On the Averaged Green’s Function of an Elliptic Equation with Random Coefficients

Abstract

We consider a divergence-form elliptic difference operator on the lattice $\mathbb {Z}^d$, with a coefficient matrix that is an i.i.d. perturbation of the identity matrix. Recently, Bourgain introduced novel techniques from harmonic analysis to prove the convergence of the Feshbach-Schur perturbation series related to the averaged Green’s function of this model. Our main contribution is a refinement of Bourgain’s approach which improves the key decay rate from $-2d+\epsilon$ to $-3d+\epsilon$. (The optimal decay rate is conjectured to be $-3d$.) As an application, we derive estimates on higher derivatives of the averaged Green’s function which go beyond the second derivatives considered by Delmotte–Deuschel and related works.

Appendices

A Proof of Corollary 1.5 on Derivatives of the Averaged Green’s Function

The proof is based on the standard fact that existence of derivatives in Fourier space (which we get from Theorem 1.1) can be translated to decay in physical space via integration by parts. For the endpoint case $|\alpha |=d+1$, we use a variant of the argument which only requires the Fourier transform to be Hölder continuous.

Let $d\ge 2$ and assume that $\alpha$ is a multi-index such that $|\alpha |>2-d$. This condition ensures that the symbol of $\nabla ^{\alpha } G$ (and $\nabla ^{\alpha } G^{\mu }$) is integrable on $\mathbb {T}^d$. We shall prove the first statement for $d\ge 3$ as the proof of the second statement is identical.

Fix $0<\epsilon <1$ and let $0<\delta <c\epsilon$, where c is the constant $\tilde{c}_d$ from Theorem 1.1. Note that the operator $\mathcal {L}$ is a convolution operator whose symbol is given by

$$\begin{aligned}m(\theta ) = (1+\delta \mathbb {E}\sigma ) \sum _{j=1}^d 2(1-\cos \theta _j) + \sum _{1\le j,k\le d} (e^{-i\theta _j}-1) \widehat{K_{j,k}^{\delta }}(\theta ) (e^{i\theta _k}-1)\end{aligned}$$

for $\theta \in \mathbb {T}^d$. By Theorem 1.1, we have

$$\begin{aligned} ||{\widehat{K_{j,k}} }||_{C^{2d-1,1-\epsilon } (\mathbb {T}^d) }\le C\delta ^2. \end{aligned}$$

In particular, we may find $0<c_d \le c\epsilon$ such that for any $0<\delta <c_d$, we have the lower bound

$$\begin{aligned} |m(\theta )| \ge C |\theta |^2 \end{aligned}$$

(A.1)

for some constant $C>0$ for any $\theta$ in $\mathbb {T}^d$ which we identify with $[-\pi ,\pi ]^d$.

Next, let $m^{\alpha }$ be the symbol of $\nabla ^{\alpha }$, i.e. $m^{\alpha }(\theta ) = \prod _{j=1}^d (e^{i\theta _j}-1)^{\alpha _j }.$ Since $|m^{\alpha }(\theta )|\le \prod _{j=1}^d |\theta _j|^{\alpha _j } \le |\theta |^{|\alpha |}$, we see that

$$\begin{aligned} \left| {\frac{m^{\alpha }(\theta )}{m(\theta )}}\right| \le C |\theta |^{|\alpha |-2}, \end{aligned}$$

(A.2)

which is integrable on $\mathbb {T}^d$ provided that $|\alpha |>2-d$.

The kernel $\nabla ^{\alpha } G(x)$ is the Fourier inverse of $m^{\alpha }(\theta )[m(\theta )]^{-1}$. We estimate $\nabla ^{\alpha } G(x)$ using a dyadic decomposition of $\mathbb {T}^d$ as follows. Let $\varphi$ be a smooth even function compactly supported on $[-2,2]$ and $\varphi (r) = 1$ for $r\in [-1,1]$. Let $\psi (r) := \varphi (r) - \varphi (2r)$ and $\psi _l(r) := \psi (2^{l} r)$ for $l\ge 1$ and $\psi _0(r) := 1-\varphi (2r)$. Note that $\sum _{l\ge 0} \psi _l(r) = 1$ for any $r\ne 0$. We write $\nabla ^{\alpha } G(x) = \sum _{l\ge 0} \nabla ^{\alpha } G_l(x)$, where we denote by $\nabla ^{\alpha } G_l(x)$ the Fourier inverse of $\psi _l(|\theta |) m^{\alpha }(\theta )[m(\theta )]^{-1}$.

