Application of log-Sobolev inequality to the stochastic dynamics of unbounded spin systems on the lattice^{1}

Nobuo Yoshida^{2}

Abstract

We consider a ferromagnetic spin system with unbounded interactions on the ddimensional integer lattice (d ≥ 1) and investigate the convergence rate for the associated stochastic dynamics which is sometimes called the Glauber dynamics. We prove that the following two conditions are equivalent;

(1) The log-Sobolev inequality for the finite volume Gibbs states holds uniformly in both the volume and the boundary condition.

(2) The finite volume Glauber dynamics relaxes to equilibrium exponentially fast, uniformly in the volume whenever it starts from a tempared configuration.

The condition (1) above is known to be true, for example when d = 1, or the coupling constants are sufficiently small ([BH98, Y97, Z96]).

1 Introduction

We begin by introducing the standard setup of the model.

The lattice: We will work on the d-dimensional integer lattice Z^{d} = {x = (x^{i})_{i=1}^{d} : x^{i} ∈ Z}

on which we consider the l_{∞}-metric; d(x_{1}, x_{2}) = max_{1≤i≤d} |x_{1}^{i} − x_{2}^{i}| (x_{1}, x_{2} ∈ Z^{d}). For a set Λ ⊂ Z^{d},

diamΛ and |Λ| stand respectively for its diameter and the cardinality. We write Λ ⊂⊂ Z^{d} when

1 ≤ |Λ| < ∞. The distance between two subsets Λ_{1} and Λ_{2} of Z^{d} will be denoted by d(Λ_{1}, Λ_{2}).

For R ≥ 1, the R-boundary of a set Λ is defined by

∂_{R}Λ = {x ∉ Λ; d(x, Λ) ≤ R}.  (1.1)

The value of R will eventually be chosen as the range R(J) of the interaction we consider (See

(1.17) below).

The configuration spaces: The configuration spaces are defined as follows;

R^{Λ} = {σ = (σ_{x})_{x∈Λ} ; σ_{x} ∈ R}, Λ ⊂ Z^{d},

1Last updated on November 27, 1998 and printed out on January 5, 1999.

2Division of Mathematics, Graduate School of Science, Kyoto University, Kyoto 606-8502, Japan, [email]nobuo@kusm.kyoto-u.ac.jp, [URL]http://www.kusm.kyoto-u.ac.jp/ nobuo

Ω = R^{Z^{d}},
S = ⋂_{n≥1}{σ ∈ Ω; sup_{x∈Z^{d}}(1 + d(x, 0))^{n}|σ_{x}| < ∞},
S′ = ⋃_{n≥1}{σ ∈ Ω; sup_{x∈Z^{d}}(1 + d(x, 0))^{−n}|σ_{x}| < ∞}.

A configuration in S and S′ is called respectively, a rapidly decresing configuration and a tempared configuration. The functions of the configuration: Function spaces C and C_{Λ} (Λ ⊂ Z^{d}) on the configuration space Ω are introduced as follows;

C = {f : Ω ⟶ R | f satisfies the properties (C1) and (C2) below }.  (1.2)

(C1) There is Λ ⊂⊂ Z^{d} such that f depends only on (σ_{x})_{x∈Λ} and is of C^{1} with respect to these variables.

(C2)

|||f||| =^{def.} ∑_{x∈Z^{d}} ∥∇_{x}f∥ < ∞,  (1.3)

where

∥f∥ = sup_{σ∈Ω} |f(σ)|, ∇_{x}f(σ) = \frac{∂}{∂σ_{x}} f(σ)  (1.4)

For f ∈ C, we denote by S_{f} the minimal set among those Λ’s which satisfy the property referred to in (C1) above. We define

C_{Λ} = {f ∈ C ; S_{f} ⊂ Λ}, Λ ⊂ Z^{d}.  (1.5)

The Hamiltonian: We introduce a function U : R → R which satisfies (U1)–(U4) below.

(U1) For any m > 0, there exist V, W ∈ C^{2}(R → R) and C_{1.8} ∈ (0, ∞) such that

U(s) = V (s) + W (s) for all s ∈ R,  (1.6)
inf_{s} V′′(s) ≥ m,  (1.7)
sup_{s} |W (s)| ≤ C_{1.8},  (1.8)

(U2)

pq \frac{∂^{2}U}{∂p∂q}(p, q) ≥ 0 for all (p, q) ∈ R^{2},  (1.9)

where

U(p, q) = U (\frac{q+p}{\sqrt{2}}) + U (\frac{q−p}{\sqrt{2}}) .  (1.10)

(U3) There exists C_{1.11} ∈ (2, ∞) such that

inf_{s∈R}(U′(s)^{2} − C_{1.11}U′′(s) − ms^{2}) > −∞.  (1.11)

for any m ∈ (0, ∞).

(U4) There is an integer N_{1.12} ≥ 2 such that

sup_{s∈R}(1 + |s|)^{−N_{1.12}}|U(s)| < ∞.  (1.12)

A typical example of U which satisfies (U1) –(U4) is given by the following polynomial;

U(s) = ∑_{ν=1}^{N} a_{2ν}s^{2ν},  (1.13)

where N ≥ 2, a_{2} ∈ R, a_{4} ≥ 0, …, a_{2(N−1)} ≥ 0 and a_{2N} > 0. Since a_{2} can be large negative value, U in (1.13) may have arbitrarily deep double wells. Note that (U1) implies that inf_{s} U′′(s) > −∞ and hence that there exist C_{1.14}, C_{1.15} ∈ (0, ∞) such that for any (s, t) ∈ R^{2},

sU′(s) ≥ −C_{1.14}(1 + s^{2}),  (1.14)
(t − s)(U′(t) − U′(s)) ≥ −C_{1.15}(t − s)^{2}.  (1.15)

For Λ ⊂⊂ Z^{d} and ω ∈ Ω, we define a function H^{Λ,ω} : Ω → R, by;

H^{Λ,ω}(σ) = −\frac{1}{2} ∑_{x,y∈Λ} J_{x,y}σ_{x}σ_{y} + ∑_{x∈Λ} (U(σ_{x}) − h_{x}σ_{x} − ∑_{y∉Λ} J_{x,y}σ_{x}ω_{y}) .  (1.16)

Here, J = (J_{x,y} ∈ R ; x, y ∈ Z^{d}) and h = (h_{z} ∈ R ; z ∈ Z^{d}) are such that

R(J) =^{def.} sup{d(x, y) ; J_{x,y} ≠ 0} < ∞,  (1.17)
∥J∥ =^{def.} sup_{x} ∑_{y} |J_{x,y}| < ∞,  (1.18)

J_{x,y} = J_{y,x} ≥ 0 if x ≠ y,  (1.19)
λ(J) =^{def.} inf_{x}\{−\frac{1}{2} ∑_{y} J_{xy}\} > 0,  (1.20)
∥h∥ =^{def.} sup_{x} |h_{x}| < ∞,  (1.21)

Note that we have from (1.19) and (1.20) that for any Λ ⊂⊂ Z^{d} and σ ∈ R^{Λ}

−\frac{1}{2} ∑_{x,y∈Λ} J_{x,y}σ_{x}σ_{y} ≥ λ(J) ∑_{x∈Λ} |σ_{x}|^{2}.  (1.22)

Remark 1.1 We assume (U1) and (U2) to borrow some lemmas from [Y97] where these two assumptions are imposed. These assumptions can be relaxed in some cases. In fact, when J_{x,y} are small enough, it follows from the arguments in [BH98] that “for any m > 0” in (U1) can be replaced by “for some m > 0” and (U2) can be erased. (U3) and (U4) are more technical assumptions which will be used only in the proof of Lemma 3.4 below (cf. the proof of (8.6) and (8.14) in Section 8).

Remark 1.2 If a matrix J satisfies (1.17)–(1.19), we may also assume (1.20) without changing the model. In fact, define Ĵ_{x,y} = J_{x,y} − (∥J∥ + 1)δ_{x,y} and Û(s) = U(s) − \frac{∥J∥+1}{2}s^{2}. Then Ĵ_{x,y} satisfies (1.17)–(1.20) and Û satisfies (1.6)–(1.12) and the replacement of (J_{x,y}, U) by (Ĵ_{x,y}, Û) does not change the Hamiltonian (1.16).

The Local specifications and the DLR-state: For a topological space X, we let P(X) denote the set of Borel probability measures on X. For Λ ⊂⊂ Z^{d} and a boundary condition ω ∈ Ω, we define E^{Λ,ω} ∈ P(R^{Λ}) by;

E^{Λ,ω}(dσ_{Λ}) = \frac{exp −H^{Λ,ω}(σ)}{Z^{Λ,ω}} ∏_{x∈Λ} dσ_{x},  (1.23)

where Z^{Λ,ω} is the normalizing constant. The measure E^{Λ,ω} is called the finite volume Gibbs state and the family {E^{Λ,ω} | Λ ⊂⊂ Z^{d}, ω ∈ Ω} is called the local specification.

