4 δ ′12 34 e -γ s 4 (τ -l) δ s s 4 (a (0) s 1 a (0) s 2 ā(0) s 3 )(l) θ(ω 12 34 , ν -1 l) dl. Similar, for m ≥ 1, a (m)
s (τ ) = m 1 +m 2 +m 3 =m-1 i τ -T dl × L -d s 1 ,...,s 4 δ ′12 34 e -γ s 4 (τ -l) δ s s 4 a (m 1 ) s 1 a (m 2 ) s 2 ā(m 3 ) s 3 (l) θ(ω 12 34 , ν -1 l). (2.13)
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 θ (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 eq. (2.13), where

(2.14) θ ∶ R 2 ↦ C is a bounded measurable function.
In particular, we may choose for θ the function θ(x, t) = I {0} (x), where I {0} (x) is the indicator function of the point x = 0 (in this case θ does not depend on t). This choice of θ is used in our paper [START_REF] Dymov | The large-period limit for equations of discrete turbulence[END_REF], where we use that in some sense a properly scaled function (s 1 , s 2 , s 3 , s 4 ) ↦ exp(iν

-1 τ ω 12 34
) with τ ≠ 0, regarded as a "generalised function on (Z d L )

4 ", converges to I {0} (ω 12 34 ) as ν → 0.

We will use correlations of the process a (0) s . By (2.9), for any τ 1 , τ 2 ≥ -T and s, s

′ ∈ Z d L , Ea (0)
s (τ 1 )a (0)

s ′ (τ 2 ) = 0, Ea (0)
s (τ 1 )ā (0)

s ′ (τ 2 ) = δ s s ′ b(s) 2 γ s e -γ s |τ 1 -τ 2 | -e -γ s (2T +τ 1 +τ 2 )
(2.15) (see (2.8) in [START_REF] Dymov | Formal expansions in stochastic model for wave turbulence 1: kinetic limit[END_REF] for the calculation). a)

a (2) s (τ 1 ) a (0) s 1 a (0) s 2 ā(1) s 3 ws 4 a (0) s 5 w s 6 ā(0) s 7 ā(0) s 8 l 1 ∶ l 2 ∶ b) a (2) ξ 0 (τ 1 ) a (0) ξ 1 a (0) ξ 2 ā(1) σ 1 wσ 2 a (0) ξ 3 w ξ 4 ā(0) σ 3 ā(0) σ 4 l 1 ∶ l 2 ∶ Figure 1. A diagram D ∈ D 2 .
The notation "l i ∶" means that the vertices a (k) , ā(k) situated opposite to it are taken at the time l i .

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 θ takes the form (2.11).

Step -γ s ′ (l k -l r ) , exp(iν

-1 ω s ′ 1 s ′ 2 s ′ 3 s ′ 4 l q ) and the processes [a (0) s ′′ (l j )]
* where a * is either a or ā, with various indices 1 ≤ k, r, q, j ≤ m and s ′ , s ′ i , s ′′ ∈ {s 1 , . . . , 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 Ĩs as in (2.16) we associate a diagram D as in fig. 1(a), constructed according to a rule, explained below; we then write Ĩs = Ĩs (D). The root of a diagram D, associated with a sum Ĩs , is the vertex a s ′ (t) by an edge and one can think that its role is to couple the block with its parent. Similarly, each vertex ā(p) s ′ (t) with degree p ≥ 1 generates a block, where the three real vertices correspond to a choice of the terms ā(m 1 ) , ā(m 2 ) , a (m 3 ) in the formula, conjugated to (2.13), and the fourth vertex w s ′′ is the non-conjugated virtual vertex (corresponding to factor δ s ′ s ′′ ). We denote by D m the set of all diagrams D associated with different terms Ĩs from (2.16), sum of which gives a (m) s (τ 1 ). 7 Then in every diagram D ∈ 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 ξ 1 , . . . , ξ 2m the indices s j ∈ Z d L , enumerating the nonconjugated vertices, and by σ 1 , . . . , σ 2m -those enumerating the conjugated vertices. Then, setting ξ 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 ξ and σ.

Now the linear relations, imposed on the multi-indices ξ i , σ j with i, j ≥ 1 in a sum (2.16) take the following compact form:

(2.17) 1)

δ ′ξ 2j-1 ξ 2j σ 2j-1 σ 2j = 1 ∀j 2) indices of adjacent in D vertices are equal,
where the first relation comes from the factor δ ′12 34 in (2.13) while the second one -from the factor δ s s 4 . Moreover, we see that in this notation the frequency of a fast rotating exponent, hidden in the term (. . . ) in (2.16) and arising from the factor θ(ω

12 34 , ν -1 l) = e iν -1 ω 12 34 l in (2.13), also takes the compact form iν -1 ∑ m j=1 ω ξ 2j-1 ξ 2j σ 2j-1 σ 2j l j . We have seen that a (m) s (τ 1 ) = D∈D m Ĩs (D),
where each sum Ĩs (D) has the form (2.16). Similarly,

ā(n) s (τ 2 ) = ∑ D∈D n Ĩs ( D)
, where D n is the set of diagrams with the root ā(n) σ 0 (τ 2 ), σ 0 ∶= s, constructed by the same inductive rule as diagrams from the set D m (see fig. 3(d)). 8 Finally, we introduce the set s (τ 1 ) through the index m only. This will become more clear in Section 3, where we will construct D m carefully. 8 Equivalently, the set D n can be obtained by conjugating diagrams from the set 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 ξ j with σ j , so that again the non-conjugated vertices are enumerated by the indices ξ j while the conjugated -by the indices σ j .

D m × D n of diagrams D = D 1 ⊔ D2 ,
the root of D2 is denoted σ 0 , then all vertices of D are enumerated by 4N +2 indices ξ i , σ j , where 0 ≤ i, j ≤ 2N and always ξ 0 = σ 0 = s; see fig.

Ea (m) s (τ 1 )ā (n) s (τ 2 ) = D∈D m ×D n EI s (D).
In fact, we do not write down the sums Ĩs 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 EI s (D)

for each diagram D = D 1 ⊔ D2 ∈ D m × D n . Due to (2.16), (2.19) EI s (D) = . . . L -N d ξ 1 ,...,ξ 2N ,σ 1 ,...,σ 2N E(. . .) dl 1 . . . dl N ,
where the term in the brackets is obtained by taking the product of the terms (. . .) in the integrands of Ĩs (D 1 ) and Ĩs ( D2 ), corresponding to the leaves of the diagram D. So, the brackets in (2.19) contain a product of 2N +2 random variables of the form [a (0)

s ′ (t)]
* , multiplied by a deterministic factor. Since a

s ′ (t) are independent complex Gaussian random variables whose correlations are given by (2.15), then by the Wick theorem (see e.g. in [START_REF] Janson | Gaussian Hilbert Spaces[END_REF]) the expectation E(. . .) is a sum over different Wick-pairings of the non-conjugated variables a (0)

ξ j (l k ) with conjugated ā(0) σ r (l q ), where (2.20) Ea (0) ξ j (l k )ā (0) σ r (l q ) ≠ 0 only if ξ j = σ r .
The summands can be parametrised by Feynman diagrams F (see e.g. in [START_REF] Janson | Gaussian Hilbert Spaces[END_REF]), obtained from the diagram D by connecting with an edge every pair of the Wick-coupled vertices a (0) ξ j (l k ) and ā(0) σ r (l q ), see fig. 5. So to every diagram D correspond several Feynman diagrams 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),

Ea (m) s (τ 1 )ā (n) s (τ 2 ) = F∈F m,n J s (F),
where the sum is taken over the set F m,n of Feynman diagrams F obtained via all possible Wick-couplings in various diagrams

D ∈ D m × D n . Each (deterministic) sum J s (F) has the form (2.21) J s (F) = . . . L -N d ξ 1 ,...,ξ 2N ,σ 1 ,...,σ 2N (. . .) dl 1 . . . dl N ,
where the term in the brackets (. . . ) is obtained from those in (2.19) by a Wick-pairing associated to the diagram F. See Lemma 4.1 for an explicit form of the sums J s (F).

Step 3. The indices ξ i , σ j in (2.21) are subject to the linear relations (2.17), where 1 ≤ j ≤ N and in (2.17)(2) the diagram D is replaced by some F (see (2.20)). By (2.17)(2) (with D replaced by F), the multi-index σ = (σ 1 , . . . , σ 2N ) is a function of the multi-index ξ = (ξ 1 , . . . , ξ 2N ), so it remains to study eq. (2.17) [START_REF] Buckmaster | Effective dynamics of the nonlinear Schrödinger equation on large domains[END_REF], where we substitute σ = σ(ξ). Analysing the diagrams F, forming the class F m,n , we find an F-dependent affine parametrization of solutions ξ for this equation by poly-vectors z = (z 1 , . . . , z N ),

z j ∈ Z d L .
Then we write the normalized sum L

-N d ∑ ξ,σ in (2.21) as L -N d ∑ z , ap- proximate it by an integral over R N d with accuracy L -2 ν -2
, and accordingly find that

(2.22) J s (F) = R N dl R N d dz F F s (l, z)e iν -1 Ω F (l,z) + O(L -2 ν -2 ).
Here l = (l 1 , . . . , l N ) ∈ R N , the density function F F s (which we write down explicitly) is Schwartz in the variables (s, z), piecewise smooth and fast decaying in l, while the phase function Ω

F is quadratic in z and linear in

l: Ω F (l, z) = ∑ 1≤i,j≤N α F ij z i ⋅ z j (l i -l j ), where z i ⋅ z j is the scalar product in R d ; see Theorem 5.9. The constant skew-symmetric matrix α F = (α F ij ), α F ij ∈ {-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 ν ≪ 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 iν -1 Ω F .
The task is made complicated by the degeneracy of the quadratic form Ω F (l, ⋅). To handle it, in Sections 6, 7 and 8.5 we prove abstract theorems which allow to estimate integrals of a more general forms than (2.22).

Diagrams D and formula for products

a (m) s ā(n) s
In this section we construct the set of diagrams

D m × D n , associated to the product a (m) s (τ 1 )ā (n) s (τ 2 )
, and express the latter through the random processes a (0) k and ā(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.

