# Sharp Convergence of Nonlinear Functionals of a Class of Gaussian Random Fields

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## Abstract

We present a self-contained proof of a uniform bound on multi-point correlations of trigonometric functions of a class of Gaussian random fields. It corresponds to a special case of the general situation considered in Hairer and Xu (large-scale limit of interface fluctuation models. ArXiv e-prints arXiv:1802.08192, 2018), but with improved estimates. As a consequence, we establish convergence of a class of Gaussian fields composite with more general functions. These bounds and convergences are useful ingredients to establish weak universalities of several singular stochastic PDEs.

## Keywords

Multi-point correlation function Trigonometric polynomial Gaussian random fields## Mathematics Subject Classification

60H## 1 Introduction

### 1.1 Motivation from Weak Universalities

The study of singular stochastic PDEs has received much attention recently, and powerful theories are being developed to enhance the general understanding of this area. We refer to the excellent surveys [8, 11] and references therein for recent breakthroughs in the field.

*F*and smooth stationary Gaussian random field \(\tilde{\xi }\). The main result in [12] is that there exists \(C_{\varepsilon } \rightarrow +\infty \) such that the rescaled and re-centered height function

*P*is the heat kernel. Hairer and Xu [13] extended the result to arbitrary even functions

*F*with sufficient regularity and polynomial growth. Similar results have also been obtained in [5] for models at stationarity.

*p*th obtain moment bounds of these stochastic objects for arbitrarily large

*p*th.

When *F* is even polynomial, this heuristic indeed gives a direct proof of the convergence of the term in (1.3). However, when *F* is not polynomial, the actual proof of the convergence becomes much subtler. The main obstacle is that \(F(\sqrt{\varepsilon } \varPsi _\varepsilon )\) expands into an infinite chaos series. If we brutally control their high moments termwise as in the polynomial case, then in order for these termwise moment bounds to be summable, we need to impose very strong conditions on *F* (namely, its Fourier transform being compactly supported), which is clearly too restrictive.

Instead, in [13], the authors expanded \(F(\sqrt{\varepsilon } \varPsi _\varepsilon )\) in terms of Fourier transform, developed a procedure in obtaining pointwise correlation bounds on trigonometric functions of Gaussians, and deduced the desired convergence from those bounds.

Similar universality results are also present in the dynamical \(\varPhi ^4_3\) model. The weak universality of \(\varPhi ^4_3\) equation for a large class of symmetric phase coexistence models with polynomial potential was established in [13] for Gaussian noise and then extended in [17] to non-Gaussian noise. The extension beyond polynomial potential (even with Gaussian noise) has the same difficulties as in the KPZ case discussed above. In the recent work [2], the authors developed different methods based on Malliavin calculus to control similar objects. The methods developed in [2, 13] to treat general nonlinearities are both robust enough to cover both KPZ and \(\varPhi ^4_3\) equations as well as other similar situations.

In this article, we follow the ideas developed in [13] and prove a uniform bound in a special case considered in there. This special case is technically simpler to explain, but is also illustrative enough to reveal the main idea of the proof for the more general case. Furthermore, we obtain a better bound in this special case, thus yielding convergence results for functions *F* with lower regularity.

### 1.2 Main Statements

*x*| instead of \(|x|_{\mathfrak {s}}\). For any Gaussian random field

*X*, any function \(F: \mathbf {R}\rightarrow \mathbf {R}\) with at most exponential growth, and any integer \(m \ge 0\), we write

*n*-th Wick power of

*X*, and \(C_{n} = \frac{1}{n!} \mathbf {E}F^{(n)}(X)\) is the coefficient of \(X^{\diamond n}\) in the chaos expansion of

*F*(

*X*). In other words, \(\mathcal {H}_{m}(F(X))\) is

*F*(

*X*) with the first \(m-1\) chaos removed. We refer to [16, Chapter 1] for more details on chaos expansion of random variables. We have the following bound.