Define

$$\begin{aligned} g^{\alpha }_l(\theta ) := \frac{\phi (\theta )m^{\alpha }(2^{-l}\theta )}{m(2^{-l}\theta )}, \end{aligned}$$

where $\phi (\theta ) := \psi (|\theta |)$. Then for $l\ge 1$, we may write by a change of variable

$$\begin{aligned} \nabla ^{\alpha } G_l(x) = 2^{-ld} \int _{\mathbb {R}^d} g^{\alpha }_l(\theta ) e^{i2^{-l}x \cdot \theta } \frac{d\theta }{(2\pi )^d}. \end{aligned}$$

First note that, by (A.2), $|\nabla ^{\alpha } G_l(x)| \le C 2^{-l(d-2+|\alpha |)}$ for any $x\in \mathbb {Z}^d$. This bound may be improved when $2^{-l}|x|\ge 1$. We claim that when $|x|\ge 2^l$ and $l\ge 1$, we have

$$\begin{aligned} |\nabla ^{\alpha } G_l(x)| \le \frac{C2^{-l(d-2+|\alpha |)}}{(2^{-l}|x|)^{2d-\epsilon }}. \end{aligned}$$

(A.3)

Given the estimate (A.3), Corollary 1.5 follows quickly. First of all, one can check, using integration by parts, that

$$\begin{aligned} |\nabla ^{\alpha } G_0(x)| \le C(1+|x|)^{-(2d-1)}. \end{aligned}$$

Using this bound and (A.3), we get

$$\begin{aligned} |\nabla ^{\alpha } G(x)| \le C \sum _{l\ge 0} 2^{-l(d-2+|\alpha |)}\le C \end{aligned}$$

(A.4)

for any $x\in \mathbb {Z}^d$, since we assume $|\alpha |>2-d$. Next we assume that $|x|\ge 100$ and study the sum over $2^l > |x|$ and $2^l \le |x|$ separately. We have

$$\begin{aligned} \sum _{l\ge 0 :\; 2^l> |x|} |\nabla ^{\alpha } G_l(x)| \le C\sum _{l\ge 0 :\; 2^l > |x|} 2^{-l(d-2+|\alpha |)} \le C |x|^{-(d-2+|\alpha |)}. \end{aligned}$$

(A.5)

On the other hand, if $|\alpha |\le d+1$, we have

$$\begin{aligned} \sum _{l\ge 0 :\; 2^l \le |x|} |\nabla ^{\alpha } G_l(x)|\le & {} C|x|^{-(2d-1)} + C\sum _{l\ge 1 :\; 2^l \le |x|} 2^{l(d + 2 - |\alpha | -\epsilon )} |x|^{-2d+\epsilon }\nonumber \\\le & {} C |x|^{-(d-2+|\alpha |)}. \end{aligned}$$

(A.6)

Observe that (A.4), (A.5) and (A.6) implies Corollary 1.5.

It remains to verify (A.3). We need the following lemma:

Lemma A.1

For $0<\delta < c_d$ and $l\ge 1$, we have

$$\begin{aligned} ||{g_l^{\alpha }}||_{C^{2d-1,1-\epsilon } (\mathbb {R}^d) }\le C 2^{-l(|\alpha |-2)}. \end{aligned}$$

Proof

When $\theta \in \,\mathrm {supp}\,\phi$, $|m(2^{-l}\theta )|$ is comparable to $2^{-2l}$ as $|\theta |\sim 1$. In addition, we have the estimates

$$\begin{aligned} |[(2^{-l}\partial )^{\beta } m](2^{-l}\theta )|&\le C_\beta 2^{-2l} \\ |[(2^{-l}\partial )^{\beta } m^{\alpha } ](2^{-l}\theta )|&\le C_{\beta ,\alpha } 2^{-l|\alpha |} \end{aligned}$$

for all multi-index $\beta$ with $|\beta |\le 2d-1$. From these estimates, if follows that

$$\begin{aligned} ||{g_l^{\alpha }}||_{C^{2d-1} (\mathbb {R}^d) }\le C 2^{-l(|\alpha |-2)}. \end{aligned}$$

