Stabilization of the statistical solutions for large times for a harmonic lattice coupled to a Klein–Gordon field

Abstract

We consider the Cauchy problem for the Hamiltonian system consisting of the Klein–Gordon field and an infinite harmonic crystal. The dynamics of the coupled system is translation-invariant with respect to the discrete subgroup $\mathbb{Z}^d$ of $\mathbb{R}^d$. The initial date is assumed to be a random function that is close to two spatially homogeneous (with respect to the subgroup $\mathbb{Z}^d$) processes when $\pm x_1>a$ with some $a>0$. We study the distribution $\mu_t$ of the solution at time $t\in\mathbb{R}$ and prove the weak convergence of $\mu_t$ to a Gaussian measure $\mu_\infty$ as $t\to\infty$. Moreover, we prove the convergence of the correlation functions to a limit and derive the explicit formulas for the covariance of the limit measure $\mu_\infty$. We give an application to Gibbs measures.

Appendix A: Proof of Theorem 3.1(1)

To prove Theorem 3.1, we introduce some auxiliary notation and prove necessary bounds for the initial correlation functions.

A.1. Bounds for initial covariance

We introduce the splitting $p=k+r$, where $k\in\mathbb{Z}^d$ and $r\in\mathrm K_1^d\cup \{0\}$. In other words, $r=x-[x]\in\mathrm K_1^d$ if $p=x\in\mathbb{R}^d$, and $r=0$ if $p=k\in\mathbb{Z}^d$. Hence, (1.7) and (2.15) imply that

$$ Q_\pm^{ij}(k+r,k'+r')=:q_\pm^{ij}(k-k',r,r')\equiv \begin{pmatrix} q_\pm^{\psi^i\psi^j}(k-k'+r,r') & q_\pm^{\psi^i u^j}(k-k'+r) \\ q_\pm^{u^i\psi^j}(k'-k+r')& q_\pm^{u^iu^j}(k-k') \end{pmatrix}.$$

(A.1)

Using Zak transform (2.2), we introduce the matrices

$$\widetilde Q_\pm(\theta,r,\theta',r'):=\mathcal Z_{p\to(\theta,r)}\mathcal Z_{p'\to(-\theta',r')}[Q_\pm(p,p')],\qquad \begin{aligned} \, \theta,\theta'&\in\mathrm K^d,\\ r,r'&\in\mathcal T^d\equiv \mathbb{T}_1^d\cup \{0\}.\end{aligned}$$

Then

$$ \widetilde Q_\pm^{ij}(\theta,r,\theta',r')= (2\pi)^d\delta(\theta-\theta') \tilde q_\pm^{ij}(\theta,r,r'),\quad \theta,\theta'\in\mathrm K^d, \qquad r,r'\in\mathcal T^d,$$

(A.2)

where

$$ \tilde q_\pm^{ij}(\theta,r,r')=e^{\mathrm i (r-r')\cdot\theta} \sum_{k\in\mathbb{Z}^d} e^{\mathrm ik\cdot\theta}q_\pm^{ij}(k,r,r')= \begin{pmatrix} \tilde q_\pm^{\psi^i\psi^j}(\theta,y,y') & \tilde q_\pm^{\psi^i u^j}(\theta,y) \\ \overline{\tilde q_\pm^{u^i\psi^j}(\theta,y')} & \tilde q_\pm^{u^iu^j}(\theta)\vphantom{|^{\Big|}} \end{pmatrix}.$$

(A.3)

Lemma A.1.

• 1. Let conditions S1 and S2 hold. Then the bounds

  $$ \begin{alignedat}{3} &\int_{\mathbb{P}^d}|Q_0(p,p')|\,dp\le C<\infty, &\qquad &p'\in\mathbb{P}^d, \\ &\int_{\mathbb{P}^d}|Q_0(p,p')|\,dp'\le C<\infty, &\qquad &p\in\mathbb{P}^d \end{alignedat}$$

  (A.4)

  hold with a constant $C$ independent of $p,p'\in\mathbb{P}^d$.