### Theorem 1.1

Theorem 1.1 is the main technical ingredient to establish that if \(\{\varPsi _\varepsilon \}\) approximates a certain Gaussian random field \(\varPsi \), then a large class of nonlinear functions of \(\varPsi _\varepsilon \), after proper rescaling and re-centering, converges to certain Wick powers of \(\varPsi \). We first give the assumption on the random field \(\varPsi \).

### Assumption 1.2

^{1}

*G*satisfies the bounds

*g*such that

### Assumption 1.3

*F*satisfies

### Theorem 1.4

*F*satisfy the above assumptions, and

*g*be the limiting \(L^1\) function of \(\varepsilon ^\alpha G(\varepsilon \cdot )\) as in Assumption 1.2. Let \(\rho \) be a mollifier on \(\mathbf {R}^d\) and \(\varPsi _{\varepsilon } = \varPsi * \rho _{\varepsilon }\). For every integer

*m*, define

*m*-th Wick power of \(\varPsi \).

We will first prove the main bound (1.5) in Theorem 1.1 and then establish the convergence in Theorem1.4 by Fourier expanding *F* and applying (1.5) to \(\varPhi _\varepsilon = \varepsilon ^{\frac{\alpha }{2}} \varPsi _\varepsilon \). Note that although the bound in Theorem 1.1 holds for every integer *m*, the convergence in Theorem 1.4 requires \(m<\frac{|\mathfrak {s}|}{\alpha }\). This can be easily seen from the fact that if \(m \ge \frac{|\mathfrak {s}|}{\alpha }\), then \(\varPsi ^{\diamond m}\) would have divergent covariance and hence not well defined.

### Example 1.5

*K*of \(\mathcal {L}^{-\frac{\beta }{2}}\) is homogeneous (in the scaling \(\mathfrak {s}\)) of order \(-|\mathfrak {s}|+\beta \) in the sense that

*g*,

*G*as in Assumption 1.2, we have the expression

In the case of standard Euclidean scaling \(\mathfrak {s}= (1, \dots , 1)\), \(\varPsi \) is simply the standard fractional Gaussian field \((-\varDelta )^{-\frac{\beta }{2}} \xi \). We refer to the survey [15] for more details on fractional Gaussian fields.

### Example 1.6

*F*, all \(\mathcal {C}^{1+}\) functions with polynomial growth fall in Assumption 1.3. More precisely, if \(f \in \mathcal {C}^{1,\beta }(\mathbf {R})\) for some \(\beta >0\), and there exist \(C, M>0\) such that

*F*satisfies Assumption 1.3.

### Example 1.7

*F*even not \(\mathcal {C}^1\), but we still have

*P*is the heat kernel, then we have

### 1.3 Remarks and Possible Generalizations

Theorem 1.1 is a special case of [13, Theorem 6.4] in that it allows only one frequency variable \(\theta \) rather than multiple ones. On the other hand, it is also more general since it allows subtraction of Wiener chaos up to any order. Furthermore, the bound (1.5) is completely independent of \(\theta \), while the corresponding one in [13] is polynomial in \(\theta \). As a consequence of this improvement, the condition on *F* for the convergence in Theorem 1.4 to hold is weaker.

The main technical difference that results in this improvement, as we shall see later in Sect. 2, is that in the clustering procedure, we are able to take the clustering distance *L* being independent of \(\theta \) rather than being quadratic in \(\theta \) as in [13].

We shall note that the convergence results in this article are not sufficient to establish weak universality in general situations. These would require convergence of the products of the objects considered in Theorem 1.4, with possible subtraction of extra chaos components after taking product. The convergence of these products requires a more general bound than Theorem 1.1 and [13, Theorem 6.2]. We leave them to future work.