For ν ∈ P(Ω), we define a new measure νE^{Λ} ∈ P(Ω) by;

νE^{Λ}f = ∫ ν(dω)∫ E^{Λ,ω}(dσ)f(σ_{Λ} · ω_{Λ^{c}}),  (1.24)

where σ_{Λ} · ω_{Λ^{c}} denotes the following configuration;

(σ_{Λ} · ω_{Λ^{c}})_{x} = \{ \begin{matrix}σ_{x} & if x ∈ Λ, \\ ω_{x} & if x ∉ Λ. \end{matrix}

It is a common practice to regard the measure E^{Λ,ω}, which was originally defined as a measure on R^{Λ}, as a measure on the full configuration space Ω by identifying it with δ_{ω}E^{Λ}, where δ_{w} is the Dirac measure concentrated on ω. With this in mind, we introduce an integral operator E^{Λ} : C ⟶ C by;

E^{Λ}f(σ) = E^{Λ,σ}(f).  (1.25)

We now define two subsets G and G_{t} of P(Ω) as follows;

G = {ν ∈ P(Ω) ; νE^{Λ} = ν for any Λ ⊂⊂ Z^{d}},  (1.26)
G_{t} = G ∩ P_{t}(Ω),  (1.27)

where

P_{t}(Ω) = {ν ∈ P(Ω) ; (ν(|σ_{x}|))_{x∈Z^{d}} ∈ S′}.  (1.28)

A measure in G and G_{t} is called respectively, the DLR-state and the tempared DLR-state. The log-Sobolev constant: We define the log-Sobolev constant γ_{LS}(Λ, ω) for Λ ⊂ Z^{d} and ω ∈ Ω as the smallest γ for which the following inequality is true for all f ∈ C ;

E^{Λ,ω} (f^{2} log \frac{f^{2}}{E^{Λ,ω}(f^{2})}) ≤ γE^{Λ,ω} (∑_{x∈Λ} |∇_{x}f|^{2}) .  (1.29)

The stochastic dynamics. We introduce now for the model above, a random time evolution which is sometimes called the Glauber dynamics. Set

Θ = {θ = (θ_{t,x})_{t≥0, x∈Z^{d}} | (θ_{t,x})_{t≥0} ∈ C([0, ∞) → R) and θ_{0,x} = 0 for all x ∈ Z^{d}}(1.30)
P = the Wiener measure on Θ.  (1.31)

The second of these amounts to saying that (θ_{t,x})_{t≥0} (x ∈ Z^{d}) are independent standard Brownian motions under the probability measure P. For a set Λ ⊂ Z^{d} and ω ∈ Ω, consider the following stochastic differential equation (SDE) for the unknown process σ_{t}^{Λ,ω} = (σ_{t,x}^{Λ,ω})_{x∈Z^{d}};

σ_{t,x}^{Λ,ω} = \{ \begin{matrix}ω_{x} + θ_{t,x} − \frac{1}{2}∫ _{0}^{t}ds∇_{x}H^{Λ,ω}(σ_{s}^{Λ,ω}) & if x ∈ Λ, \\ ω_{x} & if x ∉ Λ. \end{matrix}  (1.32)

The existence and the uniqueness of the solution to (1.32) for ω ∈ S′ is well-known for Λ = Z^{d} as well as for Λ ⊂⊂ Z^{d} (cf. [DR78]). When Λ = Z^{d}, we drop the superscript Λ and the solution to (1.32) in this case will simply be denoted by σ_{t}^{ω}.

2 Results

We have the following results;

Theorem 2.1 (a) Suppose that

sup{γ_{LS}(Λ, ω) | Λ ⊂⊂ Z^{d}, ω ∈ Ω} < C_{2.1}  (2.1)

for some constant C_{2.1} ∈ (0, ∞). Then there exist t_{0} ∈ [0, ∞), C_{2.3} ∈ (0, ∞) and coefficients {B_{2.3}(f, ω) ≥ 0 | f ∈ C, ω ∈ S′} such that for all Λ ⊂⊂ Z^{d}, ω ∈ S′, f ∈ C_{Λ} and t ≥ t_{0};

σ_{Λ} ↦ B_{2.3}(f, σ_{Λ} · ω_{Λ^{c}}) ∈ L^{2}(E^{Λ,ω}),  (2.2)
|Pf(σ_{t}^{Λ,ω}) − E^{Λ,ω}f| ≤ B_{2.3}(f, ω) exp (−t/C_{2.3}) .  (2.3)

In fact, we can take t_{0} = 1, C_{2.3} = C_{2.1} and

B_{2.3}(f, ω) = C_{2.4} exp (diam(0 ∪ S_{f})) |||f||| ∑_{y∈Z^{d}} k_{y} (1 + |ω_{y}|^{2N_{1.12}}) ,  (2.4)

where C_{2.4} ∈ (0, ∞), k = (k_{x} ≥ 0)_{x∈Z^{d}} ∈ S and N_{1.12} ≥ 2 is an integer for which (1.12) is true.

(b) Conversely, if there exist t_{0} ∈ [0, ∞), C_{2.3} ∈ (0, ∞) and coefficients {B_{2.3}(f, ω) ≥ 0 | f ∈ C, ω ∈ S′} such that (2.2) and (2.3) are true, then (2.1) holds with some C_{2.1} ∈ (0, ∞).

We also have the following result, which says (2.3) and (2.4) are enough to guarantee the exponential relaxation of the process (σ_{t}^{ω})_{t≥0}, which evolves on the whole lattice Z^{d}.

Theorem 2.2 Suppose that there exist C_{2.5} ∈ (0, ∞), N_{2.5} ≥ 2, k = (k_{x} ≥ 0)_{x∈Z^{d}} ∈ S and {B_{2.5}(f) ∈ (0, ∞) | f ∈ C} such that the following holds for any t ≥ 1, Λ ⊂⊂ Z^{d}, f ∈ C_{Λ} and ω ∈ S′;

|Pf(σ_{t}^{Λ,ω}) − E^{Λ,ω}f| ≤ B_{2.5}(f) ∑_{y∈Z^{d}} k_{y} (1 + |ω_{y}|^{N_{2.5}}) exp (−t/C_{2.5})  (2.5)

Then, the set G_{t} of tempared DLR-states contains precisely one element μ and there exist C_{2.7} ∈ (0, ∞) and m = (m_{x} ≥ 0)_{x∈Z^{d}} ∈ S such that for any t ≥ 1, f ∈ C and ω ∈ S′;

|Pf(σ_{t}^{ω}) − μf| ≤ B_{2.7}(f) ∑_{y∈Z^{d}} m_{y} (1 + |ω_{y}|^{N_{2.5}}) exp (−t/C_{2.5})  (2.6)

where

B_{2.7}(f) = C_{2.7}B_{2.5}(f) + C_{2.7} exp (diam(0 ∪ S_{f})) |||f|||.  (2.7)

Remark 2.1 It is known that the uniform boundedness of the log-Sobolev constants (2.1) which we assume in Theorem 2.1 is equivalent to that there is C_{2.8} ∈ (0, ∞) such that for any Λ ⊂⊂ Z^{d}, ω ∈ Ω and x, y ∈ Λ,

|E^{Λ,ω}(σ_{x}; σ_{y})| ≤ C_{2.8} exp (−d(x, y)/C_{2.8}) .  (2.8)

Moreover, (2.1) is indeed the case at least in the following two examples; (a): d = 1 ([Z96, Theorem 4.1]), (b): J_{x,y} are small enough ([BH98, Theorem 4.1],[Y97, Theorem 2.3]). Condition (2.1), therefore, seems to be a reasonable assumption to work on.

Remark 2.2 Estimates of the type (2.6) are discussed by B. Zegarlinski [Z96, Section3] under weaker assumptions than are put in this paper. As compared with these results in [Z96], part (a) of Theorem 2.1 says that a stronger result follows from stronger, but reasonable assumptions.

Remark 2.3 If one assumes (2.1) in the compact spin case, the exponential relaxation to the equilibrium for the finite volume Glauber dynamics is uniform, not only over Λ but also over ω ([MO94, SZ92]). But when the spin space is unbounded, such extra uniformity turns out to fail even for the simplest example (U(s) = s^{2}). Therefore, an estimate of the type (2.3) seems to be more or less the best one can expect in this setting.

Remark 2.4 In both Theorem 2.1 and Theorem 2.2, the assumptions ((2.1) and (2.5)) are made uniformly over all Λ ⊂⊂ Z^{d}. However, as will be clear from the proof we present, we may relax (2.1) by requiring the uniformity only over well-shaped Λ’s (fat enough boxes, for example) to get (2.3) for such Λ’s. Also, if a sequence Λ_{n} ⊂⊂ Z^{d} (n = 1, 2, …) approximates Z^{d} regularly enough, then, (2.6) can be obtained just by requiring (2.5) uniformly over Λ = Λ_{n} (n = 1, 2, …).

3 Lemmas

In this section, we present some lemmas, which play key roles in proving Theorem 2.1 and Theorem 2.2.

Lemma 3.1 Suppose that ε ∈ (0, ∞) and p ≥ 1. Then the following hold;

(a) There exists C_{3.1} = C_{3.1}(p, ε) > 0 such that

P ( sup_{0≤s≤t} |σ_{s,x}^{Λ,ω}|^{2p})^{1/p} ≤ C_{3.1} exp (C_{3.1}t) ∑_{y∈Z^{d}} (1 + |ω_{y}|^{2}) exp −εd(x, y) (3.1)

for all t ≥ 0, x ∈ Z^{d} ω ∈ S′ and Λ ⊂⊂ Z^{d}. Furthermore, (3.1) is also true when Λ = Z^{d}.