• 2. Let conditions S1–S3 hold. Then

  $$D_{y,y'}^{\alpha,\beta} \tilde q_\pm^{\psi^i\psi^j}(\theta,y,y'),\quad D_{y}^\alpha\tilde q_\pm^{\psi^i u^j}(\theta,y),\quad D_{y'}^{\beta}\tilde q_\pm^{u^i\psi^j}(\theta,y'),\quad \tilde q_\pm^{u^iu^j}(\theta)$$

  are uniformly bounded in $(\theta,y,y')\in\mathrm K^d\times \mathbb{T}_1^d\times \mathbb{T}_1^d$, $|\alpha|\le1-i$, $|\beta|\le 1-j$.

Proof.

The bounds in (A.4) follow from condition S2. Furthermore, S1–S3 imply

$$|D_{y,y'}^{\alpha,\beta}\tilde q_\pm^{\psi^i\psi^j}(\theta,y,y')|\le \sum_{k\in\mathbb{Z}^d}|D_{y,y'}^{\alpha,\beta}q_\pm^{\psi^i\psi^j}(k+y,y')|\le C\sum_{k\in\mathbb{Z}^d}h(|k|-2\sqrt d\,)\le C<\infty.$$

The remaining bounds can be proved similarly.

Corollary A.1.

• 1. The quadratic form $\mathcal Q_0(Z,Z)$ is continuous in $\mathbf L^2:=[L^2(\mathbb{P}^d)]^2$, i.e.,

  $$|\mathcal Q_0(Z,Z)|\equiv|\langle Q_0(p,p'),Z(p)\otimes Z(p')\rangle|\le C\|Z\|^2_{\mathbf L^2},\qquad Z\in\mathbf L^2.$$

  Also, the quadratic forms with the matrix kernels $Q_\pm(p,p')$ are continuous in $\mathbf L^2$.

• 2. Let $i,j=0,1$, $\sigma,\sigma'\in\mathbb{N}$, $p^{ij}_{\pm,\sigma\sigma'}$ be defined in (2.9), $k,p\in\{-1;0\}$ for any $i,j$ or $k=1$ if $i=0$ and $p=1$ if $j=0$. Then the operators $\Omega^{k}(\theta)\widetilde{p}^{ij}_{\pm,\sigma\sigma'}(\theta) \Omega^{p}(\theta)$ satisfy the bounds

  $$\sum_{\sigma=1}^{+\infty} \int_{\mathrm K^d}\bigl((\Omega^{k}(\theta)\widetilde{p}^{ij}_{\pm,\sigma\sigma}(\theta)\Omega^{p}(\theta)) (\theta,r,r'),\widetilde Z_{e}(\theta,r)\otimes \widetilde Z_{e}(\theta,r')\bigr)\,d\theta\le C\|Z\|_{L^2(\mathbb{P}^d)}^2$$

  for all $Z\in L^2(\mathbb{P}^d)$.

• 3. The quadratic form $\mathcal Q_\infty(Z,Z)$ is continuous in $\mathbf L^2$.

Assertion 1 follows from bounds (A.4) by applying either the Schur test (see, e.g., [20]) or Young’s inequality. Assertions 2 and 3 can be proved similarly to Corollary 3.3 in [3].

A.2. Splitting of the covariance $Q_t$

Lemma A.2.

The functions $\zeta_\pm$ introduced in condition S3 admit the following representations (in the Zak transform)

$$ \tilde\zeta_{\pm,e}(\theta,r)= \pi\delta(\theta)\pm \mathrm i\operatorname{PV}\biggl(\frac{1}{2\tan(\theta/2)}\biggr)\widetilde\alpha_{\pm,e}(\theta,r),\qquad \theta\in\mathrm K^1,\quad r\in\mathcal{T}^1\equiv \mathbb{T}^1_1\cup\{0\},$$

(A.5)

where $\widetilde{\alpha}_{\pm,e}(0,r)=1$, $\alpha_\pm(p)=0$ for $|p|\ge a+1$.

Proof.