## 2 Proof of Theorem 1.1

This section is devoted to the proof of Theorem 1.1. Assumptions 1.2 and 1.3 on \(\varPsi \) and *F* are irrelevant here. We fix \(\alpha \in (0,|\mathfrak {s}|)\), and let \(\{\varPhi _{\varepsilon }\}_{\varepsilon \in (0,1)}\) be a family of Gaussian random fields with correlation functions satisfying (1.4). The following preliminary bounds on the correlation function will be used throughout the section.

### Proposition 2.1

### Proof

In what follows, we keep our notations same as in [13, Section 6]. For every finite set \(\mathcal {A}\), let \(\mathbf {N}^{\mathcal {A}}\) be the set of multi-indices on \(\mathcal {A}\). For \(\mathcal {A}\)-tuple of Gaussian random variables \(\mathbf {X}= (X_{a})_{a \in \mathcal {A}}\) and \(\mathbf {n}\in \mathbf {N}^{\mathcal {A}}\), we write Open image in new window . Similarly, we write \(\mathbf {n}! = \prod _{a \in \mathcal {A}} n_{a}!\) and \(|\mathbf {n}| = \sum _{a \in \mathcal {A}} n_a\). In general, we use standard letters for scalars and boldface ones to denote tuples.

Fix \(K \ge 1\) and \(m, r \in \mathbf {N}\). Let \([K] = \{1, \dots , K\}\). We also fix \(\theta \in \mathbf {R}\) and \(\mathbf {x}= (x_k)_{k \in [K]} \in \mathbf {R}^K\) arbitrary. Write \(\langle \theta \rangle = 1 + |\theta |\). All the constants *C* below depend on \(\varLambda \), *K*, *m*, and *r* only unless otherwise mentioned. We seek bounds that are uniform in \(\varepsilon \), \(\theta \), and \(\mathbf {x}\). We also write \(X_j = \varPhi _{\varepsilon }(x_j)\) for simplicity since the bounds will be independent of \(\varepsilon \).

### 2.1 Clustering and the First Bound

*K*,

*m*, and

*r*only, will be specified later. Let \(\sim \) be an equivalence relation on [

*K*] such that \(j \sim j'\) if there exists \(k \in \mathbf {N}\) and \(j_0, \dots , j_k \in [K]\) with \(j_0 = j\) and \(j_k = j'\) such that

*K*] into clusters obtained in this way. In other words,

*j*and \(j'\) belong to the same cluster if and only if starting from \(x_{j}\), one can reach \(x_{j'}\) by performing jumps with sizes at most \(L \varepsilon \) onto connecting points in \(\mathbf {x}\).

*C*independent of \(\varepsilon \), \(\theta \), and \(\mathbf {x}\). It then remains to show that the right-hand side of (1.5) is bounded by a constant from below. For this, we use the assumption that \(\mathscr {C}\) contains no singletons.

*L*(to be chosen later) is also independent of \(\varepsilon \), \(\theta \), and \(\mathbf {x}\), this concludes the case when \(\mathscr {C}\) contains no singleton. The rest of the section is devoted to establishing (1.5) when at least one cluster in \(\mathscr {C}\) is singleton.

### 2.2 Expansion

*N*since this simplifies the notations later. The following lemma gives control on the coefficients \(C_{\mathbf {n}_{\mathfrak {u}}}\).

### Lemma 2.2

*K*,

*m*,

*r*, and \(\varLambda \) only such that

### Proof

*r*derivatives of each \(\beta _j\) into the two terms in the parenthesis, the differentiation of the first term (\(\varvec{\beta }_{\mathfrak {u}}^{\mathbf {n}_{\mathfrak {u}}}\)) yields an additional factor which is at most \(|\mathbf {n}_{\mathfrak {u}}|^{Kr} \le C^{|\mathbf {n}_{\mathfrak {u}}|}\) for some fixed constant

*C*, while the second term is uniformly bounded both in \(\theta \) and \(\mathbf {n}_{\mathfrak {u}}\) since \(X_j\)’s all have bounded variance. This gives (2.9). The bound (2.10) follows from (2.9) and the multinomial theorem.