For the Hölder estimate, we note that when $|\beta | = 2d-1$, we may write

$$\begin{aligned} \partial ^{\beta } g_l^{\alpha }(\theta ) =\frac{\chi (\theta ) 2^{-l(2d-1)} \partial ^{\beta } m (2^{-l}\theta ) m^{\alpha } (2^{-l}\theta )}{m(2^{-l}\theta )^2} + R(\theta ), \end{aligned}$$

where $||{R}||_{C^1} \le C 2^{-l(|\alpha |-2)}$. Thus, it remains to show that the $C^{0,1-\epsilon }$ norm of the first term is $O(2^{-l(|\alpha |-2)})$. This again reduces to quantify the $C^{0,1-\epsilon }$ norm of the functions resulting from replacing $\partial ^{\beta } m(2^{-l}\theta )$ in the first term by

$$\begin{aligned} (e^{-i2^{-l}\theta _j} -1)(e^{i2^{-l}\theta _k} -1) \partial ^{\beta } \widehat{K^{\delta }_{j,k}}(2^{-l}\theta ) \end{aligned}$$

for each $1\le j,k\le d$. One can verify that $C^{0,1-\epsilon }$ norm of the resulting functions are $O(2^{-l(|\alpha |-2)})$. $\quad \square$

Finally, we may deduce (A.3) from Lemma A.1 by a standard argument. Let $|x| \ge 2^l$. Without loss of generality, we may assume that $|x_1| = \max _j |x_j|$, hence $|x_1| \sim |x|$. Using integration by parts, we see that

$$\begin{aligned} \nabla ^{\alpha } G_l(x) = \frac{C2^{-ld}}{(2^{-l}x_1)^{2d-1}} \int (\partial _1)^{2d-1} g^{\alpha }_l(\theta ) e^{i2^{-l}x\cdot \theta } d\theta . \end{aligned}$$

After the change of variable $\theta _1 \rightarrow \theta _1 + \frac{\pi }{2^{-l}x_1}$ in the integral, we also see that

$$\begin{aligned} \nabla ^{\alpha } G_l(x) = - \frac{C2^{-ld}}{(2^{-l}x_1)^{2d-1}} \int (\partial _1)^{2d-1} g^{\alpha }_l\left( \theta _1+\frac{\pi }{2^{-l}x_1}, \theta ' \right) e^{i2^{-l}x\cdot \theta } d\theta , \end{aligned}$$

where we write $\theta ' = (\theta _2,\cdots ,\theta _d)$. Estimating the average of these expressions for $\nabla ^{\alpha } G_l(x)$ using Lemma A.1, we obtain (A.3) which finishes the proof.

B Proof of Corollary 1.6 on Averaged Solutions

We have $f\in H^{-1}(\mathbb {Z}^d)$ by (2.1) and $u_\omega =L_\omega ^{-1} f$ is the unique solution in $H^1(\mathbb {Z}^d)$ to the equation $L_\omega u_\omega = f$.

By the definition of $\mathcal {L}$, we have $\mathbb {E} [u_\omega ] = \mathcal {L}^{-1} f$. In addition, we have

$$\begin{aligned} \mathcal {L} (G*f) = (\mathcal {L} G)*f = \delta _0 * f = f \end{aligned}$$

(B.1)

which yields $\mathcal {L} ^{-1} f = G*f$. We need to verify the first equality of (B.1), which is trivial when f is compactly supported. For general $f\in \ell ^{p_d}(\mathbb {Z}^d)$, it suffices to show that

$$\begin{aligned} \nabla _i^* K^{\delta }_{i,j} \nabla _{j} (G*f) =( \nabla _i^* K^{\delta }_{i,j} \nabla _{j} G)*f. \end{aligned}$$

(B.2)

To see this, first note that the sum defining the convolution $G*f$ converges absolutely since $G\in \ell ^{q_d}(\mathbb {Z}^d)$ and $f\in \ell ^{p_d}(\mathbb {Z}^d)$ and $\frac{1}{p_d}+\frac{1}{q_d} = 1$. This shows that $\nabla _j (G*f) = (\nabla _j G)*f$ with $\nabla _j G \in \ell ^{q_d}(\mathbb {Z}^d)$. In fact, $\nabla _j G\in \ell ^2(\mathbb {Z}^d)$ since $G\in H^1(\mathbb {Z}^d)$, but we do not use this fact here. Moreover, the kernel of $K_{i,j}$ belongs to $\ell ^1(\mathbb {Z}^d)$ and we have $K_{i,j} [( \nabla _j G) * f] = (K_{i,j} \nabla _j G) * f$ by Fubini’s theorem and $K_{i,j} \nabla _j G \in \ell ^{q_d}(\mathbb {Z}^d)$. The argument for $\nabla _i^*$ is the same and this establishes (B.2).