We write $\alpha^{\mathrm c}_\pm(x):=\zeta^{\mathrm c}_\pm(x)-\zeta^{\mathrm c}_\pm(x\mp1)$, $x\in\mathbb{R}$. Then

$$\zeta^{\mathrm c}_\pm(k+y)=\sum_{\pm l\le k}\alpha^{\mathrm c}_\pm(l+y).$$

By (2.11), the functions $\alpha^{\mathrm c}_\pm(x)$ have the properties

$$\alpha^{\mathrm c}_\pm\in C^\infty(\mathbb{R}),\qquad \alpha^{\mathrm c}_\pm(x)=0\;\,\text{for}\;\,|x|\ge a+1,\qquad \sum_{l\in\mathbb{Z}}\alpha^{\mathrm c}_\pm(l+y)=1.$$

Therefore, the functions $\zeta^{\mathrm c}_\pm(x)$ have the form

$$\zeta^{\mathrm c}_\pm(k+y)=\sum_{l\in\mathbb{Z}}H(\pm l)\alpha^{\mathrm c}_\pm(k-l+y),$$

where $H(l)$ is the Heaviside function of the interval $[0,+\infty)$. Applying the Zak transform gives (A.5). Similarly, if we set $\alpha^{\mathrm d}_\pm(k):=\zeta^{\mathrm d}_\pm(k)-\zeta^{\mathrm d}_\pm(k\mp1)$, $k\in\mathbb{Z}$, then we obtain (A.5) for $\zeta^{\mathrm d}_\pm$.

We apply the Zak transform to the initial covariance $Q_0(p,p')$:

$$ \widetilde Q_0(\theta,\theta',r,r'):=\mathcal Z_{p\to(\theta,r)}\mathcal Z_{p'\to (-\theta',r')}\bigl[Q_0(p,p')\bigr],\qquad\theta,\theta'\in\mathrm K^d.$$

(A.6)

We let $\widetilde Q_0(\theta,\theta')$ denote the integral operator with the kernel $\widetilde Q_0(\theta,\theta',r,r')$. Using the equality $\widetilde{fg}(\theta)=(2\pi)^{-2d}\tilde f*\tilde g(\theta)$ for $\theta\in\mathbb{R}^{2d}$, representation (2.12), and equality (A.2), we obtain

$$\begin{aligned} \, \widetilde Q_0(\theta,\theta')&:=\mathcal Z_{p\to(\theta,r)}\mathcal Z_{p'\to(-\theta',r')} \biggl[\sum_{\pm}\zeta_\pm(p_1)\zeta_\pm(p'_1)Q_\pm(p,p')\biggr]= \nonumber\\ &\;=\frac{1}{(2\pi)^{2d}}\sum_{\pm} \bigl(\mathcal Z_{p\to(\theta,r)}[\zeta_{\pm}(p_1)]\mathcal Z_{p'\to(-\theta',r')}[\zeta_{\pm}(p'_1)]\bigr)* (2\pi)^{d}\tilde q_\pm(\theta)\delta(\theta-\theta')= \nonumber\\ &\;=(2\pi)^{d-2}\delta(\bar\theta-\bar\theta')\sum_{\pm}\int_{[-\pi,\pi]}\tilde\zeta_{\pm,e}(\theta_1-\xi,r_1) \overline{\tilde\zeta_{\pm,e}(\theta'_1-\xi,r'_1)}\,\tilde q_\pm(\xi,\bar\theta,r,r')\,d\xi. \end{aligned}$$

(A.7)

Here, $\theta=(\theta_1,\bar\theta)\in\mathrm K^d$ and $\bar\theta=(\theta_2,\ldots,\theta_d)$ $*$ stands for the convolution in $\theta$ and $\theta'$.

In the Zak transform, the solution of problem (1.5) has form (2.3), where

$$ \widetilde{\mathcal G}_t(\theta)=e^{\widetilde{\mathcal A}(\theta)t}=\cos(\Omega(\theta)t)\,\mathrm I+\sin(\Omega(\theta)t)\,C(\theta).$$

(A.8)