### 2.3 Representative Point

For \(|{\mathfrak {u}}| \ge 2\), the corresponding term in the expectation on the right-hand side of (2.7) is a Wick product of multiple Gaussian random variables. We aim to reduce it to the Wick product of a single variable by choosing a representative point from each cluster.

For every \({\mathfrak {u}}\in \mathscr {C}\), choose \(u^{*}({\mathfrak {u}}) \in {\mathfrak {u}}\) arbitrary. The choice for \(u^*\) is unique if \({\mathfrak {u}}\) is singleton. We have the following proposition.

### Proposition 2.3

### Proof

If \(|\mathbf {n}| = \sum _{{\mathfrak {u}}\in \mathscr {C}} |\mathbf {n}_{\mathfrak {u}}|\) is odd, then both sides of (2.12) are 0, so we only need to consider the situation when \(|\mathbf {n}|\) is even.

In this case, the left-hand side is the sum of products of pairwise expectations \(\mathbf {E}(X_j X_{j'})\) for *j* and \(j'\) belonging to different clusters. The right-hand side (without the factor \(C^{|\mathbf {n}|}\)) is the same except that each instance of \(X_j\) for \(j \in {\mathfrak {u}}\) is replaced by \(X_{u^*({\mathfrak {u}})}\). It then suffices to control the effects of such replacements.

From now on, we restrict ourselves to the situation when \(|\mathcal {S}| \ge 1\), and recall the notation \(\mathcal {U}= \mathscr {C}{\setminus } \mathcal {S}\). We need to split the product on the right-hand side of (2.7) into sub-products in \(\mathcal {S}\) and in \(\mathcal {U}\). For this, we introduce multi-indices \(\mathbf {k}=(k_s)_{s \in \mathcal {S}}\) and \(\varvec{\ell }= (\ell _{\mathfrak {u}})_{{\mathfrak {u}}\in \mathcal {U}}\) and write \(|\mathbf {k}| = \sum _{s \in \mathcal {S}} k_s\) and \(|\varvec{\ell }| = \sum _{{\mathfrak {u}}\in \mathcal {U}} \ell _{\mathfrak {u}}\). We then have the following proposition.

### Proposition 2.4

### Proof

### 2.4 Graphic Representation

It remains to control the term involving the expectation on the right-hand side of (2.13). Since all \(X_j\)’s are Gaussian, it can be written as a sum over products of pairwise expectations. The number of terms in each product (and hence the total power) can be arbitrarily large since *N* will be summed over all integers. Following [13], we introduce graphic notations to describe these objects.

Given a set \(\mathbb {V}\), we write \(\mathbb {V}_2\) for the set of all subsets of \(\mathbb {V}\) with exactly two elements. A (generalized) graph is a triple \(\varGamma = (\mathbb {V}, \mathbb {E}, R)\). Here, \(\mathbb {V}\) is the set of vertices, and \(\mathbb {E}: \mathbb {V}_2 \rightarrow \mathbf {N}\) is the set of edges with multiplicities. More precisely, each edge \(\{x,y\} \in \mathbb {V}_2\) has multiplicity \(\mathbb {E}(x,y) = \mathbb {E}(y,x)\). We do not allow self-loops, so \(\mathbb {E}(x,x)=0\) for all \(x \in \mathbb {V}\). Finally, \(R: \mathbb {V}_2 \rightarrow \mathbf {R}\) is a function that assigns a value to each pair of vertices.