1 c (0) 2 c(0) 1 w2 c (0) 3 c (0) 4 c(0) 3 w4 b) c (2) 0 c (0) 1 c (1) 2 c(0) 1 w2 c (0) 3 c (0) 4 c(0) 3 w4 c) c (2) 0 c (0) 1 c (0) 2 c(1) 1 w2 c (0) 3 w 4 c(0) 3 c(0) 4 Figure 2. The set of diagrams D 2 .
We start by constructing the set D m , construction of D n is similar. For example, the set D 1 consists of a unique diagram, which can be obtained from the diagram in fig. 3(a deg ĉ(k) j = k, and each vertex is either virtual or real. If a vertex c j , cj or ĉj is virtual, we may write it as w j , wj or ŵj . Each virtual vertex ŵj has zero degree, deg ŵj = 0, 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 ≥ 1, is {c 2k-1 , c 2k , c2k-1 , c2k }. Among these four vertices either the vertex c 2k or the vertex c2k is virtual, while the other are real. If the vertex c 2k is virtual then we denote this block by B k ,

(3.1) B k = {c 2k-1 , w 2k , c2k-1 , c2k },
and call it the non-conjugated k-th block. If the vertex c2k is virtual, we denote Non-conjugated vertices c j generate conjugated blocks Bk while conjugated vertices cj generate non-conjugated blocks B k . The numeration of the blocks (i.e. the dependence of k as a function of (ĉ 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 Bk , but fixes only their sum. So, for m ≠ 0, 1 the diagram D obtained by the inductive procedure above is not unique. We define D m as the set of all possible diagrams obtained by this procedure.

(3.2) Bk = {c 2k-1 , c 2k , c2k-1 , w2k }, a) c (1) 0 w2 c(0) 1 c (0) 2 c (0) 1 c(0) 0 b) c (3) 0 c (2) 1 c (0) 2 c(0) 1 w2 c (0) 3 c (1) 4 c(0) 3 w4 c (0) 5 c (0) 6 c(0) 5 w6 c) c (3) 0 c (1) 1 c (0) 2 c(1) 1 w2 c (0) 3 c (0) 4 c(0) 3 w4 c (0) 5 w 6 c(0) 5 c(0) 6 d) c(1) 0 w 2 c(0) 1 c(0) 2 c (0) 1
The set of diagrams D n is constructed by a similar procedure. The only difference is that in diagrams D ∈ D n the root vertex is conjugated and has the form c(n) 0 , see fig. i , w j }, where the degree p ≥ 1.

D m × D n = {D 1 ⊔ D2 ∶ D 1 ∈ D m , D2 ∈ D n }.
• An edge which joins c i with cj is denoted

(c i , cj ) (either c i or cj is a virtual vertex). Each diagram D ∈ D m × D n has N edges.
• Recall that a real vertex ĉ(0) j of zero degree is called a leaf. There is no vertex in D, adjacent to a leaf. The number of conjugated leaves equals to the number of non-conjugated leaves and equals to N + 1.

where ξ i , σ j ∈ Z d L , so (ξ, σ) ∈ Z (4N +2)d L
, and we recall that 4N + 2 is the number of vertices in D. These will be the indices for summation in a formula for a

(m) s (τ 1 )ā (n) s (τ 2 )
, where ξ j 's are indices for the non-conjugated variables a s and σ j -for the conjugated variables ās in the following sense.

We regard a real vertex c

j (l, ξ, σ) = a (p) ξ j (t), c(p) j (l, ξ, σ) = ā(p) σ j (t ′ ),
where t and t ′ are components of the time-vector l in (3. ξ j (t) and ā(p) σ j (t ′ ). For virtual vertices w j and wj we say that they take values w ξ j and wσ j . . At some stages of our constructions it will be necessary to put restrictions on the (ξ, σ) and substitute in the diagram only the corresponding multi-indices which we call admissible. The set of all admissible multi-indices (ξ, σ) is denoted

A s (D) ⊂ R (4N +2)d ,
where we take s, ξ i , σ j in R d rather than in Z d L to be able to extend the correlation from Theorem 2.2 to a function on R d = {s}. The set A s (D) is defined by the following three relations:

• We have (3.6) ξ 0 = σ 0 = s. • For all 1 ≤ i ≤ N , (3.7) δ ′ξ 2i-1 ξ 2i σ 2i-1 σ 2i = 1.
• If the vertices c i and cj are adjacent, the corresponding indices are equal:

(3.8) c i ∼ cj ⇒ ξ i = σ j .
We will also use the corresponding discrete subset (3.9)

A L s (D) = A s (D) ∩ Z (4N +2)d L ,
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 ∈ Z • Let (ξ, σ) ∈ A s (D) be an admissible index-vector. Then ξ i = σ j if c i ∼ cj . Denoting by ϕ an edge ϕ = (c i , cj ), we set

(3.10) γ ϕ (ξ, σ) ∶= γ ξ i = γ σ j
(γ s 's are defined in (1.4)).

• Recall that in each pair of adjacent vertices in D always one vertex is real while another is virtual. Then, by l ϕ r and l ϕ w we denote the times assigned to the real and virtual vertices, connected by the edge ϕ. So each l ϕ r,w is a mapping from E(D) to the set {τ 1 , τ 2 , l 1 , . . . , l N }.

• By l ĉi we denote the time, assigned to a vertex ĉi .

• By

G D (τ, l, ξ, σ) ∶= c D ϕ∈E(D) e -γ ϕ (ξ,σ) (l ϕ r -l ϕ w ) I {-T ≤l ϕ w ≤l ϕ r } (τ, l) j∈L(D) a (0) ξ j (l c j ) j∈ L(D) ā(0) σ j (l cj ), (3.
c D = (-1) #w D i N ,
where #w D is the number of non-conjugated virtual vertices of the diagram D, N = m + n and i = -1 is the imaginary unit. Set

(3.14) ω j (ξ, σ) = ω ξ 2j-1 ξ 2j σ 2j-1 σ 2j
and ω = (ω 1 , . . . , ω N ),

we recall that the quadratic form ω s 1 s 2 s 3 s is defined in (2.5). For vectors ⃗ x = (x 1 , . . . , x N ) and ⃗ t = (t 1 , . . . , t N ) set also

(3.15) Θ(⃗ x, ⃗ t) = N j=1 θ(x j , t j ),
where θ is as in (2.14).

Lemma 3.1. For any integers m, n ≥ 0 satisfying

N ∶= m + n ≥ 1, s ∈ Z d L
and τ 1 , τ 2 ≥ -T , we have

(3.16) a (m) s (τ 1 )ā (n) s (τ 2 ) = D∈D m ×D n I s (D),
where

I s (D) = I s (τ ; D), (3.17) I s (τ ; D) = R N dl L -N d (ξ,σ)∈A L s (D) G D (τ, l, ξ, σ) Θ(ω(ξ, σ), ν -1 l).
Components of the formula (3.17) are quite simple. Indeed, the factor ∏ j∈L(D) a (0) The factor θ(ω j , ν -1 l j ), entering the function Θ(ω, ν -1 l), comes from the iteration of the Duhamel formula, generating the j-th block of D.

ξ j (l c j ) ∏ j∈ L(D) ā(0) σ j (l cj ), entering G D (τ, l, ξ, σ) (see (3. 12
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 a (1)

s (τ 1 )ā (0) s (τ 2 ) = i τ 1 -T dl 1 L -d ξ 1 ,ξ 2 ,σ 1 ,σ 2 δ ′ξ 1 ξ 2 σ 1 σ 2 δ s σ 2 e -γ σ 2 (τ 1 -l 1 ) a (0) ξ 1 a (0) ξ 2 ā(0) σ 1 (l 1 )ā (0) s (τ 2 ) θ(ω 1 , ν -1 l 1 ), (3.18)
where we use notation (3.14). The set D 1 × D 0 consists of a unique diagram, given in fig. 3 has the form

A L s (D 1,0 ) = (ξ 0 , ξ 1 , ξ 2 , σ 0 , σ 1 , σ 2 ) ∈ Z 6d L ∶ ξ 0 = σ 0 = s, δ ′ξ 1 ξ 2 σ 1 σ 2 = 1, ξ 0 = σ 2 .
Then we rewrite the r.h.s. of (3.18) as

i τ 1 -T dl 1 L -d (ξ,σ)∈A L s (D 1,0 ) e -γ σ 2 (τ 1 -l 1 ) a (0) ξ 1 a (0) ξ 2 ā(0) σ 1 (l 1 )ā (0) σ 0 (τ 2 ) θ(ω 1 , ν -1 l 1 ).
The unique edge of the diagram D

1,0 is ϕ ∶= (c (1)
0 , w2 ), so using the introduced in Section 3.1.6 notation we write e

-γ σ 2 (τ 1 -l 1 ) = e -γ ϕ (l ϕ r -l ϕ w ) and ∫ τ 1 -T dl 1 = ∫ R I {-T ≤l ϕ w ≤l ϕ r } dl 1 .
Then, noting that the term a (0) diagrams F (see [START_REF] Janson | Gaussian Hilbert Spaces[END_REF] for the notion), which are obtained from the diagram D by connecting the Wick paired leaves by edges, see fig. 5.

ξ 1 a (0) ξ 2 ā(0) σ 1 ā(0) σ 0 is
1 c (0) 2 c(0) 1 w2 c (0) 3 c (0) 4 c(0) 3 w4 b) c (2) 0 c(0) 0 c (1) 1 c (0) 2 c(0) 1 w2 c (0) 3 c (0) 4 c(0) 3 w4 f F f F f F f F
Next we give a detailed definition of the Feynman diagrams and discuss related notions. Let F(D) be the set of all Feynman diagrams, which may be obtained from D using different partitions of the set of leaves to pairs. We define

(4.1) F m,n = ⋃ D∈D m ×D n F(D).
This is the set of all Feynman diagrams corresponding to the product a

ξ f F = (ξ f F (0) , . . . , ξ f F (2N ) ).
ξ 0 = σ 0 = s. • For any 1 ≤ i ≤ N , (4.6) δ ′ξ 2i-1 ξ 2i σ 2i-1 σ 2i = 1.
In other words, we require (4.5) and (4.6) to be satisfied by the multiindices ξ, σ, where σ is obtained from ξ accordingly to the edges of diagram F: We also consider the discrete set

σ j = ξ i if a vertex cj is coupled with the vertex c i . In particular, if ξ ∈ A ′ s (F) then (ξ, ξ f F ) ∈ A s (D F ),
A ′L s (F) = A ′ s (F) ∩ 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 EI(D) can be written as a sum over the Feynman diagrams F,

EI s (D) = F∈F(D) J s (F).
Each summand J s (F) has the form (3.17) where G D is replaced by a function obtained by the corresponding to F Wick pairing of the values a (0) ξ i , ā(0) σ j of leaves. In this formula for J s (F) we should take summation only over those multi-indices (ξ, σ) ∈ A L s (D) for which ξ i = σ j once the vertices c i and cj are adjacent in F, that is over vectors of the form (ξ, ξ f F ), where ξ ∈ A ′L s (F).

Formula for the expectation Ea

(m) s ā(n)
s . Below we introduce some more notation. Let F ∈ F m,n be a Feynman diagram.