Here, $\mathrm I$ is the identity matrix and $C(\theta)$ is introduced in (3.7). Applying the Zak transform to the matrix $Q_t(p,p')$ defined in (3.1) yields

$$\widetilde Q_t(\theta,\theta'):=\mathcal Z_{p\to (\theta,r)}\mathcal Z_{p'\to(-\theta',r')}[Q_t(p,p')]= \widetilde{{\mathcal G}}_t(\theta)\widetilde Q_0(\theta,\theta') \widetilde{{\mathcal G}}_t^{\kern1.5pt\mathrm T}(-\theta'), \qquad\theta,\theta'\in\mathrm K^d.$$

Using the equality $\widetilde{{\mathcal G}}^{\kern1.5pt\mathrm T}_t(-\theta')=\widetilde{{\mathcal G}}^*_t(\theta')$ and representation (A.7), for $Z\in\mathcal D$, we obtain

$$\mathcal Q_t(Z,Z):=\langle Q_{t}(p,p'),Z(p)\otimes Z(p')\rangle= \frac{1}{(2\pi)^d}\sum_{\pm}\int_{\mathrm K^{d}} \bigl(\tilde q_{\pm}(\theta),I^Z_{\pm,t}(\theta,\,{\cdot}\,)\otimes I^Z_{\pm,t}(\theta,\,{\cdot}\,)\bigr)\,d\theta,$$

where $I^Z_{\pm,t}(\theta,r)$ stands for the inner vector-valued integral of the form

$$ I^Z_{\pm,t}(\theta,r):= \frac{1}{2\pi}\int_{[-\pi,\pi]}\tilde\zeta_{\pm,e}(\eta,r_1) \widetilde{\mathcal G}^{\kern1.5pt\mathrm T}_t(\theta_1+\eta,\bar\theta)\, \overline{\widetilde Z_e(\theta_1+\eta,\bar\theta,r)}\,d\eta,\qquad r\in\mathcal{T}^d.$$

(A.9)

A.3. Stabilization of correlation matrices of $\mu_t$

We introduce a set $\mathcal D^0\subset \mathcal D$ as

$$\mathcal D^0=\bigcup_{N}\mathcal D_{N},\quad \mathcal D_{N}:=\biggl\{Z\in\mathcal D\,\bigg|\, \begin{aligned} \, &\Pi_\sigma \widetilde Z_e(\theta,\,{\cdot}\,)=0\;\,\text{for all}\;\,\sigma\ge N,\;\,\theta\in\mathrm K^d,\\ &\widetilde Z_e(\theta,\,{\cdot}\,)=0\;\,\text{ in a neighborhood of the set }\;\,\mathcal C\cup\partial \mathrm K^d \end{aligned}\biggr\},$$

where the set $\mathcal{C}$ is defined in (2.8). We recall that $\mathrm{mes}\,\mathcal{C}=0$.

Lemma A.3.

Let convergence (3.8) hold for any $Z\in\mathcal D^0$. Then (3.8) holds for all $Z\in\mathcal D$.

This lemma can be proved similarly to Lemma 5.2 in [3].

Let $Z\in\mathcal D^0$ and $\operatorname{supp}\widetilde Z_e$ stand for the closure of the set

$$\bigl\{\theta\in\mathrm K^d\colon\widetilde Z_e(\theta,y)\not\equiv 0,\,y\in\mathbb{T}_1^d\bigr\}.$$

We note that $\operatorname{supp}\widetilde Z_e\cap (\mathcal C\cup\partial \mathrm K^d)=\varnothing$. Hence, for any point $\Theta\in\operatorname{supp}\widetilde Z_e$, there is a neighborhood $\mathcal O(\Theta)\subset \mathrm K^d\setminus(\mathcal C \cup \partial \mathrm K^d)$ with the properties from Lemma 2.1. Thus, $\operatorname{supp} \widetilde Z_e\subset\bigcup_{m=1}^M \mathcal O(\Theta_m)$, where $\Theta_m\in\operatorname{supp} \widetilde Z_e$. Therefore, there exists a finite partition of unity

$$ \sum_{m=1}^M g_m(\theta)=1,\qquad \theta\in\operatorname{supp} \widetilde Z_e\subset\mathrm K^d\setminus(\mathcal C\cup\partial\mathrm K^d).$$

(A.10)