The pointwise estimate is a direct consequence of Corollary 1.5. $\quad \square$

C Proof of the Deterministic Bound in Lemma 2.2

We closely follow [8] and provide some details. Recall that

$$\begin{aligned} T^n(x_0,x_n) = \sum _{\underline{x}} K^1(x_0,x_1)K^2(x_1,x_2)\ldots K^n(x_{n-1},x_n), \end{aligned}$$

where $\underline{x}= (x_1,x_2,\ldots ,x_{n-1})\in (\mathbb {Z}^d)^{n-1}$. When $x_0\ne x_n$, we may write

$$\begin{aligned} \sum _{\underline{x}} = \sum _{m\ge 0} \sum _{\underline{x}:\; 2^m \le \max _j{|x_j-x_{j+1}|} < 2^{m+1}} = \sum _{j_0=0}^{n-1} \sum _{m\ge 0} \sum _{\underline{x}\in S_{j_0}^{m}}, \end{aligned}$$

where $S_{j_0}^{m}$ is defined in (3.6). When $x_0 =x_n$, this yields a decomposition for the sum $\sum _{\underline{x}}$ except for $\underline{x}= (x_0,\cdots ,x_0)$ for which we may invoke the bound $|K^1(x_0,x_0)\cdots K^n(x_0,x_0)|\le A^n$.

Let $\mathbf {I}_m = [0,2^m)$. Observe that

$$\begin{aligned}&\sum _{\underline{x}\in S^m_{j_0} } K^1(x_0,x_1)\ldots K^n(x_{n-1},x_n) \nonumber \\&\quad = \sum _{\underline{x}\in (\mathbb {Z}^d)^{n-1}} \prod _{j=1}^{j_0} K^j_{\mathbf {I}_m} (x_{j-1},x_j) K^{j_0+1}_{\mathbf {I}_{m+1}\setminus \mathbf {I}_{m}}(x_{j_0}, x_{j_0+1}) \prod _{j=j_0+2}^{n} K^j_{\mathbf {I}_{m+1}}(x_{j-1},x_j) \nonumber \\&\quad = \sum _{x_{j_0}, x_{j_0+1} } T^{j_0}_{\mathbf {I}_m}(x_0,x_{j_0}) K^{j_0+1}_{\mathbf {I}_{m+1}\setminus \mathbf {I}_{m} }(x_{j_0}, x_{j_0+1}) \widetilde{T}^{j_0}_{\mathbf {I}_{m+1}} (x_{j_0+1},x_n), \end{aligned}$$

(C.1)

where $T^{j_0}_{\mathbf {I}_m} := \prod _{j=1}^{j_0} K^j_{\mathbf {I}_m}$ and $\widetilde{T}^{j_0}_{\mathbf {I}_{m+1}} :=\prod _{j=j_0+2}^{n} K^j_{\mathbf {I}_{m+1}}$. Here, the sum over $x_{j_0}$ is in fact a finite sum; $|x_0 - x_{j_0}| \le |x_0- x_1| + \ldots +|x_{j_0-1}-x_{j_0}| \le n 2^{m+1}.$ Similarly, the sum over $x_{j_0+1}$ is a finite sum over $|x_{j_0+1}-x_n|\le n2^{m+1}$. From this, Hölder’s inequality, and Assumption (i),

$$\begin{aligned}&|(C.1)|\\&\quad \le A2^{-md} \sum _{x_{j_0}} |T^{j_0}_{\mathbf {I}_m} (x_0,x_{j_0})| \sum _{x_{j_0+1}} |\widetilde{T}^{j_0}_{\mathbf {I}_{m+1}} (x_{j_0+1},x_n)| \\&\quad \le C A2^{-md} (n2^m)^{2d(p-1)/p} \Big (\sum _{x_{j_0}}| T^{j_0}_{\mathbf {I}_m} (x_0,x_{j_0})|^p\Big )^{1/p} \Big (\sum _{x_{j_0+1}}|\widetilde{T}^{j_0}_{\mathbf {I}_{m+1}} (x_{j_0+1},x_n)|^p \Big )^{1/p} \end{aligned}$$

for $p>1$ selected by $2d(p-1)=\epsilon$.