### Definition 2.5

For each \(\mathbf {k}\in \mathbf {N}^{\mathcal {S}}\) and \(\varvec{\ell }\in \mathbf {N}^{\mathcal {U}}\), the set \(\varOmega _{\mathbf {k},\varvec{\ell }}\) consists of graphs with \(\mathbb {V}\) and *R* specified above, and such that \(\deg (x_s)=m+k_s\) for all \(s \in \mathcal {S}\), \(\deg (x_{u^*({\mathfrak {u}})})=\ell _{\mathfrak {u}}\) for all \({\mathfrak {u}}\in \mathcal {U}\), and \(\deg (x)=0\) for all other \(x \in \mathbb {V}\).

- 1.
\(\varGamma \in \varOmega _{\mathbf {k},\varvec{\ell }}\) for some \(\mathbf {k}\in \mathbf {N}^{\mathcal {S}}\) and \(\varvec{\ell }\in \mathbf {N}^{\mathcal {U}}\) with the restriction that \(k_s \in \{0,1\}\) for every \(s \in \mathcal {S}\) and \(\ell _{\mathfrak {u}}\le m+1\) for each \({\mathfrak {u}}\in \mathcal {U}\).

- 2.
If \(\ell _{\mathfrak {u}}\ge 1\) for some \({\mathfrak {u}}\in \mathcal {U}\), then there exists \(s \in \mathcal {S}\) such that \(\mathbb {E}(x_s, x_{u^*({\mathfrak {u}})}) = \ell _{\mathfrak {u}}\).

### Remark 2.6

The first requirement for \(\varOmega ^*\) above is equivalent to that \(\deg (x_s) \in \{m,m+1\}\) for all \(s \in \mathcal {S}\), \(\deg (x_{u^*({\mathfrak {u}})}) \le m+1\) for all \({\mathfrak {u}}\in \mathcal {U}\), and is 0 for all other points. The second requirement says that if \(x_{u^*({\mathfrak {u}})}\) has a nonzero degree, then all its edges must be connected to a single point \(x_s\) for some \(s \in \mathcal {S}\). We will see later that the definition of \(\varOmega ^*\) corresponds to “minimal graphs” after the reduction procedure in the next subsection.

### Remark 2.7

The clustering depends on the choice of *L*, and hence so do the definitions of \(\varOmega _{\mathbf {k},\varvec{\ell }}\) and \(\varOmega ^*\). On the other hand, these are just intermediate steps and our final bound (1.5) does not involve clustering at all. Furthermore, the choice of *L* later [in (2.18)] is also independent of the location of \(\mathbf {x}\). Hence, we omit the dependence of the clustering on *L* here for notational simplicity.

### 2.5 Reduction

We now start to control the right-hand side of (2.13). If \(m|\mathcal {S}| + |\mathbf {k}| + |\varvec{\ell }|\) is odd, then the term with the expectation is 0. So we only need to deal with the case when \(m|\mathcal {S}| + |\mathbf {k}| + |\varvec{\ell }|\) is even.

We start with the reduction step. This is where we need to choose the clustering distance *L* sufficiently large, which will ensure the uniform in \(\theta \) bound after summing over \(\mathbf {k}\) and \(\varvec{\ell }\). We first give the following proposition, which reduces graphs in \(\varOmega _{\mathbf {k},\varvec{\ell }}\) to those in \(\varOmega ^*\).

### Proposition 2.8

*C*does not depend on the choice of

*L*, though the clusters and the definitions of \(\varOmega _{\mathbf {k},\varvec{\ell }}\) and \(\varOmega ^*\) do.

### Proof

To see the existence of such a \(\bar{\varGamma }\) when \(\varGamma \in \varOmega _{\mathbf {k},\varvec{\ell }} {\setminus } \varOmega ^*\), we consider the two situations where either one of the two conditions for \(\varOmega ^*\) in Definition 2.5 is violated. We first consider the violation of Condition 1. Since \(\varGamma \in \varOmega _{\mathbf {k},\varvec{\ell }}\), failure of Condition 1 means there exists \(j \in \mathcal {S}\cup \{u^*({\mathfrak {u}}): {\mathfrak {u}}\in \mathcal {U}\}\) such that \(\deg (x_j) \ge m+2\). We fix this *j*, and there are two possibilities in this situation.