Here, $g_m$ are nonnegative functions from $C_0^\infty(\mathrm K^d)$ with $\operatorname{supp}g_m\subset\mathcal O(\Theta_m)$ and the eigenvalues $\omega_\sigma(\theta)$ and the projectors $\Pi_\sigma(\theta)$ are real-analytic functions in $\theta\in\operatorname{supp}g_m$ for every $m$ (we do not label the functions by the index $m$ to not overburden the notation). Let $Z\in\mathcal D_N$ with some $N\in\mathbb{N}$. Using decomposition (2.7) and formulas (A.8)–(A.10), we obtain

$$ \mathcal Q_t(Z,Z)=\sum_{\sigma,\sigma'=1}^N\sum_{m=1}^M\sum_{\pm} \frac{1}{(2\pi)^d}\int_{\mathrm K^{d}}g_m(\theta) \bigl(\tilde q_{\pm}(\theta),I^Z_{\pm,t,\sigma}(\theta,\,{\cdot}\,)\otimes I^Z_{\pm,t,\sigma'}(\theta,\,{\cdot}\,)\bigr)\,d\theta,$$

(A.11)

where

$$ I^Z_{\pm,t,\sigma}(\theta,r):=\frac{1}{2\pi} \int_{[-\pi,\pi]}\tilde\zeta_{\pm,e}(\eta,r_1) \widetilde{\mathcal G}^{\kern1.5pt\mathrm T}_{t,\sigma}(\theta_1+\eta,\bar\theta)\Pi_\sigma(\theta_1+\eta,\bar\theta)\, \overline{\widetilde Z_e(\theta_1+\eta,\bar\theta,r)}\,d\eta;$$

(A.12)

here, $r\in\mathcal{T}^d$ and

$$ \widetilde{\mathcal G}_{t,\sigma}(\theta):=\cos\omega_\sigma(\theta)t\,\mathrm I+\sin\omega_\sigma(\theta)t\,C_\sigma(\theta),\qquad C_\sigma(\theta):=\begin{pmatrix} 0 & \omega^{-1}_\sigma(\theta) \\ -\omega_\sigma(\theta) & 0 \end{pmatrix}.$$

(A.13)

The next lemma follows from Proposition A.4 i), ii) in [11].

Lemma A.4.

Let $\chi\in C^1(\mathrm K^d)$ and $\partial_{\theta_1} \omega_\sigma(\theta)\neq 0$ for $\theta\in\operatorname{supp}\chi\subset (\mathrm K^d\setminus\mathcal{C})$. Then for $\theta\in\operatorname{supp}\chi$,

$$\begin{aligned} \, P_\sigma(\theta,t)& :=\operatorname{PV}\int_{[-\pi,\pi]} \frac{e^{\pm\mathrm i\omega_\sigma(\theta_1+\eta,\bar\theta)t}}{\tan(\eta/2)}\chi(\theta_1+\eta,\bar\theta)\,d\eta= \\ &\;\,=\pm 2\pi\mathrm i\,\chi(\theta)e^{\pm\mathrm i\omega_\sigma(\theta)t}\operatorname{sgn}(\partial_{\theta_1}\omega_\sigma(\theta))+o(1), \end{aligned}$$

as $t\to+\infty$, where $\bar\theta=(\theta_2,\dots,\theta_d)$. Moreover, $\sup_{t\in\mathbb{R},\,\theta\in\mathrm K^d} |P_\sigma(\theta,t)|<\infty$.

Corollary A.2.

Let $Z\in\mathcal D_N$ with some $N\in\mathbb{N}$. Then, by (A.12), (A.13) and (A.5), as $t\to+\infty$, we obtain

$$ I^Z_{\pm,t,\sigma}(\theta,r)=\frac{1}{2}\widetilde{\mathcal G}^{\kern1.5pt\mathrm T}_{t,\sigma}(\theta) [\mathrm I\pm \mathrm i\operatorname{sgn}(\partial_{\theta_1}\omega_\sigma(\theta))C_\sigma(\theta)]^{\mathrm T} \Pi_\sigma(\theta)\overline{\widetilde Z_e(\theta,r)}+o(1).$$

(A.14)