• We denote by E(F) the set of edges of the diagram F, then E(D F ) ⊂ E(F) (see (4.2)). We also set

E D (F) ∶= E(D F ), E L (F) ∶= E(F) \ E D (F) so E(F) = E D (F) ∪ E L (F).
• For an edge ϑ ∈ E(F) which couples some vertices c i and cj , we set c ϑ ∶= c i , cϑ ∶= cj , and denote by l c ϑ and l cϑ the times assigned to the vertices c ϑ and cϑ . We also set

γ ϑ (ξ) ∶= γ ξ i and b ϑ (ξ) ∶= b(ξ i ) if ϑ couples c i and c j ,
where γ s is defined in (1.4) and b(s) enters the definition of the random force η ω .

• If ϕ ∈ E D (F), we define the times l

ψ∈E L (F) e -γ ψ |l c ψ -l cψ | -e -γ ψ (l c ψ +l cψ +2T ) (b ψ ) 2 γ ψ , (4.7)
where

γ ϕ = γ ϕ (ξ), γ ψ = γ ψ (ξ), b ψ = b ψ (ξ).
We also associate to F the quadratic forms

(4.8) ω F j (ξ) ∶= ω j (ξ, ξ f F ) = ω ξ 2j-1 ξ 2j ξ f F (2j-1) ξ f F (2j) and set ω F = (ω F 1 , . . . , ω F
s (τ 1 )ā (n) s (τ 2 ) = F∈F m,n c F J s (F),
where J s (F) = J s (τ ; F)

(4.10) J s (τ ; F) = R N dl L -N d ξ∈A ′L s (F) F F (τ, l, ξ) Θ(ω F (ξ), ν -1 l).
The density F F , given by (4.7), is real, smooth in ξ ∈ R (2N +1)d , piecewise smooth in l ∈ R N and such that for any vector κ ∈ Z d(2N +1) +

we have: ā(n) s = 0. Indeed, in this case F m,n = ∅, since every partition of the set of leaves into pairs couples vertices from the same block. Accordingly, we will often assume that m + n ≥ 2. ā(n) s , however, the structure of the set of multi-indices A ′L s (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.

(4.11) |∂ κ ξ F F (τ, l, ξ)| ≤ C # κ (ξ) e -δ ∑ m i=1 |τ 1 -l i |+∑ N i=m+1 |τ 2 -l i | , for any ξ ∈ A ′ s (F), l ∈ R N and

Change of coordinates and final formula for expectations Ea

By (4.6) the vectors ξ ∈ A ′L s satisfy the system of equations (5.1)

ξ 2k-1 + ξ 2k = σ 2k-1 + σ 2k , σ ∶= ξ f F , 1 ≤ k ≤ N.
We start with constructing coordinates x = (x 0 , . . . , x 2N ), where x 0 = ξ 0 and for every 1

≤ k ≤ N we set either x 2k-1 = ξ 2k-1 -σ 2k-1 and x 2k = ξ 2k -σ 2k or x 2k-1 = ξ 2k-1 -σ 2k and x 2k = ξ 2k -σ 2k-1
, so that equations (5.1) take trivial form x 2k = -x 2k-1 , ∀k. We will make a choice (different for different k and dependent on the diagram F) between the two possibilities above in such a way that the transformation ξ ↦ x is invertible. Then, components This leads to an explicit formula for the sum J s (F) and next in Sections 6, 7 allows to analyse its asymptotic behaviour as ν → 0, L → ∞. Next we explain construction of the transformation ξ ↦ x in detail. Proof of Lemma 5.1, given in Appendix 8.2, follows from the fact that the diagram, made by the blocks of F, is connected. In general, the Hamilton cycle in Lemma 5.1 is not unique, so we fix one for each diagram U F and denote it as C F . Examples of the Hamilton cycles C F are given in fig. 6(b). It is straightforward to see that, by construction of the diagrams U F , the cycles C F contain all solid edges of U F (which are the edges of 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 C F . On the cycle C F the dashes and solid edges alternate.

z k ∶= x 2k-1 , 1 ≤ k ≤ N ,
We orient the dashed edges of C 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, C F becomes an oriented cycle, see fig. 6(b).

Let us denote by ĈF the reduced oriented cycle C F , that is the cycle C F in which we keep only non-conjugated vertices and "forget" all conjugated ones, see fig. 7. The cycle ĈF defines a cyclic permutation π F of the set of x 0 ∶= ξ 0 , x j ∶= ξ j -ξ π F (j) ∀1 ≤ j ≤ 2N, and define the (2N + 1)-vector x = (x j ) 0≤j≤2N , x j ∈ R d . Since π F is a cyclic permutation of the set {0, . . . , 2N }, then the transformation ξ ↦ x is a bijection of (R d ) 2N +1 . Indeed, it is invertible since iterating (5.2) we get

c 0 c 3 c 2 c 4 c 1 π F π F A 1 A 2
ξ j = x j + ξ π F (j) = x j + x π F (j) + ξ π 2 F (j) = . . . = k i=0 x π i F (j) + ξ 0 ,
where k = k(j) denotes minimal positive integer satisfying π k+1 F (j) = 0. In other words, since ξ 0 = x 0 , (5.3) (5.4) α

ξ j = i∶ c i ∈[c j ,c 0 ] x i , 0 ≤ j ≤ 2N,
F ij = ⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩ 1 if c 2j-1 ∈ [c 2i-1 , c 2i ], c 2j ∉ [c 2i-1 , c 2i ], -1 if c 2j-1 ∉ [c 2i-1 , c 2i ], c 2j ∈ [c 2i-1 , c 2i ], 0 otherwise.
It is straightforward to see that the matrix α F is skew-symmetric, so it is S-regular in the following sense:

Definition 5.2. A square matrix α = {α ij } is called S-regular if it is skewsymmetric and all its elements α ij are such that α ij ∈ {-1, 0, 1}. If in addition all rows and columns of α 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.

(5.7) ξ 2k-1 (x) -ξ π F (2k) (x) = N i=1 α F ki x 2i-1 .
Proof. In this proof we omit the lower index F in the notation π F and f F . We start by noting that for every k ≥ 1

f (2k-1) , ξ f (2k) } = {ξ π(2k-1) , ξ π(2k) }, so δ ′ξ 2k-1 ξ 2k ξ f (2k-1) ξ f (2k) = δ ′ξ 2k-1 ξ 2k
ξ π(2k-1) ξ π(2k) . Then eq. (4.6) is equivalent to δ ′ξ 2k-1 ξ 2k ξ π(2k-1) ξ π(2k) = 1, that is to the equation (5.9)

ξ 2k-1 + ξ 2k = ξ π(2k-1) + ξ π(2k)
jointly with

(5.10)

{ξ 2k-1 , ξ 2k } ∩ {ξ π(2k-1) , ξ π(2k) } = ∅ (see (1. 10
)). Equation (5.9) is equivalent to (5.5)(b). The equality ξ 0 = s from (4.5) is equivalent to (5.5)(a). The equality σ 0 = s, in view of (5.3), takes the form (5.11)

s = σ 0 = ξ f (0) = i∶ c i ∈[c f (0) ,c 0 ] x i . Since f (0) = π(0), then [c f (0) , c 0 ] = [c π(0) , c 0 ]
, and this interval contains all vertices of the cycle ĈF . 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 (5.12)

ξ 2k-1 ≠ ξ π(2k-1) and ξ 2k-1 ≠ ξ π(2k) .
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),

ξ 2k-1 -ξ π(2k) = i∶ c i ∈[c 2k-1 ,c 0 ] x i - i∶ c i ∈[c π(2k) ,c 0 ] x i .
Let us first assume that c π(2k) ∈ [c 2k-1 , c 0 ] and c π(2k) ≠ c 2k-1 . Then the identity above takes the form

ξ 2k-1 -ξ π(2k) = i∶ c i ∈[c 2k-1 ,c 2k ] x i ,
where we have used that the vertex situating in ĈF before c π(2k) is c 2k . In view of the assumption above, c 0 ∉ [c 2k-1 , c 2k ]. Then, due to cancellations provided by (5.5)(b), we get (5.7).

The case

c π(2k) ∉ [c 2k-1 , c 0 ] can be considered similarly. If c π(2k) = c 2k-1 , that is π(2k) = 2k -1,
ξ j (z) = s + i∶ c i ∈[c j ,c 0 ) (-1) i+1 z ⌈i/2⌉ , 1 ≤ j ≤ 2N,
where [c j , c 0 ) ∶= [c j , c 0 ]\{c 0 }, and ξ 0 = s. Here we emphasize that the linear mapping z ↦ ξ depends on F (through the interval [c j , c 0 )) and that ξ is an affine vector-function of the parameter s ∈ R d . Thus, in the z-coordinates

(5.14) A ′ s (F) = {ξ(z) ∶ z ∈ Z(F)}
, where, by (5.6), (5.15)

Z(F) = {z ∈ R dN ∶ z j ≠ 0 and N i=1 α F ji z i ≠ 0 ∀1 ≤ j ≤ N }.
Remark 5.4. Since the choice of the Hamilton cycle C F in general is not unique, the obtained parametrization z ↦ ξ is not unique as well. However, one can show that if z ′ ↦ ξ is another parametrization, obtained by the procedure above, and α F ′ is the associated incidence matrix, then for each

j we have either z ′ j (ξ) = z j (ξ) or z ′ j (ξ) = ∑ N i=1 α F ji z i .
In the latter case we also have the symmetric relation

z j (ξ) = ∑ N i=1 α F ′ ji z ′ i .
Next we write the quadratic forms ω F j (ξ), defined in (4.8), in the zcoordinates.

Proposition 5.5. Functions ω F j (ξ), restricted to the set A ′ s (F) ∋ ξ and written in the z-coordinates, take the form (5. [START_REF] Hörmander | The Analysis of Linear Partial Differential Equations[END_REF])

ω F j (z) = 2z j ⋅ N i=1 α F ji z i ,
where the S-regular incidence matrix α F = (α F ij ) 1≤i,j≤N is defined in (5.4). Proof. In this proof we again skip the lower index F in the notation π F . Due to (5.8) and their definition (3.14),(4.8), the functions ω

F j (ξ) have the form ω F j (ξ) = |ξ 2j-1 | 2 -|ξ π(2j-1) | 2 + |ξ 2j | 2 -|ξ π(2j) | 2 .
Using (5.5)(b), we obtain

ω F j (ξ) = x 2j-1 ⋅ (ξ 2j-1 + ξ π(2j-1) ) + x 2j ⋅ (ξ 2j + ξ π(2j) ) = x 2j-1 ⋅ (ξ 2j-1 + ξ π(2j-1) -ξ 2j -ξ π(2j) ) = x 2j-1 ⋅ (2ξ 2j-1 -x 2j-1 -2ξ π(2j) -x 2j ) = 2x 2j-1 ⋅ (ξ 2j-1 -ξ π(2j) ).
Thus, by (5.7), ω

F j (z) = 2z j ⋅ ∑ N i=1 α F ji z i .
Finally, we note that instead of considering the set of Feynman diagrams F m,n it suffices to study only its subset. Denote by

F true m,n ⊂ F m,n
the subset of diagrams F for which the corresponding incidence matrix α 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 F ∉ F true m,n , then Z(F) = ∅. If F ∈ F true m,n , then Z(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 R dN .