(b) For any M ∈ (0, ∞), there exist B_{3.2} = B_{3.2}(p, ε, M) ∈ (0, ∞) and C_{3.3} = C_{3.3}(p, ε, M) ∈ (0, ∞) such that if x ∈ Γ ⊂ Λ ⊂⊂ Z^{d}, ξ ≡ η on Γ and

d(x, Γ^{c}) > R + B_{3.2}t,  (3.2)

then

P ( sup_{0≤s≤t} |σ_{s,x}^{Γ,ξ} − σ_{s,x}^{Λ,η}|^{2p})^{1/p}

≤ C_{3.3} exp (−Md(x, Γ^{c})) ∑_{y∈Z^{d}} (1 + |ξ_{y}|^{2} + |η_{y}|^{2}) exp −εd(x, y).  (3.3)

Furthermore, (3.3) is also true when Λ = Z^{d}.

Estimates of above type for compact spin cases are standard and even for unbounded spin cases, they do not seem to be entirely new (cf. [Z96, Section1]). However, we could not find in the litrature results which were stated strongly enough to be used in this paper, and for this reason we are going to present a self-contained proof of Lemma 3.1 in Section5.

We will need the following bounds for integrals with respect to Gibbs measures.

Lemma 3.2 (a) There exists s = (s_{y} ≥ 0)_{y∈Z^{d}} ∈ S such that

E^{Λ,ω} (|σ_{x}|^{p}) ≤ C_{3.4} + C_{3.4} ∑_{y∉Λ} s_{x−y}|ω_{y}|^{p}.  (3.4)

for all Λ ⊂⊂ Z^{d}, x ∈ Λ and p = 1, 2, …, where C_{3.4} ∈ (0, ∞) depends only on U, J, h and p = 1, 2, . . .. Furthermore, for any μ ∈ G_{t} and p = 1, 2, …,

sup_{y∈Z^{d}} μ(|ω_{y}|^{p}) < ∞.  (3.5)

(b) There exists ε_{0} ∈ (0, ∞) such that for any ε ∈ (0, ε_{0})

sup {∫ E^{Λ,ω} ⊗ E^{Λ,ω}(dσ_{Λ}dσ̃_{Λ}) exp (ε|σ_{x} − σ̃_{x}|^{2}) | Λ ⊂⊂ Z^{d}, ω ∈ Ω} < ∞.  (3.6)

It is possible to prove all estimates contained in Lemma 3.2 by repeating the argument in [BH82, Theorem III.2]. However, in Section6, we try to present another proof of these estimates by combinning results in [AMS94, AS94, BH82], which we hope is quicker than repeating the same argument in [BH82] and clearer than just referring the readers to [BH82].

Remark 3.1 It follows from [Y97, lemma 3.1] that (3.6) is true not only for small ε > 0 but for all ε > 0. But we do not need this observation in this paper.

Lemma 3.3 For ν ∈ P(R^{Λ}), define P_{t}^{Λ,ω} ∈ P(R^{Λ}) and νP_{t}^{Λ,ω} ∈ P(R^{Λ}) by

P_{t}^{Λ,ω} = P(σ_{t}^{Λ,ω} ∈ ·),  (3.7)
νP_{t}^{Λ,ω} = ∫ ν(dσ_{Λ})P_{t}^{Λ,σ_{Λ}·ω_{Λ^{c}}}  (3.8)

Then, there exists C_{3.9} ∈ (0, ∞) such that

|νP_{t}^{Λ,ω}f − E^{Λ,ω}f| ≤ C_{3.9}|||f||| (H(ν|E^{Λ,ω}) + H(ν|E^{Λ,ω})^{1/2}) exp −\frac{t}{γ_{LS}(Λ, ω)}  (3.9)

for any t > 0 and f ∈ C_{Λ}, where H(ν|μ) for probability measures μ, ν denotes the relative entropy;

H(ν|μ) = \{ \begin{matrix}ν(log \frac{dν}{dμ}), & if ν is absolutely continuous with respect to μ, \\ +∞, & otherwise. \end{matrix}  (3.10)

Inequality (3.9) with |||f||| replaced by ∥f∥ is well known and easy to prove (Just combine [DS89, page 76, (3.2.24)] and [DS89, page 250, (6.1.36)]). To make the estimate depend only on |||f|||, so that it can be applied to all f ∈ C, we have to work a little harder. The proof of Lemma 3.3 will be presented in Section7.

Lemma 3.4 For any ε ∈ (0, ∞), there exists C_{3.11} ∈ (0, ∞) such that

H(P_{t}^{Λ,ω}|E^{Λ,ω}) ≤ (1 + t + log(1/t)) C_{3.11}|Λ|
+ C_{3.11}|Λ| exp (C_{3.11}t + εdiam(0 ∪ Λ)) ∑_{y∈Z^{d}} (1 + |ω_{y}|^{2N_{1.12}}) exp −εd(0, y) (3.11)

for all Λ ⊂⊂ Z^{d} and t > 0, where N_{1.12} is an integer for which (1.12) holds.

Lemma 3.4 will be proved in Section8. To prove (3.11), we will make use of a rather explict expression of the Radon-Nikodym derivative \frac{dP_{t}^{Λ,ω}}{dE^{Λ,ω}} which will be obtained via Girsanov transformation (cf. (8.13) below). The idea of applying Girsanov’s theorem to the relative entropy can also be found in works of A. Ramirez [R98], A. Ramirez and S. Varadhan [RV96], however, for somewhat different purposes.

4 Proof of the theorems

In this section, we prove Theorem 2.1 and Theorem 2.2 by using some lemmas presented in Section3.

Proof of Theorem 2.1, part (a): Suppose that C_{2.1} ∈ (0, ∞) is such that (2.1) holds. We take f ∈ C and Λ ⊂⊂ Z^{d} such that S_{f} ⊂ Λ. We then consider the following subset of Λ:

Λ(t) = {x ∈ Λ | d(S_{f}, x) ≤ R + B_{4.1}t},  (4.1)

where B_{4.1} is a large enough constant which will be specified later in the proof. We begin by decomposing the left-hand-side of (2.3) as follows;

|Pf(σ_{t}^{Λ,ω}) − E^{Λ,ω}f| ≤ I_{1}(ω) + I_{2}(ω) + I_{3}(ω),  (4.2)

where

I_{1}(ω) = |Pf(σ_{t}^{Λ,ω}) − Pf(σ_{t}^{Λ(t),ω})|
I_{2}(ω) = |Pf(σ_{t}^{Λ(t),ω}) − E^{Λ(t),ω}f|
I_{3}(ω) = |E^{Λ(t),ω}f − E^{Λ,ω}f| .

To estimate I_{1}(ω), we will use Lemma 3.1, part (b). We take B_{3.2} and C_{3.2} in Lemma 3.1 for p = ε = M = 1 and set B_{4.1} = B_{3.2} + C_{2.1}^{−1} in (4.1). We then have by definition of Λ(t) that d(x, Λ(t)^{c}) ≥ R + B_{3.2}t for all x ∈ S_{f} and hence by Lemma 3.1 that

P (|σ_{t,x}^{Λ(t),ω} − σ_{t,x}^{Λ,ω}|) ≤ C_{3.3} exp (−d(x, Λ(t)^{c})) ∑_{y∈Z^{d}} (1 + ω_{y}^{2}) exp −d(x, y)
≤ C_{3.3} exp (d(0, x) − B_{4.1}t) ∑_{y∈Z^{d}} (1 + ω_{y}^{2}) exp −d(0, y)
≤ C_{3.3} exp (d(0, x) − \frac{t}{C_{2.1}} ) ∑_{y∈Z^{d}} (1 + ω_{y}^{2}) exp −d(0, y).

Therefore, we obtain

I_{1}(ω) ≤ ∑_{x∈S_{f}} ∥∇_{x}f∥P (|σ_{t,x}^{Λ(t),ω} − σ_{t,x}^{Λ,ω}|)
≤ C_{3.3} ∑_{x∈S_{f}} ∥∇_{x}f∥ exp (d(0, x) − \frac{t}{C_{2.1}} ) ∑_{y∈Z^{d}} (1 + ω_{y}^{2}) exp −d(0, y)
≤ C_{3.3}|||f||| exp (diam(0 ∪ S_{f}) − \frac{t}{C_{2.1}} ) ∑_{y∈Z^{d}} (1 + ω_{y}^{2}) exp −d(0, y).  (4.3)

We next turn to the estimate of I_{2}(ω). We will use Lemma 3.3 and Lemma 3.4 as follows. We have from the Chapman-Kolmogorov equation that

Pf(σ_{t}^{Λ(t),ω}) = ∫ P_{1}^{Λ(t),ω}(dσ)P_{t−1}^{Λ(t),σ}(f)  (4.4)

for t ≥ 1. We see from this and (3.9) that

I_{2}(ω) ≤ C_{3.9}|||f||| (H(P_{1}^{Λ(t),ω}|E^{Λ(t),ω}) + H(P_{1}^{Λ(t),ω}|E^{Λ(t),ω})^{1/2}) exp −\frac{t − 1}{γ_{LS}(Λ(t), ω)}. (4.5)

On the other hand, we have from Lemma 3.4 that

H(P_{1}^{Λ(t),ω}|E^{Λ(t),ω})
≤ C_{4.6}|||f||| exp (εdiam(0 ∪ S_{f}) + 4εB_{4.1}t) ∑_{y∈Z^{d}} (1 + |ω_{y}|^{2N_{1.12}}) exp −εd(0, y), (4.6)

where C_{4.6} ∈ (0, ∞) depends only on d, B_{4.1} and ε. Plugging this into (4.5) and taking ε ∈ (0, 1) so small that 2εB_{4.1} − (sup_{Λ,ω} γ_{LS}(Λ, ω))^{−1} ≤ −C_{2.1}^{−1}, we have

I_{2}(ω) ≤ C_{4.7}|||f||| exp (εdiam(0 ∪ S_{f}) − \frac{t}{C_{2.1}} ) ∑_{y∈Z^{d}} (1 + |ω_{y}|^{2N_{1.12}}) exp −εd(0, y)/2. (4.7)

To get an upper bound for I_{3}(ω), we further devide it into three parts as follows;

I_{3}(ω) ≤ ∫ E^{Λ,ω}(dσ_{Λ})I_{4}(σ_{Λ}) + ∫ E^{Λ,ω}(dσ_{Λ})I_{5}(σ_{Λ}) + ∫ E^{Λ,ω}(dσ_{Λ})I_{6}(σ_{Λ}),  (4.8)

where

I_{4}(σ_{Λ}) = |E^{Λ(t),ω}f − Pf(σ_{t}^{Λ(t),σ_{Λ(t)}·ω_{Λ(t)^{c}}})| ,
I_{5}(σ_{Λ}) = |Pf(σ_{t}^{Λ(t),σ_{Λ(t)}·ω_{Λ(t)^{c}}}) − Pf(σ_{t}^{Λ,σ_{Λ}·ω_{Λ^{c}}})| ,
I_{6}(σ_{Λ}) = |Pf(σ_{t}^{Λ,σ_{Λ}·ω_{Λ^{c}}}) − E^{Λ,ω}f| .