We return to the proof of convergence in (3.8). We write

$$ M_{\sigma\sigma'}(\theta):=\frac{1}{4}\sum_{\pm}[\mathrm I\pm \mathrm i\operatorname{sgn}(\partial_{\theta_1}\omega_\sigma(\theta)) C_\sigma(\theta)] p_{\pm,\sigma\sigma'}(\theta) [\mathrm I\mp \mathrm i\operatorname{sgn}(\partial_{\theta_1}\omega_{\sigma'}(\theta))C^*_{\sigma'}(\theta)],$$

(A.15)

where $p_{\pm,\sigma\sigma'}$ is defined in (2.9). Then (A.11), (A.14) and (A.15) imply

$$\begin{aligned} \, \mathcal Q_t(Z,Z)= \sum_{\sigma,\sigma'=1}^N\sum_{m=1}^M\frac{1}{(2\pi)^d} \int_{\mathrm K^{d}}g_m(\theta) \bigl(\widetilde Z_e(\theta,\,{\cdot}\,),\widetilde{\mathcal G}_{t,\sigma}(\theta)M_{\sigma\sigma'}(\theta) \widetilde{\mathcal G}^*_{t,\sigma'}(\theta)\widetilde Z_e(\theta,\,{\cdot}\,)\bigr)\,d\theta+o(1) \end{aligned}$$

(A.16)

as $t\to\infty$. Using decomposition (A.13) and notation (A.15), we have

$$ \widetilde{\mathcal G}_{t,\sigma}(\theta)M_{\sigma\sigma'}(\theta) \widetilde{\mathcal G}^*_{t,\sigma'}(\theta)= \sum_{\pm}\bigl[\cos(\omega^\pm_{\sigma\sigma'}(\theta)t) A^\mp_{\sigma\sigma'}(\theta) +\sin(\omega^\pm_{\sigma\sigma'}(\theta)t) B^\pm_{\sigma\sigma'}(\theta)\bigr],$$

(A.17)

where $\omega^\pm_{\sigma\sigma'}(\theta):=\omega_\sigma(\theta)\pm\omega_{\sigma'}(\theta)$ and

$$ \begin{aligned} \, &A^\pm_{\sigma\sigma'}(\theta):=\frac{1}{2}[M_{\sigma\sigma'}(\theta)\pm C_\sigma(\theta)M_{\sigma\sigma'}(\theta)C^*_{\sigma'}(\theta)], \\ &B^\pm_{\sigma\sigma'}(\theta):=\frac{1}{2}[C_\sigma(\theta)M_{\sigma\sigma'}(\theta)\pm M_{\sigma\sigma'}(\theta)C^*_{\sigma'}(\theta)]. \end{aligned}$$

(A.18)

The oscillatory integrals in (A.16) with $\omega^\pm_{\sigma\sigma'}(\theta)\not\equiv\text{const}$ vanish as $t\to\infty$ by the Lebesgue–Riemann theorem, because all integrands in (A.16) are summable by Corollary A.1(2). Moreover, the identities $\omega^\pm_{\sigma\sigma'}(\theta)\equiv\text{const}_\pm$ with the $\text{const}_\pm\neq 0$ are impossible by R4' (see Remark 2.3 (ii)). If condition R4 is satisfied, then in the case where $\omega^\pm_{\sigma\sigma'}(\theta)\equiv\text{const}_\pm$ (with $\text{const}_\pm\ne 0$), the matrix-valued coefficients $A^\mp_{\sigma\sigma'}$ and $B^\pm_{\sigma\sigma'}$ in (A.17) are equal to zero. Thus, only the integrals with $\omega^{-}_{\sigma\sigma'}(\theta)\equiv 0$ contribute to the limit, because $\omega^{+}_{\sigma\sigma'}(\theta)\equiv 0$ would imply $\omega_\sigma(\theta)\equiv\omega_{\sigma'}(\theta)\equiv 0$, which is impossible by condition R2 (and also by R3). Therefore, as $t\to\infty$,

$$ \mathcal Q_t(Z,Z)=\sum_{\sigma=1}^N \sum_{m=1}^M\frac{1}{(2\pi)^d} \int_{\mathrm K^{d}} g_m(\theta) \bigl(\widetilde Z_e(\theta,\,{\cdot}\,), A^{+}_{\sigma\sigma}(\theta)\widetilde Z_e(\theta,\,{\cdot}\,)\bigr)\,d\theta+o(1).$$