*Case 1.*There exist \(i \ne i'\) such that \(\mathbb {E}(x_j, x_i) \ge 1\) and \(\mathbb {E}(x_j, x_{i'}) \ge 1\). In this case, we let \(\bar{\varGamma }\) be the graph obtained from \(\varGamma \) by performing the following operations:

*x*and simply write the indices to denote vertices. Since all the other parts of the graph remain unchanged, this operation gives a desired \(\bar{\varGamma }\) with (2.15).

*Case 2. *

- (a)
either \(x_{u^*({\mathfrak {u}})}\) is connected to two other different points \(x_i\) and \(x_{i'}\);

- (b)
or \(x_{u^*({\mathfrak {u}})}\) is connected to \(x_{u^*({\mathfrak {u}}')}\) for some \({\mathfrak {u}}' \in \mathcal {U}\).

Since the above cases have covered all the possibilities for \(\varGamma \in \varOmega _{\mathbf {k},\varvec{\ell }} {\setminus } \varOmega ^*\), we have completed the proof of the proposition. \(\square \)

The following proposition is then a simple consequence.

### Proposition 2.9

*K*,

*m*, and

*r*only such that for every \(\theta \in \mathbf {R}\), every location \(\mathbf {x}\in (\mathbf {R}^d)^K\), and every \(\varepsilon \in (0,1)\), we have the bound

### Remark 2.10

The bound is completely independent of \(\theta \) and \(\varepsilon \), and its dependence on the location of \(\mathbf {x}\) is via \(\varOmega ^*\) only. Also note that the clustering, and hence \(\varOmega ^*\), depends on the choice of *L*.

### Proof of Proposition 2.9

*L*sufficiently large depending on \(C_0\) and \(\varLambda \) only such that

*K*,

*m*, and

*r*only, so does

*L*. Finally, \(L^{\frac{\alpha }{2} \cdot \deg (\varGamma ^*)}\) is also uniformly bounded since graphs in \(\varOmega ^*\) have degrees at most \((m+1)K\). This completes the proof. \(\square \)

### Remark 2.11

The reason why we need to choose *L* large is to ensure the exponential and hence the whole right-hand side of (2.17) being uniformly bounded in \(\theta \). As we see now, the Gaussian factor \(\mathrm{e}^{-\frac{\theta ^2}{2 \varLambda }}\) in (2.11) allows us to choose such *L* being independent of \(\theta \). Together with the enhancement procedure in Sect. 2.6, this ensures the bounds in Proposition 2.9 and hence in Theorem 1.1 are completely independent of \(\theta \).

Without the Gaussian factor, one would need to take *L* quadratic in \(\theta \) to make the exponential in (2.17) bounded, and the enhancement procedure in below would produce a bound that is polynomial in \(\theta \) with its degree depending on *m* and *K*.

### 2.6 Enhancement and Conclusion of the Proof

From now on, we fix the choice of *L* in (2.18). We need to control the right-hand side of (2.16) by that of (1.5). To achieve this, we enhance every \(\varGamma \in \varOmega ^*\) to a graph \(\mathrm{Enh}(\varGamma )\) where \(\deg (x_j) \in \{m,m+1\}\) for every \(j \in [K]\), which matches the pairing occurring in the desired upper bound. The enhancement procedure will also be performed in such a way that \(|\mathrm{Enh}(\varGamma )|\) is an upper bound for \(|\varGamma |\) up to some proportionality constant, which is uniform in \(\varepsilon \), \(\theta \), and \(\mathbf {x}\) subject to \(|\mathcal {S}| \ge 1\). This will lead to bound (1.5). The procedure is similar to the one used to obtain (2.6) when \(|\mathcal {S}|=0\).