To handle the first two integrals on the right-hand side of (4.8), we will use Lemma 3.2. Note that I_{4}(σ_{Λ}) = I_{2}(σ_{Λ(t)} · ω_{Λ(t)^{c}}). We then have from (4.7) that

I_{4}(σ_{Λ}) ≤ C_{4.7}(f) exp (−t/C_{2.1}) ∑_{y∈Λ(t)} (1 + |σ_{y}|^{N_{1.12}}) exp −εd(0, y)/2
+ C_{4.7}(f) exp (−t/C_{2.1}) ∑_{y∉Λ(t)} (1 + |ω_{y}|^{N_{1.12}}) exp −εd(0, y)/2.  (4.9)

By (3.4), we see that

∑_{y∈Λ} E^{Λ,ω} (|σ_{y}|^{N_{1.12}}) exp −εd(0, y)/2

≤ C_{4.10} + C_{4.10} ∑_{y∈Λ} (∑_{z∉Λ} s_{y−z}|ω_{z}|^{N_{1.12}} ) exp −εd(0, y)/2
= C_{4.10} + ∑_{z∈Z^{d}} t_{z}|ω_{z}|^{N_{1.12}},  (4.10)

where we have defined (s_{z})_{z∈Z^{d}} ∈ S by

t_{z} = C_{4.10} ∑_{y∈Z^{d}} s_{y−z} exp −εd(0, y)/2.

From this and (4.9), we obtain

∫ E^{Λ,ω}(dσ_{Λ})I_{4}(σ_{Λ}) ≤ C_{4.7}(f) exp (−t/C_{2.1}) (C_{4.11} + ∑_{z∈Z^{d}} t_{z}|ω_{z}|^{N_{1.12}} )
+ C_{4.7}(f) exp (−t/C_{2.1}) ∑_{y∈Z^{d}} (1 + |ω_{y}|^{N_{1.12}}) exp −εd(0, y)/2. (4.11)

Finally, we consider the second integral on the right-hand side of (4.8). We see from the proof of (4.3) that

I_{5}(σ_{Λ}) ≤ C_{3.3}|||f||| exp (diam(0 ∪ S_{f}) − \frac{t}{C_{2.1}} ) ∑_{y∈Λ(t)} (1 + σ_{y}^{2}) exp −d(0, y)
+ C_{3.3}|||f||| exp (diam(0 ∪ S_{f}) − \frac{t}{C_{2.1}} ) ∑_{y∉Λ(t)} (1 + ω_{y}^{2}) exp −d(0, y). (4.12)

Therefore, repeating the argument which we pass from (4.9) to (4.11) by, we end up with the following estimate;

∫ E^{Λ,ω}(dσ_{Λ})I_{5}(σ_{Λ})

≤ C_{3.3}|||f||| exp (diam(0 ∪ S_{f}) − \frac{t}{C_{2.1}} ) ∑_{y∈Z^{d}} (C_{4.13} + ∑_{z∈Z^{d}} u_{z}|ω_{z}|^{2})
+ C_{3.3}|||f||| exp (diam(0 ∪ S_{f}) − \frac{t}{C_{2.1}} ) ∑_{y∈Z^{d}} (1 + ω_{y}^{2}) exp −d(0, y),  (4.13)

where we have defined (u_{z})_{z∈Z^{d}} ∈ S by

u_{z} = C_{3.3} ∑_{y∈Z^{d}} s_{y−z} exp −d(0, y).

The third integral on the right-hand side of (4.8) can be estimated as follows;

∫ E^{Λ,ω}(dσ_{Λ})I_{6}(σ_{Λ}) ≤ (∫ E^{Λ,ω}(dσ_{Λ})I_{6}(σ_{Λ})^{2})^{1/2}
≤ E^{Λ,ω} (|f − E^{Λ,ω}f|^{2})^{1/2} exp (−t/C_{2.1}) .  (4.14)

Furthermore, we have by (3.6) that

E^{Λ,ω} (|f − E^{Λ,ω}f|^{2})^{1/2} = (\frac{1}{2}∫ E^{Λ,ω} ⊗ E^{Λ,ω}(dσ_{Λ}dσ̃_{Λ}) (f(σ) − f(σ̃))^{2})^{1/2}
≤ ∑_{x∈S_{f}} ∥∇_{x}f∥ (\frac{1}{2}∫ E^{Λ,ω} ⊗ E^{Λ,ω}(dσ_{Λ}dσ̃_{Λ}) |σ_{x} − σ̃_{x}|^{2})^{1/2}
≤ C_{4.15}|||f|||,  (4.15)

and therefore that

∫ E^{Λ,ω}(dσ_{Λ})I_{6}(σ_{Λ}) ≤ C_{4.15}|||f||| exp (−t/C_{2.1}) .  (4.16)

We now obtain (2.3) by (4.2), (4.3), (4.7), (4.8), (4.11), (4.13) and (4.16). □

Proof of Theorem 2.1, part (b): Suppose that (2.2) and (2.3) hold. We then have that

∫ E^{Λ,ω}(dσ_{Λ}) |Pf(σ_{t}^{Λ,σ_{Λ}·ω_{Λ^{c}}}) − E^{Λ,ω}f|^{2} ≤ exp (−2t/C_{2.3}) ∫ E^{Λ,ω}(dσ_{Λ})B_{2.3}(f, σ_{Λ} · ω_{Λ^{c}})^{2}.

This and the argument in [Ho85, Lemma 1.13] implies that the following inequality is true for all Λ ⊂⊂ Z^{d}, f ∈ C_{Λ} and ω ∈ Ω;

E^{Λ,ω} (f; f) ≤ C_{2.3}E^{Λ,ω} (∑_{x∈Λ} |∇_{x}f|^{2}) .  (4.17)

By the result of [Y98], (4.17) is equivalent to (2.1) for some C_{2.1} ∈ (0, ∞). □

Proof of Theorem 2.2: Suppose (2.5) is true. We are going to prove that (2.6) holds for any μ ∈ G_{t}, from which the uniqueness of G_{t} also follows.

The proof of (2.6) is somehow, similar to that of (2.3). We take f ∈ C and ω ∈ S′ arbitrarily. We then consider the following subset of Z^{d}:

Λ(t) = {x ∈ Z^{d} | d(S_{f}, x) ≤ R + B_{4.18}t},  (4.18)

where B_{4.18} is a large enough constant which will be specified later in the proof. We begin by decomposing the left-hand-side of (2.6) as follows;

Pf(σ_{t}^{ω}) − μf = J_{1}(ω) + J_{2}(ω) + J_{3}(ω),  (4.19)

where

J_{1}(ω) = Pf(σ_{t}^{ω}) − Pf(σ_{t}^{Λ(t),ω})
J_{2}(ω) = Pf(σ_{t}^{Λ(t),ω}) − E^{Λ(t),ω}f
J_{3}(ω) = E^{Λ(t),ω}f − μf.

To estimate J_{1}(ω), we will use part(b) of Lemma 3.1. We take B_{3.2} and C_{3.2} in Lemma 3.1 for p = M = 1 and set B_{4.18} = B_{3.2} + C_{2.5}^{−1} in (4.18). Then, by the argument which led to (4.3), we obtain that

|J_{1}(ω)|
≤ C_{3.3}|||f||| exp (diam(0 ∪ S_{f}) − \frac{t}{C_{2.5}} ) ∑_{y∈Z^{d}} (1 + ω_{y}^{2}) exp −d(0, y).  (4.20)

On the other hand, we have by (2.5) that

|J_{2}(ω)| ≤ B_{2.5}(f) ∑_{y∈Z^{d}} k_{y} (1 + |ω_{y}|^{N_{2.5}}) exp (−t/C_{2.5})  (4.21)

To get an upper bound for J_{3}(ω), we further devide it into three parts as follows;

J_{3}(ω) = ∫ μ(dσ)J_{4}(σ) + ∫ μ(dσ)J_{5}(σ) + ∫ μ(dσ)J_{6}(σ),  (4.22)

where

J_{4}(σ) = E^{Λ(t),ω}f − Pf(σ_{t}^{Λ(t),σ_{Λ(t)}·ω_{Λ(t)^{c}}})
J_{5}(σ) = Pf(σ_{t}^{Λ(t),σ_{Λ(t)}·ω_{Λ(t)^{c}}}) − Pf(σ_{t}^{Λ(t),σ})
J_{6}(σ) = Pf(σ_{t}^{Λ(t),σ}) − μf