(A.19)

Finally, using (2.9) and formula for $C_\sigma$ in (A.13), we rewrite the matrix $A^{+}_{\sigma\sigma}$ as $A^{+}_{\sigma\sigma}(\theta) =\Pi_\sigma(\theta)\tilde q_\infty(\theta)\Pi_\sigma(\theta)$, where $\tilde q_\infty(\theta)$ is defined in (3.6). Assertion 1 of Theorem 3.1 is proved.

Appendix B: The existence of Gibbs measures

We let $\mathbb{E}_\beta$ denote the expectation w.r.t. the measure $g_\beta$. By the Minlos theorem, to prove Lemma 4.1, it suffices to verify that for all $s,\alpha<-d/2$,

$$ \mathbb{E}_\beta(\|Y\|^2_{s,\alpha})\equiv\int\|Y\|^2_{s,\alpha}\, g_\beta(dY)\le C<\infty.$$

(B.1)

Because $g_\beta$ is translation invariant under shifts in $\mathbb{Z}^d$ and $\alpha<-d/2$, it follows that

$$\begin{aligned} \, &\mathbb{E}_\beta^{}(\|Y\|^2_{s,\alpha})\le C(\alpha,d)e^s_\beta, \\ &e^s_\beta:=\mathbb{E}_\beta^{}\biggl[\,\int_{\mathrm K_1^d}\bigl(|\Lambda^{s+1}\psi(y)|^2+|\Lambda^s\pi(y)|^2\bigr)\,dy +|u(0)|^2+|v(0)|^2\bigg], \end{aligned}$$

where the operators $\Lambda^s$ are introduced in (2.1).

Let $Y(p)=(\psi(x),u(k),\pi(x),v(k))$ be a random function with the distribution $g_\beta$. We let $C^s_\beta$ denote the correlation operator of the random function

$$Y^s:=\bigl((\Lambda^{s+1}\psi)(y)|_{y\in\mathrm K_1^d},u(0),(\Lambda^s\pi)(y)|_{y\in\mathrm K_1^d},v(0)\bigr)\in\mathcal H_0:= [\mathrm H^0(\mathrm K_1^d)\oplus \mathbb{R}^n]^2.$$

Then $e^s_\beta$ is equal to the trace of the operator $C^s_\beta$ in $\mathcal H_0$. We note that $C^s_\beta=\operatorname{Op}(q^s_\beta(0,r,r'))$ is an integral operator with the kernel $q^s_\beta(0,r,r')$, where

$$ q^s_\beta(0,r,r')=\frac{1}{(2\pi)^d}\int_{\mathrm K^d} e^{-\mathrm i\theta\cdot(r-r')}\tilde q^s_\beta(\theta,r,r')\,d\theta,\qquad r,r'\in\mathrm K_1^d\cup\{0\}.$$

(B.2)

We let $\tilde q^s_\beta(\theta):=\operatorname{Op}(\tilde q^s_\beta(\theta,r,r'))$ denote an integral operator with the kernel $\tilde q^s_\beta(\theta,r,r')$, where $\tilde q^s_\beta(\theta)=\bigl(\tilde q^{s,ij}_\beta(\theta)\bigr)_{i,j=0,1}$. Then $\tilde q^{s,ij}_\beta(\theta)=0$ for $i\neq j$ and

$$\tilde q^{s,ii}_\beta(\theta)=\widetilde{\mathbf{\Lambda}}^{s+1-i}(\theta)\tilde q^{ii}_\beta(\theta)\widetilde{\mathbf{\Lambda}}^{s+1-i}(\theta),\qquad i=0,1,$$