Let $\delta _y$ be the delta function on $\mathbb {Z}^d$; $\delta _y(x)$ is equal to 1 if $x=y$ and 0 otherwise. Note that the product of two $\ell ^p$ sums in the last inequality is bounded by

$$\begin{aligned} ||{(T^{j_0}_{\mathbf {I}_m} )^* \delta _{x_0}}||_{\ell ^p(\mathbb {Z}^d)} ||{\widetilde{T}^{j_0}_{\mathbf {I}_{m+1}} \delta _{x_n}}||_{\ell ^p(\mathbb {Z}^d)}, \end{aligned}$$

which is bounded by $[A/(p-1)]^{j_0} [A/(p-1)]^{n-j_0-1} =\left( \frac{2dA}{\epsilon }\right) ^{n-1}$ by Assumption (2). Therefore, we get

$$\begin{aligned} |(C.1)| \le C n^{\epsilon } \left( \frac{2dA}{\epsilon }\right) ^{n} \epsilon 2^{-m(d-\epsilon )} . \end{aligned}$$

(C.2)

It only remains to sum (C.2) over $m\ge 0$ and $j_0$. For this, we distinguish the cases $|x_0-x_n|\ge 2n$ and $|x_0-x_n|< 2n$. For the first case, the sum over m is restricted to

$$\begin{aligned} 2^m \ge \frac{|x_0-x_n|}{ 2n}, \end{aligned}$$

which follows from (given that $\max _j|x_j-x_{j+1}| < 2^{m+1}$)

$$\begin{aligned} |x_0-x_n| \le \sum _{j=0}^{n-1} |x_j-x_{j+1}| \le n 2^{m+1}. \end{aligned}$$

In the first case, therefore, summing (C.2) over m and $j_0$ yields

$$\begin{aligned} |T^n(x_0,x_n)|&\le C n^{1+\epsilon }\left( \frac{2dA}{\epsilon }\right) ^{n} \epsilon \sum _{m: \; 2^m \ge |x_0-x_n|/ (2n)} 2^{-m(d-\epsilon )} \\&\le (C_d A/\epsilon )^n \epsilon \langle {x_0-x_n}\rangle ^{-(d-\epsilon )}. \end{aligned}$$

When $|x_0-x_n|< 2n$, we sum (C.2) over $m\ge 0$ and $j_0$. Then we get

$$\begin{aligned} |T^n(x_0,x_n)| \le C n^{1+\epsilon } \left( \frac{2dA}{\epsilon }\right) ^{n} \epsilon , \end{aligned}$$

which completes the proof since $1 \le Cn \langle {x_0-x_n}\rangle ^{-1}$.

D Proof of Lemma 2.7 on Disjointification

Since $S'=\emptyset$ when $E_l \cap F_l \ne \emptyset$ for some l, we may assume that $E_l \cap F_l = \emptyset$ for all $1\le l\le m$.

We closely follow the argument given in Section 4 of [8]. We introduce an additional set of variables (“Steinhaus system”) on the torus $\mathbb {T}=\mathbb {R}/2\pi \mathbb {Z}$

$$\begin{aligned} \overline{\theta }:=(\overline{\theta }^1, \overline{\theta }^2, \ldots , \overline{\theta }^m ), \text { where } \overline{\theta }^l := \left\{ {\theta ^l_x \in \mathbb {T}}\; : \;{x\in \mathbb {Z}^d}\right\} . \end{aligned}$$

We use these variables to define the complex-valued functions $e_j:\mathbb {Z}^d\rightarrow \mathbb {C}$,

$$\begin{aligned} e_j(x,\overline{\theta })&:=\prod _{l=1}^m \exp \left( i \nu ^l_j \theta ^l_{x}\right) ,\\ \text {where }\quad \nu ^l_j&:={\left\{ \begin{array}{ll} 1,\qquad \;\text { if }j\in E_l,\\ -1,\qquad \text {if }j\in F_l,\\ 0,\qquad \;\text { otherwise}. \end{array}\right. } \end{aligned}$$

Note that $\Vert e_j(\cdot ,\overline{\theta })\Vert _{\infty } \le 1$ for all $\overline{\theta }$.