Due to (5.14), Proposition 5.6 implies that the integrals J s (F) from (4.10) with F ∉ F true m,n vanish.

Final formula for expectations

Ea (m) s ā(n) s . Let F F s (τ, l, z) be the function F F (τ, l, ξ)
, defined in (4.7), restricted to the set A ′ s (F) and written in the z-coordinates, (5.17)

F F s (τ, l, z) = F F τ, l, ξ(z) .
The function F F s depends on s ∈ R d through the transformation z ↦ ξ, see (5.13). Changing in the sums J s (F) the coordinates ξ 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 ≥ 0 satisfying

N = m + n ≥ 1, s ∈ Z d L
and τ 1 , τ 2 ≥ -T , we have

(5.18) Ea (m) s (τ 1 )ā (n) s (τ 2 ) = F∈F true m,n c F J s (F),
where F true m,n is a finite set of diagrams, defined in Section 5.2, the constants c F ∈ {±1, ±i} are defined in Section 4.2 and

(5.19) J s (τ ; F) = R N dl L -N d z∈Z(F)∩Z N d L F F s (τ, l, z) Θ(ω F (z), ν -1 l).
Here Θ(ω

F , ν -1 l) = ∏ N j=1 θ(ω F j , ν -1 l j ),
(5.20) |∂ µ s ∂ κ z F F s (τ, l, z)| ≤ C # µ,κ (s)C # µ,κ (z) e -δ ∑ m i=1 |τ 1 -l i |+∑ N i=m+1 |τ 2 -l i | with a suitable δ = δ N > 0, for any vectors µ ∈ Z d + , κ ∈ Z dN + , and any s ∈ R d , z ∈ R N d .
Since the density F Subsequent analysis of the sums J s (F) is based on the lemma below, where the function Θ(ω F (z), ν -1 l) is defined as in (3.13) with θ(y, t) = e ity (cf. (2.11)).

Lemma 5.8. If θ(y, t) = e ity , then

(5.21) Θ(ω F (z), ν -1 l) = e iν -1 Ω F (l,z) ,
where the phase function Ω F is given by

(5.22) Ω F (l, z) = 1≤i,j≤N α F ij (l i -l j ) z i ⋅ z j .
Proof. Since θ(x, t) = e itx , the function Θ is given by (5.21) with Ω

F (l, z) = ∑ N j=1 l j ω F j (z). From (5.16) it follows that Ω F = 2 ∑ 1≤i,j≤N l j α F ji z j ⋅ z i . Now (5.
J s (F) - R N dl R dN dz F F s (τ, l, z)e iν -1 Ω F (l,z) ≤ C # (s)L -2 ν -2 ,
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 [START_REF] Dymov | Formal expansions in stochastic model for wave turbulence 1: kinetic limit[END_REF]. Together with (5.18) this estimate implies the main result of the section: Theorem 5.9. Assume that d ≥ 2 and the function θ is as above. Then for any integers m, n ≥ 0 satisfying

N = m + n ≥ 1, any s ∈ R d and τ 1 , τ 2 ≥ -T ,
we have

(5.23) Ea (m) s (τ 1 )ā (n) s (τ 2 ) - F∈F true m,n c F Js (F) ≤ C # (s)L -2 ν -2 ,
where

(5.24) Js (τ ; F) = R N dl R dN dz F F s (τ, l, z)e iν -1 Ω F (l,z) .
The constant c F = c D F is defined in (3.13) and the diagram D F -in (4.2).

The real valued density F F s is given by (5.17) together with (4.7) and satisfies the estimate (5.20). The phase function Ω F is given by (5.22) where the incidence matrix α 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 Ea (m) s ā(n) s with arbitrary m and n, up to an error term of the size O(L -2 ν -2 ). In Section 8.4 of the arXiv version [10] of this paper we consider an example when m = n = 2.

Since the functions F F and Ω 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 8.4 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 Js (F). Next we estimate the integrals Js (F) from Theorem 5.9.

Theorem 6.1. For any integers m, n ≥ 0, satisfying

N = m + n ≥ 2, any F ∈ F true m,n and s ∈ R d , (6.1)
| Js (F)| ≤ C # (s)ν min(⌈N /2⌉,d) χ N d (ν), uniformly in τ 1 , τ 2 ≥ -T , wherethe function χ N d 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. 10 We deduce (6.1) from an abstract theorem below where we estimate integrals of the following more general form. Let (6.2)

J s = R K dl R dM dz F s (l, z)e iν -1 Q(l,z) , where M ≥ 2, K ≥ 1, d ≥ 1, l = (l 1 , . . . , l K ) and z is the polyvector (z 1 , . . . , z M ) with z i = (z 1 i , . . . , z d i ) ∈ R d . The phase function Q(l, z) is
assumed to be linear in l and quadratic in z,

Q(l, z) = Q(l)z ⋅ z = M i,j=1 (q ij ⋅ l) (z i ⋅ z j ) = M i,j=1 q ij ⋅ l d m=1 z m i z m j .
Here

q ij = q ji = (q k ij ) 1≤k≤K are K-vectors, so for l ∈ R K , Q(l) = q ij ⋅ l 1≤i,j≤M is a real symmetric M × M -matrix. Denoting by Q k , 1 ≤ k ≤ K, the real symmetric matrices Q k = (q k ij ) 1≤i,j≤M , we write Q(l) as ∑ k Q k l k . We denote by R ≤ K the rank of the system of vectors q ij ∈ R K , 1 ≤ i, j ≤ M.
It equals to the rank of the system of 10 We recall that F true m,n = ∅ for N = 1, see (4.12), so we study only the case N ≥ 2.

matrices {Q k , 1 ≤ k ≤ K} in the space of M × M matrices. Indeed, consider the row

(Q 1 , . . . , Q K ),
interpreting it as a matrix with columns Q j (regarded as vectors in 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 Ω 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 ∈ R K , z ∈ R dN , smooth in z, depend on the parameter s ∈ R d , and satisfy

(6.3) |∂ κ z F s (l, z)| ≤ C # κ (s)C # κ (z)h κ (|l -l 0 |), for any κ ∈ Z dM +
and some l 0 ∈ R K . Each function h κ is assumed to be monotone decreasing, bounded and such that the function

h κ (|l|) is integrable over R K . Then (6.4) 0 ≤ h κ ≤ C κ , ∞ 0 h κ (r)r K-1 dr ≤ C κ ,
for appropriate constants C κ > 0. For k ∈ N we introduce the functions (6.5)

ψ k d (ν) = -ln ν if k = d, 1
otherwise.

Theorem 6.2. Assume that tr Q k = 0 for each k. Then, under assumption (6.3)

(6.6) |J s | ≤ C # (s)ν min(R,d) ψ R d (ν).
The function C # depends on F s only through the functions C # κ and the constants C κ 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 8.5 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 Js (F) by explicit integration over R N l . That result is not needed for the present paper, however we believe that it is of independent interest. It is related to [START_REF] Kuksin | Asymptotical expansions for some integrals of quotients with degenerated divisors[END_REF].

In view of (5.20), the integrals Js (F) satisfy the assumptions of Theorem 6.2 with the functions C # κ and the constants C κ independent from τ 1 , τ 2 . Here, in view of (5.22), we have q ij ⋅ l = α F ij (l i -l j ) and it can be shown that the rank R satisfies R ≥ ⌈N /2⌉ (this follows solely from the fact that the matrix α 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

χ N d is replaced by ψ R d .
In the case R = d = ⌈N /2⌉ (which holds true for certain matrices α F ) this estimate is slightly weaker than (6.1). Indeed, it

gives | Js (F)| ≤ C # (s)ν ⌈N /2⌉ ln ν -1 ,
so compare to (6.1) we get an additional factor ln ν -1 , unless N = 2, 3. This is insufficient for the purposes of [START_REF] Dymov | Formal expansions in stochastic model for wave turbulence 1: kinetic limit[END_REF]. To overcome this difficulty, in Theorem 7.1 we prove that if the tensor (q k ij ) 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 λ 1 (l), . . . , λ M (l) be (real) continuous in l eigenvalues of the symmetric matrix Q(l), enumerated in decreasing order:

(6.7) |λ 1 (l)| ≥ |λ 2 (l)| ≥ . . . ≥ |λ M (l)| for any l ∈ R K .
The corresponding normalized eigenvectors are vector-functions of l, forming an orthonormal basis in R M for every l, and analytic outside an analytic subset of R K l of positive codimension (while as functions of l ∈ R k they even are not continuous). Denoting by w(l, z) = w = (w 1 , . . . , w M ), w i ∈ R d , the polyvector z = (z 1 , . . . , z M ), written in this basis, we write function Q as

Q(l, z) = M i=1 λ i (l)|w i | 2 .
Since the transformation z ↦ w is orthogonal for every l, then dz = dw and (6.8)

J s = R K dl R dM dw H s (l, w)e iν -1 ∑ M i=1 λ i (l)|w i | 2 .
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 × M -matrix, M ≥ 2,
such that tr A = 0, and let λ 1 , . . . , λ M be its eigenvalues. Then (6.9)

i,j a 2 ij = -2 i<j λ i λ j .
Proof. Let ∥A∥ HS be the Hilbert-Schmidt norm of the matrix A. From one hand, ∥A∥ 2 HS equals to the l.h.s. of (6.9). From another hand,

∥A∥ 2 HS = M i=1 λ 2 i = M i=1 λ i 2 -2 i<j λ i λ j . It remains to note that ∑ M i=1 λ i 2 = (tr A) 2 = 0.
Applying Lemma 6.3 to the matrix Q(l), we find 1 2

i,j

q ij ⋅ l 2 = - i<j λ i (l)λ j (l) =∶ K(l) ≥ 0.
Let us consider the sets

N ν ∶= {l ∈ R K ∶ K(l) ≥ ν 2 } and N c ν ∶= R K \ N ν .
Below for a set V ⊂ R K l we denote by ⟨J s , V ⟩ the integral (6.8) with the domain of integration R K l 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 |⟨J s , N c ν ⟩| ≤ C # (s)ν R .
Proof of Lemma 6.4 is given at the end of this section. It remains to establish estimate (6.6) for the integral ⟨J s , N ν ⟩. In view of (6.7), (6.10)

|λ 1 λ 2 | ≥ -2 i<j |λ i λ j | ≥ M -2 K. So for l ∈ N ν we have |λ 1 (l)λ 2 (l)| ≥ M -2 ν 2 .
It is known (see [START_REF] Dimassi | Spectral Asymptotic in the Semi-Classical Limit[END_REF][START_REF] Hörmander | The Analysis of Linear Partial Differential Equations[END_REF]) that the (generalized) Fourier transform of the function

