The log-Sobolev inequality for weakly coupled lattice fields^{1}

Nobuo Yoshida^{2}

July 17, 1998

Abstract We consider a ferromagnetic spin system with unbounded interactions on the ddimensional integer lattice (d ≥ 1). Under mild assumptions on the one-body interactions (so that arbitrarily deep double wells are allowed), we prove that if the coupling constants are small enough, then the finite volume Gibbs states satisfy the log-Sobolev inequality uniformly in the volume and the boundary condition.

1 Introduction

In this paper we address a question of understanding the ergodic property of unbounded lattice spin systems. We will consider a random field on Z^{d} described by the formal Hamiltonian

H(σ) = −\frac{1}{2} ∑_{x,y∈Z^{d}} J_{x,y}σ_{x}σ_{y} + ∑_{x∈Z^{d}} (U(σ_{x}) − h_{x}σ_{x}) ,  (1.1)

where σ_{x} ∈ R is the spin at the site x ∈ Z^{d}, J_{x,y} are finite range, ferromagnetic coupling constants (cf. (1.18)–(1.21) below), h_{x} ∈ R and U(s) is a function which diverges to +∞ faster than any constant multiple of s^{2} as |s| ↗ ∞.

Literatures on the equilibrium statistical mechanics for models of this kind are vast and their phase structure are fairly well understood in some cases (See [FSS76], [COPP78] and [BH82] for example and references therein). On the other hand, from the view point of probability theory, it seems very natural to be interested in the time evolution of the model, or more specifically, the associated stochastic dynamics called Glauber dynamics. Glauber dynamics is a natural model to describe the way a given configuration relaxes to equilibrium as time goes by. Therefore, its ergodic properties are very interesting subject to work on. The study of such dynamical theory for unbounded spin systems seems to be in a much more primitive stage as compared with that for models with compact spin spaces, but it begins to capture the attentions of many probabilists in recent years.

A key to rapid progresses in the study of dynamical lattice spin systems with compact spin spaces was the log-Sobolev inequality (See [SZ92], [LY93],[MO94] and references therein). Although the proof of the log-Sobolev inequality for unbounded spin systems are much harder, it has become increasingly feasible in view of some successful examples. The simplest case is when one can apply the Bakry-Emery Γ_{2} criterion ([BE85]) of log-Sobolev inequality. The Γ_{2} criterion is applicable to continuous spin systems when the Hamiltonian of the model is a strictly convex function. This is in particular the case with the Hamiltonian (1.1) when inf U′′ is positive ( To be precise, inf U′′ > −λ(J) is enough. cf. (1.21) below). In [Z96], the log-Sobolev inequality is extended beyond the applicability region of the Γ_{2}-criterion. In fact, the results in [Z96] are applicable to (1.1) even when U′′ can take large negative values and we see that the log-Sobolev inequality is true in at least the following two cases; (a) d = 1 ( cf. [Z96, Theorem 4.1]),

1Mathematical Subject Classification (1997): 60K35, 82B20

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

(b) d ≥ 2, J_{x,y} are large enough and U is a small perturbation from a strictly convex function in the sense of the sup-norm (cf. paragraphs following [Z96, Proposition 5.2]).

In this article, we mainly consider the restriction of the lattice field described by the Hamiltonian (1.1) to a finite set Λ in the lattice by imposing a boundary condition ω ∈ R^{Λc}. We prove that, if J_{x,y}’s (x ≠ y) are small enough, then the log-Sobolev inequality holds uniformly in Λ and ω. This implies in particular that the (unique) infinite volume tempared DLR-state also satisfies the log-Sobolev inequality. In view of the corresponding phase structure (See Theorem 1.1 below), our assumption seems to be one of the most natural example to be investigated. Heuristically speaking, the proof of log-Sobolev inequality under our assumption is possible for the following reason. As is well known, the log-Sobolev constant is invariant under independent product of underlying measure space ([G94, Theorem 2.3]) and, since the interaction is weak under our assumption, spins on each site in the lattice behave almost independently.

The proof is divided into three steps:

First of all, we prepare a theorem which says that if a certain mixing condition for the Gibbs measure is satisfied, then the spectral gap is uniformly positive in the volume Λ and the boundary condition ω (Theorem 2.2). Here, we will use the method of [LY93] as the basic strategy, as well as ideas from [Z96].

We next prove that the mixing condition mentioned above implies the log-Sobolev inequality which is uniform in Λ and ω (Theorem 2.1). We will do this with the help of the bound on spectral gap obtained in Theorem 2.2. Here, the basic strategy is again the one in [LY93].

Finally, we show that the mixing condition referred to above holds assuming that the coupling constants are small enough (Theorem 2.4). This is carried out by using a formula for the Vassershtein distance ([COPP78]) and a “constructive criterion” (Proposition 2.3), which is very similar to the famous condition for the phase uniqueness presented in [DSh85].