$\tilde q^{ii}_\beta(\theta)$ is introduced in (4.1) and (4.2), the operator $\widetilde{\mathbf{\Lambda}}^s(\theta)\colon\mathrm H_1^s\to \mathrm H^0_1$ is defined by the rule

$$\begin{aligned} \, &\widetilde{\mathbf{\Lambda}}^s(\theta)(\psi(\,{\cdot}\,),u):=(\widetilde{\Lambda}^s(\theta)\psi,u),\qquad (\psi,u)\in\mathrm H^s_1\equiv\mathrm H^s(\mathbb{T}_1^d)\oplus\mathbb{C}^n, \\ &(\widetilde{\Lambda}^s(\theta)\psi)(y):=F^{-1}_{l\to y}[\langle2\pi l+\theta\rangle^s\psi^\#(l)], \\ &\psi^\#(l):=F_{y\to l}[\psi(y)]=\int_{\mathbb{T}_1^d}e^{2\pi\mathrm i\,l\cdot y}\psi(y)\,dy,\qquad l\in\mathbb{Z}^d. \end{aligned}$$

Equation (B.2) implies that

$$e^s_\beta=\operatorname{tr}_{\mathcal H_0} C^s_\beta= \frac{1}{(2\pi)^d} \int_{\mathrm K^d}\operatorname{tr}_{\mathcal H_0}[e^{-\mathrm ir\cdot\theta}\tilde q^s_\beta(\theta)e^{\mathrm ir'\cdot \theta}]\,d\theta.$$

The operator $e^{-\mathrm ir\cdot\theta}\colon(\psi(y),u)\to (e^{-\mathrm iy\cdot\theta}\psi(y),u)$ is bounded in $\mathrm H_1^0$. In turn, $\tilde q^{s,ii}_\beta(\theta)$, $i=0,1$, are nonnegative self-adjoint operators in $\mathrm H_1^0$, because

$$\int_{\mathrm K^d} \bigl(\tilde q^{s,ii}_\beta(\theta),\widetilde Z_e(\theta,\,{\cdot}\,)\otimes\widetilde Z_e(\theta,\,{\cdot}\,)\bigr)\,d\theta= C\int|\langle\mathbf{\Lambda}^{s+1-i}Y,Z\rangle|^2\,g_\beta^{i}(dY)\ge 0$$

for all $Z\in D_{\mathrm F}\oplus D_{\mathrm L}$. Hence,

$$e^s_\beta\le C \int_{\mathrm K^d}\operatorname{tr}_{\mathrm H_1^0}[\tilde q^s_\beta(\theta)]\,d\theta= C \int_{\mathrm K^d} \operatorname{tr}_{\mathrm H^0_1} \bigl[\widetilde{\mathbf{\Lambda}}^s(\theta) \bigl(\widetilde{\mathbf{\Lambda}}(\theta)\tilde q^{00}_\beta(\theta)\widetilde{\mathbf{\Lambda}}(\theta) +\tilde q^{11}_\beta(\theta)\bigr) \widetilde{\mathbf{\Lambda}}^s(\theta)\bigr]\,d\theta,$$

where $\tilde q^{ii}_\beta(\theta)$ is defined in (4.1) and (4.2). The operator

$$\widetilde{\mathbf{\Lambda}}(\theta)\tilde q^{00}_\beta(\theta)\widetilde{\mathbf{\Lambda}}(\theta)= \beta^{-1}\widetilde{\mathbf{\Lambda}}(\theta)\widetilde{\mathcal H}^{-1}(\theta)\widetilde{\mathbf{\Lambda}}(\theta)$$

is bounded in $\mathrm H_1^0$ (uniformly in $\theta\in\mathrm K^d$) because $\|\widetilde{\mathcal H}^{-1}(\theta)Y\|_{\mathrm H_1^1}\le C\|Y\|_{\mathrm H_1^{-1}}$ for all $Y\in\mathrm H_1^{-1}$ (see Lemma 9.2 in [3]). In turn,

$$\int_{\mathrm K^d} \operatorname{tr}_{\mathrm H^0_1}[\widetilde{\mathbf{\Lambda}}^{2s}(\theta)]\,d\theta\le C\int_{\mathbb{R}^d}\langle\xi\rangle^{2s}\,d\xi<\infty,\qquad s<-\frac{d}{2}.$$

Then $e^s_\beta\le C\beta^{-1}<\infty$ by Theorem 1.6 in [21]. This implies the bound in (B.1).