R 2d ∋ (w 1 , w 2 ) ↦ e -iν -1 (λ 1 |w 1 | 2 +λ 2 |w 2 | 2 ) is ζ(πν) d (|λ 1 λ 2 |) -d/2 e iν(λ - 1
1 |ξ 1 | 2 +λ - 1
2 |ξ 2 | 2 )/4 ,
where ζ is a complex constant of unit norm; here we define the Fourier transform of a regular function u(x) as ∫ e -ix⋅ξ u(x) dx. Inspired by the stationary phase method [START_REF] Dimassi | Spectral Asymptotic in the Semi-Classical Limit[END_REF][START_REF] Hörmander | The Analysis of Linear Partial Differential Equations[END_REF], we apply the Parseval's identity to the partial integral over w 1 , w 2 and get

R 2d H s (l, w)e iν -1 (λ 1 |w 1 | 2 +λ 2 |w 2 | 2 ) dw 1 dw 2 = ζ (ν/4π) d |λ 1 λ 2 | d/2 R 2d Ĥ1,2 s (l, ξ 1 , ξ 2 , w ≥3 )e -i ν 4 (λ - 1
1 |ξ 1 | 2 +λ - 1
2 |ξ 2 | 2 ) dξ 1 dξ 2 ,
for any w ≥3 = (w 3 , . . . , w M ), where Ĥ1,2 s stands for the Fourier transform of function H s with respect to the variables w 1 , w 2 . 11 Since H s satisfies assumption (6.3) (where l 0 = 0), the r.h.s. above is bounded in absolute value by

C # (s)C # (w ≥3 ) ν d |λ 1 λ 2 | d/2 h(|l|),
11 See Section 12.2 in [START_REF] Dymov | Formal expansions in stochastic model for wave turbulence 1: kinetic limit[END_REF] for justification of this formula without using the generalized Fourier transform.

where the function h ≥ 0 satisfies (6.4). So

|⟨J s , N ν ⟩| ≤ N ν C # (s)ν d h(|l|) |λ 1 (l)λ 2 (l)| d/2 dl R (M -2)d C # (w ≥3 ) dw ≥3 ≤ C # 1 (s)ν d N ν h(|l|) |λ 1 (l)λ 2 (l)| d/2 dl.
Then, due to (6.10), we have

|⟨J s , N ν ⟩| ≤ C # (s)ν d N ν h(|l|) (K(l)) d/2 dl =∶ C # (s)I ν .
Lemma 6.5. Assume that the function h satisfies assumption (6.4). Then

I ν ≤ Cν min(R,d) ψ R d (
ν). 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

|⟨J s , N c ν ⟩| ≤ C # (s) N c ν h 0 (|l|) dl.
Let us denote by v 1 , . . . , v R a maximal collection of linearly independent vectors from the set (q ij ) 1≤i,j≤M . In the integral ⟨J s , N c ν ⟩ we make a linear (non-degenerate) change of variable from l to t = (t 1 , . . . , t K ) in such a way that t k = v k ⋅ l for k ≤ R. Let t < = (t 1 , . . . , t R ) and t > = (t R+1 , . . . , t K ). Due to the identity K(l) = 1 2 ∑ q ij ⋅ l 2 , (6.11)

|t < | 2 2 ≤ K(l(t)) ≤ C|t < | 2 .
So denoting by N c ν,t the set N c ν written in the t-coordinates, we obtain

N c ν,t ⊂ {t ∶ |t < | < 2ν}. Since the function h 0 is decreasing, inequalities |l| ≥ C|t| and |t| ≥ |t > | imply that h 0 (|l|) ≤ h 0 (C|t|) ≤ h 0 (C|t > |). Then |⟨J s , N c ν ⟩| ≤ C # (s) t < ∈R R ∶ |t < | < 2ν dt < R K-R h 0 (C|t > |) dt > ≤ C # 1 (s)ν R R K-R h 0 (C|t > |) dt > .
The last integral is bounded by C ∫ ∞ 0 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 ↦ t as in the proof of Lemma 6.4 and use the notation t < , t > , introduced there. Denoting by N ν,t the set N ν written in the t-coordinates, in view of (6.11)

2 d/2 ν d t∈R K ∶ Cν≤|t < |≤1 h(C 1 |t > |) |t < | d dt ≤ C 2 ν d t < ∈R R ∶ Cν≤|t < |≤1 1 |t < | d dt < ,
where we have integrated out the variable t > . Passing to the spherical coordinates, we see that the latter integral is bounded by

C 3 ν d 1 Cν r R-1 r d dr ≤ ⎧ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎩ C 4 ν d if d ≤ R -1 C 4 ν d ln ν -1 if d = R C 4 ν R if d > R.
The r.h.s. of the last inequality can be rewritten as

C 4 ν min(R,d) ψ R d (ν).
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 ≥ 2. We consider a decomposition of the set {1, . . . N } into p ≥ 1 non-empty disjoint subsets We assume that the phase function Q admits a block-decomposition

Q(l, z) = p k=1 U k (l, z), U k (l, z) = i,j∈I k (q ij ⋅ l k ) (z i ⋅ z j ),
where

q ij = q ji ∈ R |I k | for i, j ∈ I k .
Concerning the density F s we assume that it satisfies the estimate

(7.2) |∂ κ z F s (l, z)| ≤ C # κ (s)C # κ (z) p i=1 h i κ (|l i -l i 0 |),
for any κ and some vectors l

i 0 ∈ R |I i | .
The functions h i κ are assumed to satisfy the same assumption that we imposed on the functions h κ in (6.3), where in (6.4) we replace K by |I i |. Denote by R k the rank of the system of

|I k |-vectors (q ij ) i,j∈I k . Theorem 7.1. Assume that ∑ i∈I k q ii = 0 ∈ R |I k | for any 1 ≤ k ≤ p. Then,
under assumption (7.2) we have

|J s | ≤ C # (s) p k=1 ν min(R k ,d) ψ R k d (ν),
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 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∈I p and set l < = (l i ) i∉I p . Similarly, we denote z p = (z i ) i∈I p and z < = (z i ) i∉I p . Let N p ∶= |I p | and

I s (l < , z < ) = R N p dl p R dN p dz p F s (l, z)e iν -1 U p (l,z) ,
where we recall that the function U p (l, z) depends only on l p and z p . Then,

∂ κ z < I s (l < , z < ) = R N p dl p R dN p dz p ∂ κ z < F s (l, z) e iν -1 U p (l,z) ,
for any κ. Literally repeating the proof of Theorem 6.2, we get

|∂ κ z < I s (l < , z < )| ≤ C # (s)C # (z < ) p-1 i=1 h i (|l i -l i 0 |)g(ν),
where g(ν) = ν min(R p ,d) ψ R p d (ν) and the functions h i satisfy the same assumptions that the functions h i κ from (7.2). Thus, we see that the function G s (l < , z < ) = (g(ν)) -1 I s (l < , z < ) satisfies assumption (7.2) with p = p -1. Note that

J s = g(ν) R N -N p dl < R d(N -N p ) dz < G s (l < , z < )e iν -1 ∑ p-1 k=1 U k (l,z) ,
where U k (l, z) depend only on l < , z < . Then, by the induction hypothesis,

|J s | ≤ C # (s)g(ν) p-1 k=1 ν min(R k ,d) ψ R k d (ν) = C # (s) p k=1 ν min(R k ,d) ψ R k d (ν).
q ij ⋅ l = α F ij (l i -l j ).
Lemma 7.2. Let A = (a ij ) be an irreducible n × n-matrix, n ≥ 2, and the vectors r ij ∈ R n be given by r ij = a ij (e i -e j ), where (e i ) 1≤i≤n is a basis in R n . Then rank R of the system of n-vectors (r ij ) 1≤i,j≤n equals n -1.

Proof of Lemma 7.2 is a simple exercise from linear algebra and we postpone it to Appendix 8.3. Set N k = |I k |. Since the N k × N k -matrices α k are skew-symmetric and non-zero, we have N k ≥ 2. Applying Lemma 7.2 to the system of vectors (q ij ) i,j∈I k , we see that its rank R k equals to N k -1. Then by Theorem 7.1 we get the estimate

(7.3) | Js (F)| ≤ C # (s) p i=1 ν min(N i -1,d) ψ N i -1 d (ν).
Let us show that (7.3) implies the desired estimate (6.1). Assume first that N i -1 < d for every 1 ≤ i ≤ p. Then, in view of (7.3) and the identity ∑

p i=1 N i = N , we have | Js (F)| ≤ C # (s)ν N -p .
Since N i ≥ 2 for any i, the number of blocks p satisfies p ≤ ⌊N /2⌋, where ⌊⋅⌋ denotes the (lower) entire part. Then Np ≥ ⌈N /2⌉, so we get ν N -p ≤ ν ⌈N /2⌉ which implies (6.1). Now we assume that (7.4) N j -1 ≥ d for some 1 ≤ j ≤ p, so ν min(N j -1,d) = ν d . Since for every i we have ν min(N i -1,d) ≤ ν, by (7.3) we find

| Js (F)| ≤ C # (s)ν d+p-1 p i=1 ψ N i -1 d (ν) ≤ C # 1 (s)ν d+p-2 .
In the case p ≥ 2 this implies (6.1). Now it remains to consider the case when (7.4) holds with p = 1. Then N -1 ≥ d. If N -1 > d then (7.3) with p = 1 implies (6.1). Otherwise N -1 = d, so

(7.5) | Js (F)| ≤ C # (s)ν N -1 ln ν -1 . If N -1 > ⌈N /2⌉ then ν N -1 ln ν -1 < Cν ⌈N /2⌉
and we are done. The opposite

situation is possible only if N ≤ 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 = ⌈N /2⌉ = d, which is 2 or 1 correspondingly, so again (7.5) implies (6.1).

s (τ 1 )ā (n) s (τ 2 ) = F∈F true m,n Js (F) + O C # (s)L -2 ν -2 ,
where we recall that the integrals Js (F) are given by (5.24) and F true m,n is a certain subset of the set of Feynman diagrams F (which are defined in Section 4), associated with the product a (m) s ā(n)

s . Each integral Js (F) satisfies the estimate (6.1), and this implies the estimate in Theorem 2.2. Since ν ln ν -1 ≤ C these estimates admit the following slightly weaker forms:

| Js (F)| ≤ C # (s)ν min(N /2,d) (8.2) |Ea (m) s (τ 1 )ā (n) s (τ 2 )| ≤ C # (s) ν -2 L -2 + ν min(N /2,d) .
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

(8.3) | Js (F)| ≤ C # (s)ν N /2 , |Ea (m) s (τ 1 )ā (n) s (τ 2 )| ≤ C # (s) ν -2 L -2 + ν N /2 ,
for any N ≥ 2. We discussed the motivation for this question in Section 1.4. Let F 2 m,n ⊂ F true m,n be a subset of Feynman diagrams F for which there is q = q(F) such that the matrix α F from (5.22) satisfies α F ij = 0 if both i ≠ q and j ≠ q (so, all elements of the matrix α F outside the q-th line and column are zero). It can be shown that F

ρ i v (i)
s 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 | ∑ F∈F 2 m,n Js (F)| ≤ C # (s)ν N -1 for N ≥ 2. Since ν N -1 ≤ ν N /2 ,
(F) ∼ ν d R N h F s (l) dl |λ 1 (l)λ 2 (l)| d/2 if N > 2d,
λ 2 = -1 2 ∑ 1≤i,j≤N (α F ij ) 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 F ∈ F 2 m,n have the same graph which we denote by G 2 .