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}.  (1.2)

The value of R will eventually be chosen as the range R(J) of the interaction we consider (See (1.18) below). We say Λ ⊂⊂ Z^{d} is a generalized box with size (n_{1}, …, n_{d}), if it can be decomposed as follows;

Λ =Λ^{∘} ∪δΛ,  (1.3)

where

Λ^{∘} = {x ∈ Z^{d} ; v^{i} ≤ x^{i} < v^{i} + n_{i}, i = 1, …, d} for some (v^{i})_{i=1}^{d} ∈ Z^{d},
δΛ ⊂ {x ∉Λ^{∘} ; ∑_{i=1}^{d} |x^{i} − y^{i}| =1 for some y ∈Λ^{∘} }.

Thus a generalized box is a box with (or without) “dust” on its faces. We call min_{1≤i≤d} n_{i} in the

above definition the minimum side-length of Λ. We set

A = {Λ ; Λ ⊂⊂ Z^{d}},  (1.4)
B(n_{0}) = all generalized boxes with the minimum side-length at least n_{0},  (1.5)

where n_{0} ≥ 1.

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

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

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 }.  (1.6)

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

(C2) There are positive constants B_{1.7} and C_{1.7} such that

|f(σ)| + |∇_{Λ}f(σ)| ≤ B_{1.7} exp (C_{1.7} ∑_{x∈Λ} |σ_{x}|)  (1.7)

for any σ ∈ Ω, where

|∇_{Λ}f(σ)|^{2} = ∑_{x∈Λ} |\frac{∂}{∂σ_{x}} f(σ)|^{2} .  (1.8)

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}.  (1.9)

The Hamiltonian: We introduce a function U : R → R which satisfies (U1) and (U2) below.

(U1) For any m > 0, there exist V, W ∈ C^{∞}(R → R) and C_{1.12} ∈ (0, ∞) such that

U(s) = V (s) + W (s) for all s ∈ R,  (1.10)
inf_{s} V′′(s) ≥ m,  (1.11)
W (s) = 0 for |s| ≥ C_{1.12}  (1.12)
∥W ∥_{∞} ≤ C_{1.12},  (1.13)

where ∥W ∥_{∞} = sup_{s} |W (s)| (U2)

pq \frac{∂^{2}U}{∂p∂q}(p, q) ≥ 0 for all (p, q) ∈ R^{2},  (1.14)

where

U(p, q) = U (\frac{q+p}{\sqrt{2}}) + U (\frac{q−p}{\sqrt{2}}) .  (1.15)

A typical example of U is given by the following polynomial;

U(s) = ∑_{ν=1}^{N} a_{2ν}s^{2ν}  (1.16)

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 (1.16) may have arbitrarily deep double wells.

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}) .  (1.17)

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

R(J) =^{def.} sup{d(x, y) ; J_{x,y} ≠ 0} < ∞,  (1.18)
∥J∥ =^{def.} sup_{x} ∑_{y} |J_{x,y}| < ∞,  (1.19)
J_{x,y} = J_{y,x} ≥ 0 if x ≠ y,  (1.20)
λ(J) =^{def.} inf_{x}\{−\frac{1}{2} ∑_{y} J_{xy}\} > 0,  (1.21)

Note that we have from (1.20) and (1.21) 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 If a matrix J satisfies (1.18)–(1.20), we may also assume (1.21) 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 (1.18)–(1.21) and Û satisfies (1.10)–(1.14) and the replacement of (J_{x,y}, U) by (Ĵ_{x,y}, Û) does not change the Hamiltonian (1.17).

The Local specifications and the DLR-state: For a topological space X, we let M_{1}(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} ∈ M_{1}(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. E^{Λ,ω} is called the finite volume Gibbs state and the family {E^{Λ,ω} | Λ ⊂⊂ Z^{d}, ω ∈ S} is called the local specification. The following exponential integrability estimate is true; there is C_{1.24} = C_{1.24}(J, h, U) ∈ (0, ∞) such that

E^{Λ,ω} exp (λ|σ_{x}|) ≤ exp (C_{1.24} (1 + λ^{2} + λ ∑_{y∈∂_{R}Λ} |ω_{y}|)) , for λ > 0,  (1.24)

whenever x ∈ Λ ⊂⊂ Z^{d}, ω ∈ Ω and λ > 0. This follows from the arguments in Proposition III.1 and Theorem III.2 in [BH82]. The bound (1.24) implies that a function in the class C and its first derivatives have moments of all order with respect to the measure E^{Λ,ω}.

For ν ∈ M_{1}(Ω), we define a new measure νE^{Λ} ∈ M_{1}(Ω) by;

νE^{Λ}f = ∫ ν(dω)∫ E^{Λ,ω}(dσ)f(σ_{Λ} · ω_{Λc}),  (1.25)

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

(σ_{Λ} · ω_{Λc})_{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).  (1.26)

We now define two subsets G and G_{t} of M_{1}(Ω) as follows;

G = {ν ∈ M_{1}(Ω) ; νE^{Λ} = ν for any Λ ⊂⊂ Z^{d}},  (1.27)
G_{t} = G ∩ M_{1,t}(Ω),  (1.28)

where

M_{1,t}(Ω) = {ν ∈ M_{1}(Ω) ; (ν(|σ_{x}|))_{x∈Z^{d}} ∈ S′}.  (1.29)

A measure in G and G_{t} is called respectively, the DLR-state and the tempared DLR-state. It is known that the tempared DLR-state is unique if J_{x,y} (x ≠ y) are small enough.