Note that J_{4}(σ) = J_{2}(σ_{Λ(t)} · ω_{Λ(t)^{c}}). We then have from (4.21) and (3.5) that

∫ μ(dσ)|J_{4}(σ)| ≤ B_{2.5}(f) (C_{4.23} + ∑_{y∉Λ(t)} k_{y}|ω_{y}|^{N_{2.5}} ) exp (−t/C_{2.5})  (4.23)

for some C_{4.23} ∈ (0, ∞). We next consider the second integral on the right-hand side of (4.22). We see from the proof of (4.3) that

|J_{5}(σ)| ≤ C_{3.3}|||f||| exp (diam(0 ∪ S_{f}) − \frac{t}{C_{2.5}} ) ∑_{y∈Λ(t)} (1 + σ_{y}^{2}) exp −d(0, y)
+ C_{3.3}|||f||| exp (diam(0 ∪ S_{f}) − \frac{t}{C_{2.5}} ) ∑_{y∉Λ(t)} (1 + ω_{y}^{2}) exp −d(0, y) (4.24)

and therefore by (3.5) that

∫ μ(dσ)|J_{5}(σ)|
≤ C_{3.3}|||f||| exp (diam(0 ∪ S_{f}) − \frac{t}{C_{2.5}} ) (C_{4.25} + ∑_{y∉Λ(t)} (1 + ω_{y}^{2})) exp −d(0, y). (4.25)

for some C_{4.25} ∈ (0, ∞).

The estimate of the third integral on the right-hand side of (4.22) can be done by use of the DLR-equation, (4.21) and (3.5) as follows;

|∫ μ(dσ)J_{6}(σ)| = |∫ μ(dσ) (Pf(σ_{t}^{Λ(t),σ}) − E^{Λ(t),σ}f)|
= |∫ μ(dσ)J_{2}(σ)|
≤ C_{4.26}B_{2.5}(f) exp −\frac{t}{C_{2.5}} .  (4.26)

We now obtain (2.6) by (4.19), (4.20), (4.21), (4.22), (4.23), (4.25) and (4.26). □

5 Proof of Lemma 3.1

In this section, we prove Lemma 3.1. This will be done as an application of elementary stochastic calculus and the following

Lemma 5.1 For ε ∈ (0, ∞), define

l(ε) = {α = (α_{x} ≥ 0)_{x∈Z^{d}} | ∥α∥_{ε} =^{def.} ∑_{x∈Z^{d}} α_{x} exp(εd(0, x)) < ∞}  (5.1)
l(−ε) = {α = (α_{x} ≥ 0)_{x∈Z^{d}} | ∥α∥_{−ε} =^{def.} ∑_{x∈Z^{d}} α_{x} exp −εd(0, x) < ∞}  (5.2)
α ∗ β = ( ∑_{y∈Z^{d}} α_{x−y}β_{y})_{x∈Z^{d}} , for α ∈ l(ε) and β ∈ l(−ε).  (5.3)

Then the following hold;

(a) If α ∈ l(ε), β ∈ l(−ε) and f_{t} ∈ l(−ε), (t ≥ 0) are such that

sup_{0≤s≤t} ∑_{y∈Z^{d}} f_{s,y} exp −εd(x, y) < ∞,  (5.4)
f_{t,x} ≤ β_{x} + ∫ _{0}^{t}ds(α ∗ f_{s})_{x}  (5.5)

for all t ≥ 0 and x ∈ Z^{d}. Then,

f_{t,x} ≤ exp(t∥α∥_{ε}) ∑_{y∈Z^{d}} |β_{y}| exp −εd(x, y)  (5.6)

for all x ∈ Z^{d} and t ≥ 0.

(b) If α ∈ l(ε), β ∈ l(−ε), R ≥ 1, Γ ⊂⊂ Z^{d} and g_{t} = (g_{t,x})_{x∈Z^{d}} ∈ l(−ε) (t ≥ 0) are such that

α_{x} = 0, if d(0, x) ≥ R  (5.7)
g_{t,x} ≤ ∫ _{0}^{t}ds(α ∗ g_{s})_{x} if d(x, Γ^{c}) > R  (5.8)

Then, for any M > 0, there exist B_{5.10} ∈ (0, ∞) and C_{5.10} ∈ (0, ∞) which depend only on M, R and ∥α∥_{ε} such that if

d(x, Γ^{c}) > R + B_{5.9}t,  (5.9)

then

g_{t,x} ≤ C_{5.10} exp(−Md(x, Γ^{c})) sup_{0≤s≤t} ∑_{y∈Z^{d}} g_{s,y} exp −εd(x, y).  (5.10)

Proof: We begin by proving that for α ∈ l(ε), β ∈ l(−ε) and x ∈ Z^{d}

|((α∗)^{n}β)_{x}| ≤ ∥α∥_{ε}^{n} ∑_{y∈Z^{d}} β_{y} exp −εd(x, y), n = 1, 2, ….  (5.11)

where (α∗)^{n}β = \underbrace{α ∗ ⋯ ∗ α∗}_{n} β. Inequality (5.11) can be proves as follows. It is easy to prove that

∥(α∗)^{n}β∥_{−ε} ≤ ∥α∥_{ε}^{n}∥β∥_{−ε}, n = 1, 2, ….  (5.12)

We now introduce the shift transformation;

τ_{y} : (α_{x})_{x∈Z^{d}} ↦ (α_{x+y})_{x∈Z^{d}}, y ∈ Z^{d}.  (5.13)

Then, α ∗ (τ_{y}β) = τ_{y}(α ∗ β) and hence (α∗)^{n}(τ_{y}β) = τ_{y}((α∗)^{n}β). From this and (5.12), we see that

|((α∗)^{n}β)_{x}| = |(τ_{x}(α∗)^{n}β)_{0}|
≤ ∥τ_{x}(α∗)^{n}β∥_{−ε}
= ∥(α∗)^{n}τ_{x}β∥_{−ε}
≤ ∥α∥_{ε}^{n}∥τ_{x}β∥_{−ε},

which implies (5.11).

To prove (5.6), we iterate (5.5) (n − 1)-times to obtain that

f_{t,x} ≤ ∑_{m=0}^{n−1} \frac{t^{m}}{m!}((α∗)^{m}β)_{x} + ∫ _{0}^{t}dt_{n−1}∫ _{0}^{t_{n−1}}dt_{n−2} ⋯ ∫ _{0}^{t_{1}}ds((α∗)^{n}f_{s})_{x}
≤ exp(t∥α∥_{ε}) ∑_{y∈Z^{d}} |β_{y}| exp −εd(x, y)
+ (t^{n}∥α∥_{ε}^{n}/n!) sup_{0≤s≤t} ∑_{y∈Z^{d}} f_{s,y} exp −εd(x, y),  (5.14)

where we have used (5.11) to proceed to the second line. Since the second term on the righthand-side of (5.14) vanishes as n ↗ ∞, we obtain (5.6).

Now we turn to the proof of part(c). We set n = ⌊\frac{d(x,Γ^{c})}{R+1}⌋ so that

\frac{d(x,Γ^{c})}{R+1} − 1 < n < \frac{d(x,Γ^{c})}{R}.  (5.15)

By (5.7) and (5.15), we can iterate (5.8) (n − 1)-times to obtain that

g_{t,x} ≤ ∫ _{0}^{t}dt_{n−1}∫ _{0}^{t_{n−1}}dt_{n−2} ⋯ ∫ _{0}^{t_{1}}ds((α∗)^{n}g_{s})_{x}
≤ (t^{n}∥α∥_{ε}^{n}/n!) sup_{0≤s≤t} ∑_{y∈Z^{d}} g_{s,y} exp −εd(x, y),  (5.16)

where we have used (5.11) in the second line. Therefore, it remains for us to find B_{5.9} and C_{5.10} which depend only on M, R and ∥α∥_{ε} such that if (5.9) holds, then

t^{n}∥α∥_{ε}^{n}/n! ≤ C_{5.10} exp −Md(x, Γ^{c}).  (5.17)

We see from (5.9) and (5.15) that

t/n ≤ d(x, Γ^{c})/(B_{5.9}n) ≤ 2(R + 1)/B_{5.9}.  (5.18)

We now choose B_{5.9} so large that

2(R + 1)/B_{5.9} ≤ (1/e∥α∥_{ε}) exp(−(R + 1)M).

We then have by Stirling’s formula and (5.15) that

t^{n}∥α∥_{ε}^{n}/n! ≤ C_{5.19} (te∥α∥_{ε}/n)^{n} ≤ C_{5.19} exp (M(R + 1) − Md(x, Γ^{c})) ,  (5.19)

which proves (5.17) and hence (5.10). □

Proof of (3.1): We will prove (3.1) for the case Λ ⊂⊂ Z^{d}. The case Λ = Z^{d} then follows from the uniformity of the estimate. We let ∥ · ∥_{L^{p}} stand for the L^{p}(P)-norm and define f_{t} = (f_{t,x})_{x∈Z^{d}} by

f_{t,x} = 1 + ∥ sup_{0≤s≤t} |σ_{s,x}^{Λ,ω}|^{2}∥_{L^{p}}  (5.20)

To prove (3.1) by applying (5.6) to f_{t} defined above, we will check (5.4) and (5.5) for some α.