To recover the set of diagrams 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 G 2 is such that in a k-th block with any k ≠ 1, m + 1, the virtual vertex can be placed in two different positions, c 2k or c2k . Now, using the stationary phase method [START_REF] Dimassi | Spectral Asymptotic in the Semi-Classical Limit[END_REF][START_REF] Hörmander | The Analysis of Linear Partial Differential Equations[END_REF], we develop each integral Js (F) in ν. 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 a)

c 2r-1 c2r-1 • • c2r c 2r • • C 1 C 2 b) c 2r-1 c2r-1 • • c2r c 2r • • C 12 Figure
G F = G F 0 we have | Js (F)| ≤ C # (s)ν r .
It is plausible that this is indeed the case and below we explain the reason. Let k > 2 be such that F 0 ∈ F k m,n . Then it is possible to show that F ∈ F k m,n as well for any diagram F satisfying G F = G 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 λ 1 , . . . , λ k , we get

Js (F) ∼ ν kd/2 R N h F s (l) dl |λ 1 (l)⋯λ k (l)| d/2
, if the latter integral converges. Perhaps, the factor ν 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 λ 1 (l)⋯λ 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 5.1. Let us consider a graph C ′ F which is obtained from U 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 C ′ 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 C ′ F in such a way that C ′ F will become a cycle. Since in the graph C ′ F every vertex is incident to two edges, C ′ F decouples to a union of k ≥ 1 disjoint cycles C i , 1 ≤ i ≤ 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 Br = {c 2r-1 , c 2r , c2r-1 , c2r } such that some but not all of its vertices belongs to cycle C 1 .

Indeed, assume that the claim is not true, so that each block either entirely belongs to the cycle C 1 , or its intersection with C 1 is empty. 8.3. Proof of Lemma 7.2. Since the matrix A is irreducible, there exists k and indices i 1 , . . . , i k satisfying i j ≠ i j+1 ∀j, such that a i 1 i 2 a i 2 i 3 ⋯a i k-1 i k ≠ 0 and the sequence i 1 , . . . , 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 , . . . , 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 , . . . , i k are considered.

Since the sequence i 1 , . . . , i k contains every 1 ≤ j ≤ 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 ≥ 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. 8.4. Integrals Js (F) in which the time variable l is integrated out. In this appendix we integrate out the time variable l in the integrals Js (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 ν → 0 behaviour of the integrals Js (F) only in case N = 2, and to find it we have to write the integrals in such integrated in l form. See in [START_REF] Dymov | Formal expansions in stochastic model for wave turbulence 1: kinetic limit[END_REF], where we found this asymptotic and, using a particular case of Theorem 5.9, deduced from it the asymptotical as ν → 0, L → ∞ behaviour of the moment E|a 2 . Note, however, that in case of general N , even to get an upper bound for the integrals Js (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 τ 1 = τ 2 =∶ t and T = ∞. We denote by S N the set of all permutations q of the set {1, . . . , N } and let

L N q ∶= {l ∈ R N ∶ -∞ ≤ l q(N ) ≤ . . . ≤ l q(1) ≤ t}, so that R N l = ⋃ q∈S N L N q ,
and the sets L N q with different q can intersect only over sets of zero Lebesgue measure. Then, in view of (4.7), (8.6) Js (F) = if q ∈ S N F and I q s (F; z) ≡ 0 otherwise. Moreover, ∑ 0≤r<k≤j≤N γ F q(r)q(j) ≥ 1 for any 2 ≤ k ≤ N , so the denominator in (8.8) is separated from zero.

The lemma can be proven by induction. Note that outside the intersection of quadrics

Q ∶= {z ∶ ω F j (z) = 0 for 1 ≤ j ≤ N }
we have I q s (F; z) ∼ ν N -1 , while on Q we have I q s (F; z) ∼ 1. So the integrand in (8.6) asymptotically degenerates on Q as ν → 0, and the integral in (8.6) is asymptotically singular. 8.5. Estimation of asymptotically singular integrals of quotients. In Lemma 8.4 we integrated out the time variable l in integrals Js (F), defined in (5.24), and represented Js (F) as sums of integrals of quotients with large quadratic forms in divisors, see (8.6), (8.8). In Theorem 6.2 we got an upper bound for a family of integrals of quotients that have more general form than the integrals Js (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 (8.6), (8.8).

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

(Q k z) ⋅ z = M i,j=1 q k ij z i ⋅ z j , 1 ≤ k ≤ K,
where z is the polyvector z = (z 1 , . . . , z M ), z j ∈ R d , and Q k = (q k ij ) 1≤i,j≤M are real symmetric matrices. As before, we denote by R the rank of the system of matrices Q 1 , . . . , Q K in the space of M × M -matrices, equal to the rank of the system of vectors

q ij ∈ R K , 1 ≤ i, j ≤ M, q ij = (q 1 ij , . . . , q K ij ).
Let (8.9)

J ν = R dM G(z) ∏ K k=1 iν -1 (Q k z) ⋅ z + Γ k (z) dz,
where 0 < ν ≤ 1/2, G is a Schwartz function, the functions Γ k are smooth, have at most polynomial growth at infinity together with their partial derivatives of all orders, and satisfy Γ k (z) ≥ C Γ for some constant C Γ > 0 and any k and z. To establish the proposition, we use representation

1 iν -1 (Q k z) ⋅ z + Γ k (z) = 0 -∞ e l k iν -1 (Q k z)⋅z+Γ k (z) dl k ,
for every k. Then the integral J ν takes the form (6.2) and the desired estimate follows from Theorem 6.2. See [START_REF] Dymov | Asymptotic estimates for singular Integrals of fractions whose denominators contain products of block quadratic forms[END_REF] for a generalization of this result.

In [START_REF] Kuksin | Asymptotical expansions for some integrals of quotients with degenerated divisors[END_REF] the asymptotic behaviour as ν → 0 of the integral J ν was found in the case when d ≥ 2 is a pair number, K = 2, Γ 1 = Γ 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 [START_REF] Dymov | Formal expansions in stochastic model for wave turbulence 1: kinetic limit[END_REF] when analysing the asymptotic behaviour of the term n (2) from the decomposition (1.21). In particular, it was shown that J ν ∼ ν which is in accordance with the estimate (8.10). The family of integrals (8.9) is significantly more complicated than that in [START_REF] Kuksin | Asymptotical expansions for some integrals of quotients with degenerated divisors[END_REF]. We get for them only the upper bound and for the moment do not now how to examine their asymptotic behaviour.

1. 1 .

 1 The setting. Let Td L = R d /(LZ d ) be a d-dimensional torus, d ≥ 1, of period L ≥ 1.We denote by ∥u∥ the normalized L 2 -norm of a complex function u on T and write the Fourier series of u as 1

  s e 2πis⋅x , v = F(u), γ s = γ 0 (|s| 2 ).

  s ′ (τ 2 ) = 0 for any s, s ′ ∈ Z d Land that Ev

Proposition 2 . 1 .

 21 For any m, n ≥ 0 there exists a function C # such that for each s ∈ Z d L and τ 1 , τ 2 ≥ -T , (2.3) |Ev (m)

  s ′ (t), p ≥ 1, is any vertex of the diagram D, then it generates a block of four vertices, three of which correspond to a choice of the three termsa (m 1 ) , a (m 2 ) , ā(m 3 ) inthe decomposition (2.13) of a (p) s ′ (t). These three are called real vertices, and the vertex a (p) s ′ is called the parent of the block. The fourth vertex of the block, denoted by ws ′′ , is called the conjugated virtual vertex. It corresponds to the factor δ s ′ s ′′ in (2.13), is coupled with the parent a (p)

  which are obtained by drawing side by side pairs D 1 and D2 with various D 1 ∈ D m and D2 ∈ D n . Here in the diagrams D2 the vertices (except the root) are enumerated by the indices ξ 2m+1 , . . . , ξ 2N , σ 2m+1 , . . . , σ 2N , N ∶= m+n. Since 7 As we will see, structure of the set D m depends on a (m)

  4. Then setting I s (D) = Ĩs (D 1 ) Ĩs ( D2 ), where D = D 1 ⊔ D2 , we find (2.18)

3. 1 .

 1 The set of diagrams D m ×D n . For integers m, n ≥ 0 we define the set D m × D n as a direct product of the sets of diagrams D m and D n , associated, correspondingly, with the variables a

3 . 1 . 1 .

 311 ) by erasing the isolated vertex c(0) 0 . Here each c (p) j is regarded as an argument, which should be substituted by a variable a (p) ξ j (l), while c(p) j should be substituted by an ā(p) σ j (l ′ ). E.g., this substitution, applied to the diagram in fig. 2(c), gives the diagram in fig. 1(b). The set D 2 consists of three diagrams in fig. 2. The set 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. Agreements and first notation. Each diagram D ∈ 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 cj . If we do not want to indicate whether a vertex is conjugated or not, we write it as ĉj . Each vertex is characterised by its degree which we often write as an upper index:

Figure 3 . 5 w6Figure 4 .

 354 Figure 3. a) The unique diagram of the set D 1 × D 0 . b,c) Some diagrams from the set D 3 . d) The unique diagram of the set D 1 .

  3(d). The set of diagrams D n equals to the set 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 D 1 ∈ D m and D2 ∈ D n we consider the diagram D 1 ⊔ D2 (see fig. 3(a) and 4), obtained by drawing the diagrams D 1 and D2 side by side and enumerate the vertices of D2 , except the root c0 , not from 1 to 2n but from 2m + 1 to 2m + 2n. Thus, in D 1 ⊔ D2 we denote the two roots as c 0 and c0 , then enumerate the remaining vertices of the diagram D 1 , and finally those of the diagram D2 . We define

3. 1 . 3 .