Assume first that the set S is finite. Define

$$\begin{aligned}&\tilde{T}^n_S(x_0,x_n,\overline{\theta }):= \sum _{\underline{x}\in S} \tilde{K}^1(x_0,x_1)\tilde{K}^2(x_1,x_2)\ldots \tilde{K}^n(x_{n-1},x_n) \\&\quad = \sum _{\underline{x}\in S} K^1(x_0,x_1) \ldots K^n(x_{n-1},x_n) \prod _{l=1}^{m} \exp \left( i \left( \sum _{j\in E_l} \theta ^l_{x_j} -\sum _{k\in F_l} \theta ^l_{x_k} \right) \right) , \end{aligned}$$

where we introduced the operators $\tilde{K}^j(x,y) :=K^j(x,y)e_j(y,\overline{\theta })$ for $2\le j\le n$ and $\tilde{K}^1 (x,y):=e_0(x,\overline{\theta })K^1(x,y)e_1(y,\overline{\theta })$. It is important to observe that, by the assumption, we have

$$\begin{aligned} |\tilde{T}^n_S(x_0,x_n,\overline{\theta })|\le M(x_0,x_n) \end{aligned}$$

(D.1)

for all $\overline{\theta }$.

The next step is to average the bound (D.1) over the variables $\overline{\theta }$ with respect to specific probability measures to be chosen. Define the set

$$\begin{aligned}&\mathbb {Z}^d_S := \{ x_0,x_n\}\\&\qquad \cup \{ x\in \mathbb {Z}^d: x=x_j \text { for some } (x_1,\ldots ,x_{n-1})\in S \text { and } 1\le j\le n-1\}, \end{aligned}$$

which is finite since S is finite by assumption. For each $-1<t<1$, let $P_t(\theta )$ be the Poisson kernel of the unit disk

$$\begin{aligned} P_t(\theta )=\sum _{n=-{\infty }}^{\infty } t^{|n|}e^{in\theta }. \end{aligned}$$

Note that $P_t(\theta ) \frac{d\theta }{2\pi }$ is a probability measure on $\mathbb {T}$. For each $1\le l\le m$ and $|t|<1$, consider the product measure $\mathrm {d}\mu _t^l$ on $\mathbb {T}^{\mathbb {Z}^d_S}= \prod _{x\in \mathbb {Z}^d_S} \mathbb {T}$ given by

$$\begin{aligned} \mathrm {d}\mu _t^l (\overline{\theta }^l) := \prod _{x\in \mathbb {Z}^d_S} P_t(\theta ^l_x) \frac{d\theta ^l_x}{2\pi }. \end{aligned}$$

We first average (D.2) over the probability space $\mathbb {T}^{\mathbb {Z}^d_S}$ equipped with the measure $\mathrm {d}\mu _{t}^1$. On the one hand, since (D.1) holds pointwise in $\overline{\theta }$, we have

$$\begin{aligned} \left| \int _{\mathbb {T}^{\mathbb {Z}^d_S}} \tilde{T}^n_S(x_0,x_n,\overline{\theta }) \mathrm {d}\mu _{t}^1\left( \overline{\theta }^1 \right) \right| \le M(x_0,x_n) \end{aligned}$$

(D.2)

for any $(\overline{\theta }^2, \ldots , \overline{\theta }^m)$ and $|t|<1$. On the other hand, we may write the integral above as

$$\begin{aligned}&\sum _{\underline{x}\in S} K^1(x_0,x_1)K^2(x_1,x_2)\ldots K^n(x_{n-1},x_n) \prod _{l=2}^{m} \exp \left( i \left( \sum _{j\in E_l} \theta ^l_{x_j} -\sum _{k\in F_l} \theta ^l_{x_k} \right) \right) \\&\quad \times \int _{\mathbb {T}^{\mathbb {Z}^d_S}} \exp \left( i \left( \sum _{j\in E_1} \theta ^1_{x_j} - \sum _{k\in F_1} \theta ^1_{x_k} \right) \right) \prod _{x\in \mathbb {Z}^d_S} P_{t}(\theta ^1_x) \frac{\mathrm {d}\theta _{x}^1}{2\pi }. \end{aligned}$$