Fix \(\varGamma \in \varOmega ^*\) arbitrary, so in particular, \(\varGamma \in \varOmega _{\mathbf {k},\varvec{\ell }}\) for some \(\mathbf {k}\in \mathbf {N}^\mathcal {S}\) and \(\varvec{\ell }\in \mathbf {N}^\mathcal {U}\). By the definition of \(\varOmega ^*\), \(\deg (x_s) \in \{m,m+1\}\) for all \(s \in \mathcal {S}\). For every \({\mathfrak {u}}\in \mathcal {U}\), we have \(\deg (x_{u^*})=\ell _{\mathfrak {u}}\le m+1\), and all of them are connected to one single \(x_s\) for some \(s \in \mathcal {S}\) if \(\ell _{\mathfrak {u}}\ge 1\). All other points in \({\mathfrak {u}}\) have degree 0. To construct \(\mathrm{Enh}(\varGamma )\), we add new edges to vertices in \({\mathfrak {u}}\in \mathcal {U}\) and also move around existing edges, but keep \(\deg (x_s)\) unchanged for all \(s \in \mathcal {S}\) throughout the procedure. We do this cluster by cluster and write \(u^*=u^*({\mathfrak {u}})\) for simplicity.

Fix \({\mathfrak {u}}\in \mathcal {U}\) arbitrary. To perform the enhancement operation for \({\mathfrak {u}}\), we let \(s \in \mathcal {S}\) be such that \(x_s\) is the unique singleton point connected to \(x_{u^*({\mathfrak {u}})}\) if \(\ell _{\mathfrak {u}}\ge 1\). This also includes \(\ell _{\mathfrak {u}}=0\), in which case *s* could be arbitrary. We distinguish several situations depending on the number of points in \({\mathfrak {u}}\).

*Case 1.*\(|{\mathfrak {u}}| = 2\). Let \(j \ne u^*({\mathfrak {u}})\) denote the other point in \({\mathfrak {u}}\). By definition of \(\varOmega ^*\), we have \(\deg (x_j)=0\). We then perform the following operations. We move \(\left\lfloor (\ell _{\mathfrak {u}}+1)/2 \right\rfloor \) of the \(\ell _{\mathfrak {u}}\) edges between \(x_s\) and \(x_{u^*}\) to connecting \(x_s\) and \(x_j\) and add \(m-\ell _{\mathfrak {u}}\) edges between \(x_{u^*}\) and \(x_{j}\). By clustering, we have \(|x_{u^*}-x_{j}| \le L \varepsilon \) and \(|x_{s}-x_j| \le 2|x_s-x_{u^*}|\). Hence, Proposition 2.1 gives the bounds

*L*as chosen in (2.18) is independent of \(\theta \), \(\varepsilon \), and \(\mathbf {x}\). So in graphic notation, the above operation giveswhere the gray area indicates the cluster \({\mathfrak {u}}\), and we have omitted drawing the remaining \((m-\ell _{\mathfrak {u}})\) or \((m+1-\ell _{\mathfrak {u}})\) edges from \(x_s\). We also drop \(|\cdot |\) and simply use the graph itself to denote its value. Then, \(\deg (x_s)=m\) or \(m+1\) is unchanged in the procedure. Furthermore, we have \(\deg (x_{u^*})=m\) and \(\deg (x_j)\in \{m,m+1\}\) after the operation. This also includes the situation \(\ell _{\mathfrak {u}}=0\).

*Case 2.*\(|{\mathfrak {u}}|=3\). Let

*i*,

*j*denote the two other points in \({\mathfrak {u}}\). We then perform the operationWe see \(\deg (x_s)\) is unchanged. One can also check that \(\deg (x_{u^*({\mathfrak {u}})}) = m\) or \(m+1\), and \(\deg (x_i) = \deg (x_j) = m\). So we have the correct degrees of the vertices as well as the desired bound.