 13 Properties of the set D m × D n and some terminology. Throughout the paper we use the notation N ∶= m + n. The following properties of a diagram D ∈ D m × D n are obvious. • The number of blocks of D equals to N . • The diagram D has 4N + 2 vertices, half of them are conjugated while another half are non-conjugated. • If vertices ĉi and ĉj are coupled by an edge, we call them adjacent and write ĉi ∼ ĉj . Each pair of adjacent vertices in D consists of a conjugated and non-conjugated vertex, and has either the form {c (p) i , wj } or {c (p)

3. 1 . 4 .

 14 Values of vertices ĉj . Consider a diagram D ∈ D m × D n . To every vertex ĉi of a block Bk , 1 ≤ k ≤ N , we assign the same time variable l k ≥ -T . E.g., in the diagram from fig. 4(b) the times assigned to the vertices c 2 and c6 are l 1 and l 3 , respectively. To the roots c 0 and c0 we assign the times τ 1 and τ 2 . The vectors (3.4) l = (l 1 , . . . , l N ) and l = (τ 1 , τ 2 , l) are called the vectors of times. Let ξ = (ξ 0 , . . . , ξ 2N ), σ = (σ 0 , . . . , σ 2N ),

  (p) j of the diagram D as a position to put there a component a (p) ξ j (t) of the random vector a (p) (t), and a real vertex c(p) j as a position for a component ā(p) σ j (t ′ ) of the conjugate vector, with suitable t = t(l), t ′ = t ′ (l). More precisely, for p ≥ 0 we view the real vertices c (p) j , c(p) j as random functions of the variables l, ξ, σ, defined by (3.5) c (p)

3. 1 . 5 .

 15 Restrictions on the multi-indices (ξ, σ). To calculate the value of a diagram D ∈ D m × D n we will make substitutions (3.5), using multi-indices (ξ, σ) ∈ Z (4N +2)d L

6 .

 6 More notation. Let D ∈ D m × D n . • We denote by E(D) the set of all edges of a diagram D, E(D) = {(c i , cj )}.

L

  (D), L(D) ⊂ {0, . . . 2N } we denote the sets such that {c i } i∈L(D) and {c i } i∈ L(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∈L(D) = {a (0) ξ i } and {c i } i∈ L(D) = {ā (0) σ i }. E.g., for the diagram from fig. 4(b) we have L(D) = {1, 2, 5, 6} and L(D) = {1, 3, 4, 5}.

3. 2 .

 2 Formula for the product a(m) s ā(n) s . Let us denote (3.11) τ ∶= (τ 1 , τ 2 ),where we recall that τ 1 , τ 2 are the times for the roots c 0 , c0 . Recall also that the sets E(D), L(D), L(D), the functions γ ϕ = γ ϕ (ξ, σ) and the times l ϕ r,w and l c j ,c j are defined in Section 3.1.6; in difference with the functions γ ϕ c j , l cj are independent from (ξ, σ). To each diagram D ∈ D m × D n we associate the (random) function

  [START_REF] Erdös | Linear Boltzmann equation as the weak coupling limit of a random Schrödinger equation[END_REF] which is defined on the set of admissible multi-indices (ξ, σ) ∈ A L s (D) (see (3.9)). See below for a discussion of this formula. Here I {-T ≤l ϕ w ≤l ϕ r } denotes the indicator function of the set in which -T ≤ l ϕ w ≤ l ϕ r and c D is the constant (3.13)

  )), is just the product of values of all leaves of the diagram D. The term l ϕ r -l ϕ w is a difference of the times assigned to vertices, coupled by an edge ϕ, and the corresponding exponent in G D (τ, l, ξ, σ) comes from the iteration of the Duhamel formula (2.13) that generates ϕ. It appears together with the indicator function I {-T ≤l ϕ w ≤l ϕ r } , which is there due to the fact that in (2.13) the integration is performed not over R but only over the interval [-T, τ ].

  (a), which we denote by D 1,0 . Now the total number of vertices is 4N +2 = 6, and the corresponding set of admissible multi-indices A

s (τ 2 ) 4 .s (τ 2 )

 242 a product of values of all leaves in the diagram D 1,0 , we see that (3.18) takes the form a = I s (D 1,0 ), where the sum I s (D 1,0 ) is defined in (3.17). Thus, formula (3.16) holds for m = 1, n = 0. Feynman diagrams and formula for expectations Ea . Due to (3.16), it suffices to find EI s (D) for every diagram D ∈ D m × D n . Note that the randomness enters I s (D) only via the product of values of leaves a (0)ξ i , ā(0)σ j in (3.12), which are Gaussian random variables. Since their correlations are given by (2.15), due to the Wick theorem EI s (D) is given by a sum over all Wick pairings of values a (0)ξ i (l c i ) of non-conjugated leaves in D with values ā(0) σ j (l cj ) of conjugated leaves. We parametrize this sum by Feynman

Figure 5 .

 5 Figure 5. a) A Feynman diagram F obtained from the diagram D from fig. 4(a). b) Orientation of the Feynman diagram from item (a) and associated permutation f F .

4. 1 .

 1 Feynman diagrams. Let us fix m, n ≥ 0 satisfying N = m + n ≥ 1, and consider a diagram D ∈ D m × D n . Its set of leaves consists of N + 1 conjugated leaves c(0) i and N + 1 non-conjugated leaves c (0) j . Consider any partition of this set to N + 1 non-intersecting pairs of the form (c to the same block of D (if such partition exists). We denote by F a diagram obtained from 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 (0) ξ α , ā(0) σ β , assigned to the vertices in this block as in (3.5), satisfy Ea (0) ξ α ā(0) σ β = 0 for α, β ∈ {2j -1, 2j} and any admissible mutli-index (ξ, σ) ∈ A L s (D), since ξ α ≠ σ β in view of (1.10) and (3.7).

  construction, the set of vertices of any Feynman diagram F ∈ F m,n is partitioned into pairs of adjacent vertices (c i , cj ). Each pair has either the form (c (p) i , wj ) or (w i , c(p) j ), where p ≥ 1, or the form (c (0) i , c(0) j ). Moreover, adjacent vertices never belong to the same block. To any D ∈ D m × D n the construction above corresponds many diagrams F ∈ F m,n , but the inverse mapping (4.2)F m,n → D m × D n , F ↦ D F ,is well defined: the diagram D F is obtained from 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 F m,n of Feynman diagrams by a smaller set F true m,n ⊂ F m,n . For the Wick pairings corresponding to diagrams F ∉ F true m,n the assumption {s 1 , s 2 } ≠ {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 A ′ s (F). Let us orient edges of a Feynman diagram F ∈ F m,n in the direction from conjugated vertices to nonconjugated, as in fig. 5(b), and consider a permutation f F of the set {0, . . . , 2N } such that (4.3) f F (j) = i if cj ∼ c i , i.e. an edge of F goes from cj to c i , see fig. 5(b). For a vector ξ ∈ R (2N +1)d we denote (4.4)

  be a set of all multi-indices ξ ∈ R (2N +1)d satisfying the following relations: for σ = ξ f F ,

  where A s (D) is defined in Section 3.1.5 and the diagram D F -in (4.2). Indeed, (3.8) holds by the definition (4.3) of the permutation f F while (3.6) and (3.7) are obvious.

ϕ r and l ϕ w as in Section 3 . 1 . 6 .

 316 The main result of this section is Lemma 4.1, stated below. Recalling (3.11) and (3.12), (2.15), to each Feynman diagram F we associate the density function c F F F (τ, l, ξ), where c F = c D F (see (3.13)) and F F (τ, l, ξ)

Lemma 4 . 1 .

 41 For any integers m, n ≥ 0 satisfying N = m + n ≥ 1, for s ∈ Z d L and τ 1 , τ 2 ≥ -T , we have (

s

  Lemma 4.1 gives an explicit formula for the expectation Ea(m) s

3 w4Figure 6 .

 36 Figure 6. a) The graph U F obtained from the Feynman diagram F from fig. 5. b) A Hamilton cycle C F in the graph U F from item (a).

5. 1 .Lemma 5 . 1 .

 151 Cycles C F , ĈF and permutation π F . Let us take a Feynman diagram F ∈ F m,n with N = m + n ≥ 2. By construction, if two vertices of F belong to the same block then they are not adjacent. Now inside each block Bk , k ≥ 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 , c0 and denote the resulting graph by U F . For each Feynman diagram F ∈ F m,n the graph U F has a Hamilton cycle in which solid and dashed edges alternate.

Figure 7 . 5 . 2 .

 752 Figure 7. Cycle ĈF corresponding to the cycle C F in fig. 6(b), the associated permutation π F and the arcs A 1 ∶= [c 1 , c 2 ], A 2 ∶= [c 3 , c 4 ]. Here we have α F 12 = -α F 21 = 1.

  where by [c a , c b ], a ≠ b, we denote a set (an arc) of vertices of the cycle ĈF that are situated between the vertices c a and c b according to the orientation of ĈF , including c a and c b . E.g., in fig.7 [c4 , 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 A ′ s (F) ⊂ R d(2N -1) in the coordinates x. An important role below is played by the N × N incidence matrix α F = (α F ij ) 1≤i,j≤N with components 9

Proposition 5 . 3 .

 53 For any diagram F ∈ F m,n and any s ∈ R d , the set A ′ s (F) consists of all vectors ξ = ξ(x), where ξ(x) is given by (5.3) and x ∈ (R d ) 2N +1 satisfies the following relations, for every 1 ≤ k ≤ N : (a) x 0 = s, (b) x 2k = -x 2k-1 , Moreover, for vectors x satisfying (5.5)(b) we have

(5. 8 )

 8 {π(2k -1), π(2k)} = {f (2k -1), f (2k)}. Indeed, the permutation π encodes the change of indices of non-conjugated vertices of the cycle C 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 A ′ s (F) consists of all multi-indices ξ satisfying (4.5) and (4.6) with σ = ξ f . In view of what is told above, we have 9 Formula (5.4) admits the following interpretation. An element α F ij equals ±1 if the arcs [c 2i-1 , c 2i ] and [c 2j-1 , c 2j ] are linked, and equals 0 otherwise. The sign "+" or "-" is determined by the order of linking; see fig. 7. So the incidence matrix α F describes the linking of the arcs [c 2j-1 , c 2j ] on the reduced cycle ĈF , or equivalently on the Hamiltonian cycle C F in the diagram F.

  {ξ

  then the both sides of eq. (5.7) vanish. Indeed, αF ki = 0 ∀i since the arc [c 2k-1 , c 2k ] contains all vertices c j ∈ ĈF . Let us denote z j ∶= x 2j-1 ∈ R d and z = (z j ) 1≤j≤N ∈ R N dL . Due to Proposition 5.3, vector z forms coordinates on the set A ′ s (F) and, by (5.5)(b), x i = (-1) i+1 z ⌈i/2⌉ . Then, using (5.5)(a), we see that for ξ ∈ A ′ s (F) relation (5.3) takes the form(5.13)

F s in ( 5 . 19 )

 519 is an affine function of s ∈ R d and satisfies (5.20), the sum J s (F) is a Schwartz function of s ∈ R d . Thus, the correlations Ea (m) s (τ 1 )ā (n) s (τ 2 ) extends to a Schwartz function of s ∈ 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 ∈ R d .