As was observed in [8], the integral in the above line is equal to $t^{w_{(x_0,x_1,\ldots ,x_n)}}$, where

$$\begin{aligned} w_{(x_0,\ldots ,x_n)} = \sum _{x \in \mathbb {Z}^d_S} \bigl | { | \{ j\in E_1: x_j=x \}| - | \{ k\in F_1: x_k=x \}| }\bigr | \le |E_1| + |F_1|. \end{aligned}$$

Moreover, $w_{(x_0,\ldots ,x_n)}= |E_1|+|F_1|$ if and only if $\underline{x} \in S_1$, where

$$\begin{aligned} S_1 := S\cap \{\underline{x}: \{x_j: j\in E_1 \} \cap \{x_k:k\in F_1 \} =\emptyset \}. \end{aligned}$$

Therefore, we may write $\int _{\mathbb {T}^{\mathbb {Z}^d_S}} \tilde{T}^n_S(x_0,x_n,\overline{\theta }) \mathrm {d}\mu _{t}^1(\overline{\theta }^1 )$ as a polynomial

$$\begin{aligned} f(t)= a_D t^D +a_{D-1}t^{D-1} + \ldots + a_0, \end{aligned}$$

(D.3)

where $D= |E_1| + |F_1|$ and

$$\begin{aligned} a_D= & {} \sum _{\underline{x}\in S_1} K^1(x_0,x_1) K^2 (x_1,x_2) \ldots K^n(x_{n-1},x_n) \\&\prod _{l=2}^{m}\exp \left( i \left( \sum _{j\in E_l} \theta ^l_{x_j} - \sum _{k\in F_l} \theta ^l_{x_k} \right) \right) . \end{aligned}$$

At this point, we recall a special case of the Markov brothers’ inequality.

Lemma D.1

Let f(t) be a polynomial as in (D.3). Then we have

$$\begin{aligned} |a_D| \le 2^{D-1} \max _{ -1\le t \le 1} |f(t)|. \end{aligned}$$

Combined with (D.2), we get, with $D= |E_1| + |F_1|$,

$$\begin{aligned} |a_D| \le 2^{D-1} M(x_0,x_n) \end{aligned}$$

for any $(\overline{\theta }^2, \ldots , \overline{\theta }^m)$. What comes next is a similar averaging argument for the top coefficient $a_D$ over the measure $\mathrm {d}\mu _t^2$, which yields

$$\begin{aligned}&\left| \sum _{\underline{x}\in S_2} K^1(x_0,x_1)K^2(x_1,x_2) \ldots K^n(x_{n-1},x_n) \prod _{l=3}^{m} \exp \left( i \left( \sum _{j\in E_l} \theta ^l_{x_j} - \sum _{k\in F_l} \theta ^l_{x_k} \right) \right) \right| \\&\quad \le \left( \prod _{1\le l \le 2} 2^{|E_l|+|F_l|-1} \right) M(x_0,x_n) \end{aligned}$$

for any $(\overline{\theta }^3, \ldots , \overline{\theta }^m)$, where

$$\begin{aligned} S_2 := S_1 \cap \{\underline{x}: \{x_j: j\in E_2 \} \cap \{x_k:k\in F_2 \} = \emptyset \}. \end{aligned}$$

A successive averaging over $\mathrm {d}\mu _{t}^3, \ldots , \mathrm {d}\mu _{t}^m$ finishes the proof.

Next, assume that S is not a finite set. By dominated convergence and the a priori bound (2.6), we have $T^n_{S'} (x_0,x_n)=\lim _{k\rightarrow \infty } T^n_{ S' \cap X_k}(x_0,x_n)$. Therefore, it is sufficient to show that

$$\begin{aligned} |T^n_{S' \cap X_k}(x_0,x_n)| \le 2^{\sum _{1\le l\le m} |E_l|+|F_l|} M(x_0,x_n) \end{aligned}$$

for all large $k\ge 1$, which follows from applying the result for the finite set to $S\cap X_k$. $\quad \square$