Inequality (5.5) can be seen as follows. Note first that there exists C_{5.21} ∈ (0, ∞) which depends only on ∥J∥, ∥h∥ and C_{1.14} (cf. (1.14)) such that

−σ_{x}∇_{x}H^{Λ,ω}(σ) + 1 ≤ C_{5.21} ∑_{y∈Λ_{d(x,y)≤R}}(1 + |σ_{y}|^{2}) + C_{5.21} ∑_{y∉Λ_{d(x,y)≤R}}(1 + |ω_{y}|^{2}).  (5.21)

We now define processes M_{t,x}^{Λ,ω} and A_{t,x}^{Λ,ω} by

M_{t,x}^{Λ,ω} = ∫ _{0}^{t}σ_{s,x}^{Λ,ω}dθ_{s,x},
A_{t,x}^{Λ,ω} = C_{5.21} ∑_{d(x,y)≤R} ∫ _{0}^{t}ds (1 + |σ_{s,y}^{Λ,ω}|^{2}) .

We then have by Ito’s formula and (5.21) that

|σ_{t,x}^{Λ,ω}|^{2} = ω_{x}^{2} + 2M_{t,x}^{Λ,ω} + ∫ _{0}^{t}ds (−σ_{s,x}^{Λ,ω}∇_{x}H^{Λ,ω}(σ_{s}^{Λ,ω}) + 1)
≤ ω_{x}^{2} + 2M_{t,x}^{Λ,ω} + A_{t,x}^{Λ,ω}.  (5.22)

The L^{p}-norm of M_{t,x}^{Λ,ω} can be estimated as follows;

∥ sup_{0≤s≤t} M_{s,x}^{Λ,ω}∥_{L^{p}} ≤ C_{5.23} ∥(∫ _{0}^{t}ds|σ_{s,x}^{Λ,ω}|^{2})^{1/2}∥_{L^{p}}
≤ C_{5.23} ∥(∫ _{0}^{t}ds|σ_{s,x}^{Λ,ω}|^{2})∥_{L^{p}}^{1/2}
≤ C_{5.23} (∫ _{0}^{t}ds ∥|σ_{s,x}^{Λ,ω}|^{2}∥_{L^{p}})^{1/2}
≤ \frac{1}{2}C_{5.23} + \frac{1}{2}C_{5.23}∫ _{0}^{t}ds ∥|σ_{s,x}^{Λ,ω}|^{2}∥_{L^{p}} ,  (5.23)

where we have used the Burkholder-Davis-Gundy inequality (cf.[RY91, page 151, (4.1)]) in the first line, and therefore, C_{5.23} ∈ (0, ∞) depends only on p. On the other hand, we have that

∥ sup_{0≤s≤t} |A_{s,x}^{Λ,ω}|∥_{L^{p}} ≤ C_{5.21} ∑_{d(x,y)≤R} ∫ _{0}^{t}ds (1 + ∥|σ_{s,y}^{Λ,ω}|^{2}∥_{L^{p}}) .  (5.24)

Putting (5.22), (5.23) and (5.24) together, we see that

∥ sup_{0≤s≤t} |σ_{t,x}^{Λ,ω}|^{2}∥_{L^{p}} ≤ ω_{x}^{2} + C_{5.23} + C_{5.25} ∑_{d(x,y)≤R} ∫ _{0}^{t}ds (1 + ∥|σ_{s,y}^{Λ,ω}|^{2}∥_{L^{p}}) ,  (5.25)

where C_{5.25} = C_{5.23} + C_{5.21}. This proves (5.4) for f_{t} in (5.20) with β_{x} = 1 + C_{5.23} + |ω_{x}|^{2} and α_{x} = C_{5.25}1_{{d(0,x)≤R}}.

Inequality (5.5) can also be verified by use of (5.25) as follows. By taking summation of both hand side of (5.25) over x ∈ Λ, and using the notation in (5.20), we obtain the following inequality,

∑_{x∈Λ} f_{t,x} ≤ (1 + tC_{5.26}) ∑_{y∈Λ∪∂_{R}Λ}(1 + ω_{y}^{2}) + C_{5.26} ∑_{y∈Λ} ∫ _{0}^{t}dsf_{s,y}  (5.26)

which, via Gronwall’s inequality implies that

∑_{x∈Λ} f_{t,x} ≤ (1 + tC_{5.26}) ∑_{y∈Λ∪∂_{R}Λ}(1 + ω_{y}^{2}) exp (tC_{5.26})
≤ C_{5.27} ∑_{y∈Λ∪∂_{R}Λ}(1 + ω_{y}^{2}) exp (tC_{5.27}) .  (5.27)

This implies (5.4), since

∑_{y∈Z^{d}} f_{t,x} exp −εd(x, y) = ∑_{y∉Λ} (1 + |ω_{y}|^{2}) exp −εd(x, y) + ∑_{y∈Λ} f_{t,x} exp −εd(x, y).

This completes the proof of (3.1). □

Proof of (3.3): We will prove (3.3) for the case Λ ⊂⊂ Z^{d}. The case Λ = Z^{d} then follows from the uniformity of the estimate. Define g_{t} = (g_{t,x})_{x∈Z^{d}} by

g_{t,x} = ∥ sup_{0≤s≤t} |σ_{s,x}^{Γ,ξ} − σ_{s,x}^{Λ,η}|^{2}∥_{L^{p}} .  (5.28)

We are going to prove (3.3) by applying (5.10) to g_{t} defined above. Note first that we have by (3.1) that

sup_{0≤s≤t} ∑_{y∈Z^{d}} g_{s,y} exp −εd(x, y) ≤ 2C_{3.1} exp (C_{3.1}t) ∑_{y∈Z^{d}} (1 + |ξ_{y}|^{2} + |η_{y}|^{2}) exp −εd(x, y) (5.29)

for t ≥ 0 and x ∈ Z^{d}. Therefore, the proof will be finished if we can find some α ∈ l(ε) which satisfies (5.7) and (5.8).

We assume d(x, Γ^{c}) > R from now on. We then see that there exists C_{5.30} ∈ (0, ∞) which depends only on ∥J∥ and C_{1.15} (cf. (1.15)) such that

−(σ_{x} − τ_{x})(∇_{x}H^{Γ,ξ}(σ) − ∇_{x}H^{Λ,η}(τ)) ≤ C_{5.30} ∑_{y:d(x,y)≤R} |σ_{y} − τ_{y}|^{2}.  (5.30)

Since ξ ≡ η on Γ, we have that

σ_{s,x}^{Γ,ξ} − σ_{s,x}^{Λ,η} = −\frac{1}{2}∫ _{0}^{t}ds (∇_{x}H^{Γ,ξ}(σ_{s}^{Γ,ξ}) − ∇_{x}H^{Λ,η}(σ_{s}^{Λ,η}))

and hence that

|σ_{t,x}^{Γ,ξ} − σ_{t,x}^{Λ,η}|^{2} = −∫ _{0}^{t}ds (σ_{s,x}^{Γ,ξ} − σ_{s,x}^{Λ,η}) (∇_{x}H^{Γ,ξ}(σ_{s}^{Γ,ξ}) − ∇_{x}H^{Λ,η}(σ_{s}^{Λ,η}))
≤ C_{5.30} ∑_{y:d(x,y)≤R} ∫ _{0}^{t}ds|σ_{s,y}^{Γ,ξ} − σ_{s,y}^{Λ,η}|^{2}

by (5.30). We therefore obtain

g_{t,x} ≤ C_{5.30} ∑_{y:d(x,y)≤R} ∫ _{0}^{t}dsg_{s,y},

which proves (5.8) with α_{z} = C_{5.30}1_{{d(0,z)≤R}}. □

6 Proof of Lemma 3.2

In this section, we prove Lemma 3.2. Proof of Lemma 3.2: We first remark the follwong fact which can be seen from [Y97, Lemma 3.2]; there exist C_{6.1} ∈ (0, ∞) such that

E^{Λ,ω} (f^{2} log \frac{f^{2}}{E^{Λ,ω}(f^{2})}) ≤ C_{6.1}E^{Λ,ω} (|∇_{x}f|^{2}) .  (6.1)

whenever x ∈ Λ ⊂⊂ Z^{d}, ω ∈ Ω and f ∈ C_{{x}}.

Let us next recall that the following estimate due to Bellissard and Hϕegh-Krohn [BH82], which is nothing but (3.4) for p = 1; there exists s = (s_{y} ≥ 0)_{y∈Z^{d}} ∈ S such that

E^{Λ,ω} (|σ_{x}|) ≤ C_{6.2} + C_{6.2} ∑_{y∉Λ} s_{x−y}|ω_{y}|.  (6.2)

whenever x ∈ Λ ⊂⊂ Z^{d} and ω ∈ S′, where C_{3.4} ∈ (0, ∞) depends only on U, J and h. This follows immediately from [BH82], page 304, display (III.7). An elegant moment estimate due to Aida and Stroock [AS94] allows us to conclude (3.4) for p ≥ 2 from (6.1) and (6.2) as follows. We have by [AS94, Theorem 3.4] and Poincaré inequality that

E^{Λ,ω} (|σ_{x}|^{p}) ≤ (E^{Λ,ω}(|σ_{x}|^{2}) + (p − 2)C_{6.1})^{p/2}
≤ (E^{Λ,ω}(σ_{x})^{2} + (p − 1)C_{6.1})^{p/2} .  (6.3)

Then, using (6.2) and (6.3) we obtain

E^{Λ,ω} (|σ_{x}|^{p}) ≤ (C_{6.4} ( ∑_{y∈Z^{d}} s_{x−y}|ω_{y}|)^{2} + C_{6.4})^{p/2}  (6.4)
≤ C_{6.5} ( ∑_{y∈Z^{d}} s_{x−y}|ω_{y}|)^{p} + C_{6.5}  (6.5)
≤ C_{6.6} ∑_{y∈Z^{d}} s_{x−y}|ω_{y}|^{p} + C_{6.5}.  (6.6)

Our proof of (3.6) uses an exponential integrability estimate due to Aida, Masuda and Shigekawa [AMS94]. From (6.1) and the invariance of the log-Sobolev constant under taking the product of same measures, we see that the measure E^{Λω}(σ_{x} ∈ ·) ⊗ E^{Λω}(σ̃_{x} ∈ ·) satisfies the log-Sobolev inequality with the conatant C_{6.1}. Therefore we have by [AMS94, Corollary 3.4] that

∫ E^{Λ,ω} ⊗ E^{Λω}(dσdσ̃) exp (ε|σ_{x} − σ̃_{x}|^{2}) ≤ exp (\frac{εC_{6.1}}{2(1−εC_{6.1})})  (6.7)

if 0 < ε < 1/C_{6.1}. □

7 Proof of Lemma 3.3

In this section, we prove Lemma 3.3.