*Case 3.*\(|{\mathfrak {u}}| \ge 4\). We denote the other \(|{\mathfrak {u}}|-1\) points in the cluster by \(j_{1}, \dots , j_{|{\mathfrak {u}}|-1}\). For \(|{\mathfrak {u}}|-2\) of them, say \(x_{j_1}, \dots , x_{j_{|{\mathfrak {u}}|-2}}\), we perform the same operation as in Sect. 2.1 by cyclically connecting them with edges of multiplicities \(\left\lfloor \frac{m+1}{2} \right\rfloor \). This yields the bound

*m*or \(m+1\) with a desired bound.

Since the bound when \(\mathcal {S}=\emptyset \) has already been established in (2.6), we have thus completed the proof of Theorem 1.1.

## 3 Convergence of the Fields—Proof of Theorem 1.4

We are now ready to prove Theorem 1.4. For notational simplicity, we write \(A \lesssim _{{\alpha }}B\) to denote that \(A \le C B\), where the constant *C* depends only on the parameter(s) in the subscripts of the symbol \(\lesssim \) (and in this case \(\alpha \)).

### Proposition 3.1

*G*is the correlation function of \(\varPsi \) as in Assumption 1.2. The coefficient of the

*m*-th term in the chaos expansion of \(F(\varPhi _\varepsilon )\) is given by

For the third term, it suffices to notice that assumption (1.6) on *G* guarantees that \(\sigma _{\varepsilon }^{2} \rightarrow \sigma ^2\), where \(\sigma _\varepsilon ^2\) and \(\sigma ^2\) are given by (3.3) and (1.8). Hence, we immediately have \(a_{m}^{(\varepsilon )} \rightarrow a_m\). The desired bound of the form (3.2) then follows immediately from the boundedness of \(\varPsi ^{\diamond m}\) in \(\mathcal {C}^{-\frac{m \alpha }{2}-\kappa }\).

*G*guarantees that \(\varPhi _{\varepsilon }\) satisfies the assumption (1.4) in Theorem 1.1 with some \(\varLambda >1\). Since \(\varPsi _{\varepsilon }^{\diamond m} = \varepsilon ^{-\frac{m\alpha }{2}} \varPhi _\varepsilon ^{\diamond m}\), and \(a_{m}^{(\varepsilon )}\) is precisely the

*m*-th coefficient in the chaos expansion of \(F(\varPhi _{\varepsilon })\), we have

*F*and changing the order of integration, we get the identity

*n*-th moments on both sides and using triangle inequality, we get

*M*and

*n*only. The second one is taken over an interval, so we need the following lemma to interchange it with the expectation.

### Lemma 3.2

*f*is a random \(\mathcal {C}^1\) function on an interval \(\mathcal {I}\). For every \(p \ge 1\), there exists

*C*depending on

*p*and \(|\mathcal {I}|\) only such that

### Proof

*p*-th power, we get

*C*depends on

*p*and \(\mathcal {I}\). The assertion then follows by taking expectation on both sides and noting that

*r*is taken over \(r \le M+1\) to include the one additional derivative required in the interchange. Plugging it back into the right-hand side of (3.7), we obtain

### Lemma 3.3

### Proof

*F*, we get

## Footnotes

- 1.A rigorous way of saying the correlation is
*G*is thatfor all \(\varphi , \phi \in \mathcal {C}_{c}^{\infty }(\mathbf {R}^d)\).$$\begin{aligned} \mathbf {E}\langle \varPsi , \varphi \rangle \langle \varPsi , \phi \rangle = \int _{\mathbf {R}^d} G(x-y) \varphi (x) \phi (y) \mathrm{d}x \mathrm{d}y \end{aligned}$$

## Notes

### Acknowledgements

The author acknowledges the support from the Engineering and Physical Sciences Research Council through the fellowship EP/N021568/1. I also thank the anonymous referee for carefully reading the draft version of the article and providing helpful suggestions on improving the presentation.

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