5 . 4 .

 54 22) follows from the skew symmetry of the matrix α F . Continuous approximation. From now on we always assume that θ(y, t) = e ity . Using that by Proposition 5.6 Z(F) with F ∈ F true m,n is an open dense subset of R dN , we approximate the sum L -N d ∑ z∈Z(F)∩Z N d L from (5.19) by an integral over R dN by applying Theorem 3.1 from [9]. For d ≥ 2 we get

(7. 1 )

 1 {1, . . . N } = I 1 ∪ . . . ∪ I p , and denote by l k the vector l k = (l i ) i∈I k .

7. 2 .

 2 Proof of Theorem 6.1. Recall that the phase function Ω F from the definition (5.24) of the integral Js (F) is given by (5.22) and that for F ∈ F true m,n the matrix α F is SS-regular (see Denition 5.2). Consider a partition (7.1) with p ≥ 1 such that the matrices α k = (α F ij ) i,j∈I k are irreducible and α F ij = 0 once i ∈ I k and j ∈ I r with k ≠ r. Then the integral Js (F) satisfies assumptions of Theorem 7.1 with

1 m 2 m

 12 k m,n a set of Feynman diagrams F for which the eigenvalues λ 1 (l), . . . , λ k (l) are not identically zero while λ i (l) ≡ 0 for any i ≥ k + 1. Since the matrix α F l is non-zero but has zero trace, we have F .Let us take a diagram F ∈ F 2 m,n and consider the integral over dz from the definition (5.24) of Js (F). Applying to it the Parseval identity in the eigendirections corresponding to the eigenvalues λ 1 and λ 2 , and using that for F ∈ F ,n we have λ 3 = . . . = λ N = 0, we find that Js

FF′F

  Take a block Bp belonging to C 1 . Then any vertex ĉj ∉ Bp which is adjacent with a vertex from the block Bp (by construction of C ′ such vertex exists), belongs to the cycle C 1 and, consequently, all other vertices from the block to which belongs ĉj belong to C 1 as well. Arguing in this way, by induction we obtain that the cycle C 1 contains all blocks of the diagram C ′ F , as well as the vertices c 0 and c0 . Thus, C 1 = C ′ F which contradicts to the assumption k > 1, so the claim is true. Let Br be a block as in the claim. By construction of the diagram C ′ F , its dashed edges either provide the couplings c 2r-1 ∼ c2r-1 and c 2r ∼ c2r or the couplings c 2r-1 ∼ c2r and c 2r ∼ c2r-1 . For definiteness, we assume that c 2r-1 ∼ c2r-1 ∈ C 1 and c 2r ∼ c2r ∈ C 2 , as in fig. 8(a). Let us erase the (dashed) edges that couple the vertices c 2r-1 with c2r-1 and c 2r with c2r , and instead couple the vertices c 2r-1 with c2r and c 2r with c2r-1 . Then from the diagram C ′ we obtain another diagram C ′′ F , constructed from U F in the same way as the diagram C ′ F , in which there is one cycle C 12 instead of the two disjoint cycles C 1 and C 2 in C ′ F , see fig. 8(b). So the total number of cycles in the diagram C ′′ F is k -1, where we recall that by our assumption in C there are k > 1 cycles. Then, arguing by induction, we arrive at a diagram C F which is a cycle itself.

Lemma 8 . 4 .

 84 l cψ | dl,(8.7) where bψ (z) = b ψ (ξ(z)), γ ψ (z) = γ ψ (ξ(z)) and γ ϕ (z) = γ ϕ (ξ(z)). It remains to compute the integrals I q s , and for this purpose we introduce the following notation. For 0≤ i < j ≤ N we set γ F ij ∶= ϑ∶ Bi ∼ Bj γ ϑ ≥ 0,where the sum is taken over all edges ϑ ∈ E(F) of the diagram F which couple a vertex from the block Bi with a vertex from the block Bj . Here by B0 we denote the set of roots {c 0 , c0 }.Let S N F ⊂ SN be the set of permutations q for which the relationl ∈ L N q implies l ϕ w ≤ l ϕ rfor all ϕ ∈ E D (F), cf. the indicator function in (8.7). Below we denote q(0) ∶= 0. Assume τ 1 = τ 2 = t and T = ∞. Then for any m, n ≥ 0 satisfying N ∶= m + n ≥ 2 and F ∈ F m,n the integral Js (F) is given by (8

Proposition 8 . 5 .

 85 Assume that tr Q k = 0 for every 1 ≤ k ≤ K. Then (8.10) |J ν | ≤ Cν min(R,d) ψ R d (ν), where the function ψ R d is defined in (6.5).

  and call the vector s ∈ Z

	d L its index. Since a (k) s (τ ) differs from y , then |Ea (m) s ā(n) s | = |Ey (m) s ȳ(n) s |. Thus, in view of (2.3), it (k) s (τ ) by the suffices to establish Theorem 1.1 with the processes v factor e -iν -1 τ |s| 2 (m) s replaced by a (m) s :
	Theorem 2.2. Assume that d ≥ 2. Then for any integers m, n ≥ 0 there
	exists a function C	# such that for s ∈ Z d L and any τ 1 , τ 2 ≥ -T ,
	(2.8)	|Ea	(m) s (τ 1 )ā (n) s (τ 2 )| ≤ C	# (s) ν	-2 L -2	+ ν	min(⌈N /2⌉,d) χ N d (ν) ,
	where N = m + n and χ above holds if we modify the r.h.s. of (2.8) by adding the term C N d is defined in (1.15). If d = 1, the assertion # (s)L -1 .
	Moreover, the correlation Ea of s ∈ R d ⊃ Z d L which satisfies the same estimate (2.8). (m) s (τ 1 )ā (n) s (τ 2 ) extends to a Schwartz function
	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	(i) s can be computed inductively. Namely, a

  a fictitious index s 4 and write the function Y

					s from
	(2.6) as
		Y s (a, t) = L -d	δ	′12 34 δ	s s 4 a s 1 a s 2 ās 3 θ(ω 34 , t), 12
			s 1 ,...,s 4 ∈Z d L
	where δ s 4 is the Kronecker symbol and we denote s
	(2.11)	θ(x, t) = e it x .
	Then (2.10) takes the form
	(2.12)
	a	(1)

  1. Let us write the variable a

	(m) s (τ 1 ) in the form (2.13) and then apply the Duhamel formula (2.13) to the variables a (m i ) s i (l) in its r.h.s. with the degrees m i ≥ 1 (see (2.7)). Iterating the procedure, we represent a (m) s (τ 1 )
	by a sum of terms of the form
	(2.16)	Ĩs = ⋯ L -md	(. . . ) dl 1 . . . dl m ;
		s 1 ,...,s 4m
	here and below we call terms of this form sums. The integrating zone in (2.16) is a convex polyhedron in [-T, τ 1 ] m and the summation is taken over
	the vectors s 1 , . . . , s 4m ∈ Z d L which are subject to certain linear relations, following from the factors δ ′12 34 and δ s s 4 in (2.13). The summand in brack-
	ets is a product of functions e

  τ 1 , τ 2 ≥ -T , where δ = δ N > 0.

	Below we do not use the explicit form (4.7) of the function F	F but only
	the estimate (4.11).
	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	F . An
	accurate proof of Lemma 4.1 is straightforward but rather tedious in view
	of the complexity of notation, so we give it only in the arXiv version of this
	paper [10].
	Let us note that
	(4.12)	if m + n = 1, then Ea	(m) s

  Since the function h(|t|) is integrable, the r.h.s. above is bounded by Cν d , if we take the integral only over the set where |t < | ≥ 1. Then, it remains to get the desired estimate for the integral over the set where Cν ≤ |t < | ≤ 1. Since the function h is decreasing, we have h(C 1 |t|) ≤ h(C 1 |t

							> |), so this
	integral is bounded by
							we find
	N ν,t ⊂ {t ∶ |t	< | ≥ C	′ ν}. Then, using that h(|l|) ≤ h(C 1 |t|), by (6.11) we
	get
			I	ν ≤ 2	d/2 ν	d	t∶ |t < |≥Cν	h(C 1 |t|) |t < | d dt.

  Prove that for any d ≥ 2 and m, n satisfying N = m + n ≥ 2, 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 α

	Problem 8.3. (8.4)	Js (F) ≤ C	# (s)ν	N /2 .
				F∈F true m,n
		If (8.4) is true, we get the second estimate from (8.3). In particular, under
	the "kinetic" scaling ρ = εν	-1/2 the energy spectrum corresponding to a
	truncation ∑ by a solution of the WKE (1.23) at least with the accuracy ε m i v (i) s of series (1.12) with any m ≥ 2 can be approximated i=0 ρ 2 , if L -2 ≤ -2 ν ν m .
				F l = α
				this estimate does not prevent the second estimate
	from (8.3) to be true. We expect that for N ≫ 2d the integrals Js (F) with
	F ∈ F	2 m,n are the biggest (see below) but still there are many other diagrams
	F ∉ F	2 m,n for which the corresponding integrals also are large (bigger than
	ν	N /2 ). Nevertheless, we expect that similar cancellations take place for
	them as well, but do not know if they go till the order ν	N /2 . So, we state
	the following

F ij (l i -l j ) 1≤i,j≤N , which depends on l ∈ R N (see

(5.22)

).

Since it is symmetric, it has N real eigenvalues, which we enumerate in the decreasing order

|λ 1 (l)| ≥ |λ 2 (l)| ≥ . . . ≥ |λ N (l)| ≥ 0, for any l ∈ R N . It

is possible to show that the set F 2 m,n can be equivalently defined as follows. For k ≥ 1 we denote by F

  for an appropriate integrable function h Due to Lemma 6.3, we have λ 1

	F s , which is fast decaying
	in s.

  [START_REF] Dymov | Asymptotic estimates for singular Integrals of fractions whose denominators contain products of block quadratic forms[END_REF]. Construction of the cycle C 12 from the cycles C 1 and C 2 . In this figure we do not make difference between solid and dashed edges.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 F 0 ∉ F

	2 m,n and consider
	its graph G F 0 . We expect that in the sum
	(8.5)

F∈F true m,n ∶ G F =G F 0 Js (F)

there are similar cancellations. However, the graph G 2 is "very symmetric"

(we are able to draw it explicitly), while for a general diagram F 0 there are less symmetries in the graph G 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 ν than in Proposition 8.2. Then the estimate (8.4) can be true only if the integrals Js (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 F satisfying

The scaling factor L -d/2 is convenient for our calculations.

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