Proof of Lemma 3.3: Set g_{t}^{Λ,ω} = \frac{d(νP_{t}^{Λ,ω})}{dE^{Λ,ω}} for simplicity. We then have E^{Λ,ω}g_{t}^{Λ,ω} = 1 and hence

|νP_{t}^{Λ,ω}f − E^{Λ,ω}f| = |E^{Λ,ω} (fg_{t}^{Λ,ω}) − E^{Λ,ω}f · E^{Λ,ω}g_{t}^{Λ,ω}|

≤ \frac{1}{2}∫ E^{Λ,ω} ⊗ E^{Λ,ω}(dσ_{Λ}dσ̃_{Λ}) |f(σ) − f(σ̃)| |g_{t}^{Λ,ω}(σ) − g_{t}^{Λ,ω}(σ̃)|
≤ \frac{1}{2} ∑_{x∈S_{f}} ∥∇_{x}f∥∫ E^{Λ,ω} ⊗ E^{Λ,ω}(dσ_{Λ}dσ̃_{Λ}) |σ_{x} − σ̃_{x}| |g_{t}^{Λ,ω}(σ) − g_{t}^{Λ,ω}(σ̃)|
≤ ∑_{x∈S_{f}} ∥∇_{x}f∥∫ E^{Λ,ω} ⊗ E^{Λ,ω}(dσ_{Λ}dσ̃_{Λ}) |σ_{x} − σ̃_{x}| |g_{t}^{Λ,ω}(σ) − 1| .  (7.1)

By noting (a − 1)^{2} ≤ \frac{2}{3}(a + 2)(a log a − a + 1) for a ≥ 0 and applying Schwarz inequality, we see that

∫ E^{Λ,ω} ⊗ E^{Λ,ω}(dσ_{Λ}dσ̃_{Λ}) |σ_{x} − σ̃_{x}| |g_{t}^{Λ,ω}(σ) − 1|

≤ (∫ E^{Λ,ω} ⊗ E^{Λ,ω}(dσ_{Λ}dσ̃_{Λ}) |σ_{x} − σ̃_{x}|^{2} (g_{t}^{Λ,ω}(σ) + 2))^{1/2} E^{Λ,ω} (g_{t}^{Λ,ω} log g_{t}^{Λ,ω} − g_{t}^{Λ,ω} + 1)^{1/2}
≤ (∫ E^{Λ,ω} ⊗ E^{Λ,ω}(dσ_{Λ}dσ̃_{Λ}) |σ_{x} − σ̃_{x}|^{2} g_{t}^{Λ,ω}(σ) + C_{7.2})^{1/2} H(P_{t}^{Λ,ω}|E^{Λ,ω})^{1/2},  (7.2)

where we have used (3.6) in the last line. We next use ab ≤ exp(a) + b log b for a, b ≥ 0 and (3.6) to obtain that for sufficiently small ε ∈ (0, 1),

∫ E^{Λ,ω} ⊗ E^{Λ,ω}(dσ_{Λ}dσ̃_{Λ}) |σ_{x} − σ̃_{x}|^{2} g_{t}^{Λ,ω}(σ)
≤ ∫ E^{Λ,ω} ⊗ E^{Λ,ω}(dσ_{Λ}dσ̃_{Λ}) exp (ε |σ_{x} − σ̃_{x}|^{2})
+ ε^{−1}H(νP_{t}^{Λ,ω}|E^{Λ,ω}) − ε^{−1} log ε.  (7.3)

From (7.1), (7.2), (7.3), we arrive at the following estimate;

|νP_{t}^{Λ,ω}f − E^{Λ,ω}f| ≤ C_{7.4}|||f||| (H(νP_{t}^{Λ,ω}|E^{Λ,ω}) + H(νP_{t}^{Λ,ω}|E^{Λ,ω})^{1/2})  (7.4)

On the other hand, [DS89, page 250, (6.1.36)] reads

H(νP_{t}^{Λ,ω}|E^{Λ,ω}) ≤ H(ν|E^{Λ,ω}) exp −\frac{2t}{γ_{LS}(Λ, ω)}.  (7.5)

From (7.4) and (7.5), we conclude (3.9). □

8 Proof of Lemma 3.4

In this section, we prove Lemma 3.4.

Proof of Lemma 3.4:

Step1: Define

H_{J,h}^{Λ,ω}(σ) = −\frac{1}{2} ∑_{x,y∈Λ} J_{x,y}σ_{x}σ_{y} − ∑_{x∈Λ} (h_{x}σ_{x} + ∑_{y∉Λ} J_{x,y}σ_{x}ω_{y}) ,  (8.1)
ρ_{x}^{Λ,ω} = \frac{1}{4}|∇_{x}H^{Λ,ω}|^{2} − \frac{1}{2}Δ_{x}H^{Λ,ω},  (8.2)

where Δ_{x} in (8.2) stands for (\frac{∂}{∂σ_{x}} )^{2}. We first prepare following estimates; there exist C_{8.3} = C_{8.3}(d, J, h) ∈ (0, ∞) and C_{8.5} = C_{8.5}(d, J, h, U) ∈ (0, ∞) such that

H_{J,h}^{Λ,ω}(σ) ≥ −C_{8.3}|Λ| − C_{8.3} ∑_{y∈∂_{R}Λ} |ω_{y}|^{2},  (8.3)
H_{J,h}^{Λ,ω}(σ) ≤ C_{8.3}|Λ| + C_{8.3} ∑_{x∈Λ} |σ_{x}|^{2} + C_{8.3} ∑_{y∈∂_{R}Λ} |ω_{y}|^{2},  (8.4)
log Z^{Λ,ω} ≤ C_{8.5}|Λ| + C_{8.5} ∑_{y∈∂_{R}Λ} |ω_{y}|^{2}  (8.5)
∑_{x∈Λ} ρ_{x}^{Λ,ω} ≥ −C_{8.5}|Λ| − C_{8.5} ∑_{y∈∂_{R}Λ} |ω_{y}|^{2}  (8.6)

Inequality (8.3) can be seen as follows; Set

h_{x}^{Λ,ω} = h_{x} + ∑_{y∉Λ} J_{x,y}ω_{y}.  (8.7)

We then have by (1.22) that

H_{J,h}^{Λ,ω}(σ) ≥ ∑_{x∈Λ} (λ(J)|σ_{x}|^{2} − h_{x}^{Λ,ω}σ_{x})
≥ −(1/4λ(J)) ∑_{x∈Λ} |h_{x}^{Λ,ω}|^{2}
≥ −(1/2λ(J)) ∑_{x∈Λ} |h_{x}|^{2} − (1/2λ(J)) ∑_{x∈Λ} (∑_{y∉Λ} J_{x,y}ω_{y})^{2}
≥ −(∥h∥^{2}/2λ(J))|Λ| − (∥J∥^{2}/2λ(J)) ∑_{y∈∂_{R}Λ} |ω_{y}|^{2},

which proves (8.3).

For (8.4), it is clear that

∑_{x∈Λ} |h_{x}σ_{x}| ≤ \frac{1}{2} ∑_{x∈Λ} |h_{x}|^{2} + \frac{1}{2} ∑_{x∈Λ} |σ_{x}|^{2}
≤ \frac{1}{2}∥h∥^{2}|Λ| + \frac{1}{2} ∑_{x∈Λ} |σ_{x}|^{2}.  (8.8)

On the other hand, we have for any ξ = (ξ_{x})_{x∈Z^{d}} and η = (η_{x})_{x∈Z^{d}} that

∑_{x,y∈Z^{d}} |J_{x,y}ξ_{x}η_{y}| ≤ \frac{1}{2} ∑_{x,y∈Z^{d}} |J_{x,y}||ξ_{x}|^{2} + \frac{1}{2} ∑_{x,y∈Z^{d}} |J_{x,y}||η_{y}|^{2}
≤ \frac{1}{2}∥J∥ ∑_{x∈Z^{d}} |ξ_{x}|^{2} + \frac{1}{2}∥J∥ ∑_{y∈Z^{d}} |η_{x}|^{2}

Therefore, we obtain

\frac{1}{2} ∑_{x,y∈Λ} |J_{x,y}σ_{x}σ_{y}| ≤ \frac{1}{2}∥J∥ ∑_{x∈Z^{d}} |σ_{x}|^{2}
∑_{x∈Λ,y∉Λ} |J_{x,y}σ_{x}ω_{y}| ≤ \frac{1}{2}∥J∥ ∑_{x∈Λ} |σ_{x}|^{2} + \frac{1}{2}∥J∥ ∑_{x∈∂_{R}Λ} |ω_{x}|^{2}.

These, together with (8.8), prove (8.4).

Inequality (8.5) follows from (8.3). In fact,

log Z^{Λ,ω} = log ∫_{∏x∈Λ}dσ_{x} exp (H_{J,h}^{Λ,ω}(σ) − ∑_{x∈Λ} U(σ_{x}))
≤ C_{8.3}|Λ| + C_{8.3} ∑_{y∈∂_{R}Λ} |ω_{y}|^{2} + ∑_{x∈Λ} log ∫ dσ_{x} exp (−U(σ_{x}))
≤ (C_{8.3} + log ∫ ds exp −U(s)) |Λ| + C_{8.3} ∑_{y∈∂_{R}Λ} |ω_{y}|^{2}

Let us next prove (8.6). Recall that we have assumed (1.11) and set

ε = \frac{1}{4} − \frac{1}{2C_{1.11}} > 0, P (s) = \frac{1}{2C_{1.11}} U′(s)^{2} − \frac{1}{2}U′′(s).  (8.9)

We then have that

ρ_{x}^{Λ,ω}(σ) = \frac{1}{4} (− ∑_{y∈Λ} J_{x,y}σ_{y} − h_{x}^{Λ,ω} + U′(σ_{x}))^{2} − \frac{1}{2}J_{x,x} − \frac{1}{2}U′′(σ_{x})
≥ P (σ_{x}) + εU′(σ_{x})^{2} − \frac{1}{2} (∑_{y∈Λ} J_{x,y}σ_{y} − h_{x}^{Λ,ω}) U′(σ_{x}) − \frac{1}{2}∥J∥
≥ P (σ_{x}) − (1/16ε) (∑_{y∈Λ} J_{x,y}σ_{y} − h_{x}^{Λ,ω})^{2} − \frac{1}{2}∥J∥
≥ P (σ_{x}) − C_{8.10} ∑_{y∈Λ_{d(x,y)≤R}} |σ_{y}|^{2} − C_{8.10} ∑_{y∉Λ_{d(x,y)≤R}} |ω_{y}|^{2} − C_{8.10},  (8.10)

where C_{8.10} depends only on d, ∥J∥ , ∥h∥ and C_{1.11}. Therefore,

∑_{x∈Λ} ρ_{x}^{Λ,ω}(σ) ≥ ∑_{x∈Λ} P (σ_{x}) − C_{8.10} ∑_{x∈Λ} ∑_{y∈Λ_{d(x,y)≤R}} |σ_{y}|^{2} − C_{8.10} ∑_{x∈Λ} ∑_{y∉Λ_{d(x,y)≤R}} |ω_{y}|^{2} − C_{8.10}|Λ|
≥ ∑_{x∈Λ} (P (σ_{x}) − (2R)^{d}C_{8.10}|σ_{x}|^{2}) − (2R)^{d}C_{8.10} ∑_{y∈∂_{R}Λ} |ω_{y}|^{2} − C_{8.10}|Λ|.

This proves (8.6), since P (s) − (2R)^{d}C_{8.10}|s|^{2} is bounded from below by (1.11).

Step2: The density of P_{t}^{Λ,ω} with respect to the Lebesgue measure can be computed as follows;

\frac{dP_{t}^{Λ,ω}}{dσ_{Λ}} (σ_{Λ}) = (2πt)^{−|Λ|/2} exp (−\frac{1}{2t} ∑_{x∈Λ} |σ_{x} − ω_{x}|^{2} + \frac{1}{2}H^{Λ,ω}(ω) − \frac{1}{2}H^{Λ,ω}(σ))
·P (exp (−\frac{1}{2} ∑_{x∈Λ} ∫ _{0}^{t}dsρ_{x}^{Λ,ω}(ω + θ_{s}))| (ω_{x} + θ_{t,x})_{x∈Λ} = σ_{Λ}) ,  (8.11)

where dσ_{Λ} in the denominator on the left-hand-side refers to the Lebesgue measure on R^{Λ}. The formula (8.11) is nothing but an application of Girsanov’s transformation to the SDE (1.32) as we now explain. Define a local martingale M^{Λ,ω} by

M_{t}^{Λ,ω} = −\frac{1}{2}H^{Λ,ω}(ω + θ_{t}) + \frac{1}{2}H^{Λ,ω}(ω) + \frac{1}{4} ∑_{x∈Λ} ∫ _{0}^{t}dsΔ_{x}H^{Λ,ω}(ω + θ_{s}).

We then have

M_{t}^{Λ,ω} − \frac{1}{2}⟨ M^{Λ,ω} ⟩_{t} = −\frac{1}{2}H^{Λ,ω}(ω + θ_{t}) + \frac{1}{2}H^{Λ,ω}(ω) − \frac{1}{2} ∑_{x∈Λ} ∫ _{0}^{t}dsρ_{x}^{Λ,ω}(ω + θ_{s}),

where ⟨ M^{Λ,ω} ⟩_{t} is the quadratic variation process for M^{Λ,ω};

⟨ M^{Λ,ω} ⟩_{t} = \frac{1}{4} ∑_{x∈Λ} ∫ _{0}^{t}ds |∇_{x}H^{Λ,ω}(ω + θ_{s})|^{2} .

Therefore, the following process is a local martingale;

D_{t}^{Λ,ω} = exp (−\frac{1}{2}H^{Λ,ω}(ω + θ_{t}) + \frac{1}{2}H^{Λ,ω}(ω) − \frac{1}{2} ∑_{x∈Λ} ∫ _{0}^{t}dsρ_{x}^{Λ,ω}(ω + θ_{s}))

This is in fact a martingale since we see from (8.3) and (8.5) that there exists C_{8.12} ∈ (0, ∞) such that

D_{t}^{Λ,ω} ≤ exp (\frac{1}{2}H^{Λ,ω}(ω) + (1 + t)C_{8.12}|Λ| + (1 + t)C_{8.12} ∑_{y∈∂_{R}Λ} |ω_{y}|^{2})  (8.12)

for all t ≥ 0. From this and Proposition 3.1 of [RY91, page 325], we see that the right-hand-side of (8.11) is a density function of some Markov process on R^{Λ}. Furthermore, this Markov process is identified with (σ_{t}^{Λ,ω})_{t≥0} by Proposition 3.4 of [RY91, page 326]. Note that “f”, “Lg” and “Γ(g, g)” in [RY91, Chapter VIII, Section 3] are interpreted here respectively as \frac{H^{Λ,ω}}{2} , \frac{1}{2} ∑_{x∈Λ} Δ_{x}g and ∑_{x∈Λ} |∇_{x}g|^{2}.

Step3: Conclusion of the proof. We see from (8.11) that

\frac{dP_{t}^{Λ,ω}}{dE^{Λ,ω}} (σ_{Λ}) = Z^{Λ,ω}(2πt)^{−|Λ|/2} exp (−\frac{1}{2t} ∑_{x∈Λ} |σ_{x} − ω_{x}|^{2} + \frac{1}{2}H^{Λ,ω}(ω) + \frac{1}{2}H^{Λ,ω}(σ))
·P (exp (−\frac{1}{2} ∑_{x∈Λ} ∫ _{0}^{t}dsρ_{x}^{Λ,ω}(ω + θ_{s}))| (ω_{x} + θ_{t,x})_{x∈Λ} = σ_{Λ})  (8.13)

and hence that

log \frac{dP_{t}^{Λ,ω}}{dE^{Λ,ω}} (σ_{Λ}) = log Z^{Λ,ω} + \frac{1}{2}|Λ| log(1/2πt)

−\frac{1}{2t} ∑_{x∈Λ} |σ_{x}|^{2} + \frac{1}{2}H^{Λ,ω}(ω) + \frac{1}{2}H^{Λ,ω}(σ)

+ log P (exp (−\frac{1}{2} ∑_{x∈Λ} ∫ _{0}^{t}dsρ_{x}^{Λ,ω}(ω + θ_{s}))| (ω_{x} + θ_{t,x})_{x∈Λ} = σ_{Λ})

≤ \frac{1}{2}|Λ| log(1/2πt) + (1 + t)C_{8.14} ∑_{y∈Λ∪∂_{R}Λ}(1 + |ω_{x}|^{N_{1.12}})
+C_{8.14} ∑_{x∈Λ}(1 + |σ_{x}|^{N_{1.12}}),  (8.14)

where we have used (1.12), (8.3)–(8.6). We therefore arrive at the following estimate;

H(P_{t}^{Λ,ω}|E^{Λ,ω}) = P (log \frac{dP_{t}^{Λ,ω}}{dE^{Λ,ω}} (σ_{t}^{Λ,ω}))
≤ \frac{1}{2}|Λ| log(1/2πt) + (1 + t)C_{8.14} ∑_{y∈Λ∪∂_{R}Λ}(1 + |ω_{y}|^{N_{1.12}})
+C_{8.14} ∑_{x∈Λ} (1 + P (|σ_{x}^{Λ,ω}|^{2N_{1.12}})) .  (8.15)

Note finally that for any N > 0 and ε > 0,

∑_{y∈Λ∪∂_{R}Λ} |ω_{y}|^{N} ≤ exp (εR + εdiam(0 ∪ Λ)) ∑_{y∈Z^{d}} |ω_{y}|^{N} exp −εd(0, y).  (8.16)

By combinning (3.1), (8.15) and (8.16), we conclude (3.11). □

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