Theorem 1.1 ([COPP78, DSh85]) There exists β = β(U) ∈ (0, ∞) such that G_{t} is a sin-

gleton if sup_{x} ∑_{y:y≠x} J_{x,y} ≤ β.

The inverse spectral gap and log-Sobolev constant: We define the inverse spectral gap γ_{SG}(Λ) as the smallest γ for which the following inequality is true for all f ∈ C and ω ∈ Ω;

E^{Λ,ω} (f; f) ≤ γE^{Λ,ω}(|∇_{Λ}f|^{2}).  (1.30)

Here and in what follows, the following common notaiton for the covariance of functions f and g with respect to a probability measure m is used;

m (f; g) = m(fg) − m(f) · m(g).  (1.31)

We define the log-Sobolev constant γ_{LS}(Λ) as the smallest γ for which the following inequality is true for all f ∈ C and ω ∈ Ω;

E^{Λ,ω} (f^{2} log \frac{f^{2}}{E^{Λ,ω}(f^{2})}) ≤ γE^{Λ,ω}(|∇_{Λ}f|^{2}).  (1.32)

It is well known that 2γ_{SG} ≤ γ_{LS} (cf. [DS89, Corollary 6.1.17]).

Measures E_{Λ,q}, E_{Λ,+} and E_{Λ,q}^{Λ\Δ,\overline{p}}: We now introduce some new measures on the configuration space, which plays important roles in this article. In fact, a mixing condition (2.1) we will assume to derive the log-Sobolev inequality will be described in terms of these new mesures rather than the original local specification defined by (1.23). For Λ ⊂⊂ Z^{d}, ω ∈ R^{Λc} and q ∈ R^{Λ}, we define a measure E_{Λ,q} ∈ M_{1}(R^{Λ}) by

E_{Λ,q}(dp) = E^{Λ,ω} ⊗ E^{Λ,ω} ((σ^{1}, σ^{2}) ; \frac{σ^{1} − σ^{2}}{\sqrt{2}} ∈ dp |\frac{σ^{1} + σ^{2}}{\sqrt{2}} = q )
= \frac{exp −H_{Λ,q}(p)}{Z_{Λ,q}} ∏_{x∈Λ} dp_{x},  (1.33)

where

H_{Λ,q}(p) = −\frac{1}{2} ∑_{x,y∈Λ} J_{x,y}p_{x}p_{y} + ∑_{x∈Λ} U(p_{x}, q_{x})  (1.34)

(Recall that we have defined the function U by (1.15)). It is also convenient to introduce the following measure;

E_{Λ,+}(dq) = E^{Λ,ω} ⊗ E^{Λ,ω} \{(σ^{1}, σ^{2}) ; \frac{σ^{1} + σ^{2}}{\sqrt{2}} ∈ dq\} .  (1.35)

The use of these measures can be demonstrated in the following expression of the spin-spin correlation function;

E^{Λ,ω}(σ_{x}; σ_{y}) = \frac{1}{2}∫ E^{Λ,ω} ⊗ E^{Λ,ω}(dσ^{1}dσ^{2}) (σ_{x}^{1} − σ_{x}^{2}) (σ_{y}^{1} − σ_{y}^{2})
= ∫ E_{Λ,+}(dq)E_{Λ,q} (p_{x}p_{y}) .  (1.36)

As can be seen from the above expression, if the correlation function E_{Λ,q} (p_{x}p_{y}) decays in d(x, y) uniformly in q-variable, so does E^{Λ,ω}(σ_{x}; σ_{y}), uniformly in ω and h. This idea will be used repeatedly in proofs in Section 3 (cf. Lemma 3.3 and Lemma 3.4). Furthermore, we define E_{Λ,q}^{W,\overline{p}} ∈ M_{1}(R^{W}) for W ⊂ Λ and \overline{p} ∈ R^{Λ} by

E_{Λ,q}^{W,\overline{p}}(dp_{W}) = E_{Λ,q}(dp_{Λ} | p ≡ \overline{p} on Λ\W )
= \frac{exp −H_{Λ,q}^{W,\overline{p}}(p)}{Z_{Λ,q}^{W,\overline{p}}} ∏_{x∈W} dp_{x},  (1.37)

where

H_{Λ,q}^{W,\overline{p}}(p) = H_{W,q}(p) − ∑_{x∈W,y∈Λ\W} J_{x,y}p_{x}\overline{p}_{y}.  (1.38)

The Vassershtein distance: For a metric space (X, ρ), which is separable and complete (hence is a Polish space), we define

M_{1,ρ}(X) = { μ ∈ M_{1}(X) ; ρ(x, ·) ∈ L_{1}(μ) for some x ∈ X}.  (1.39)

We introduce the Vassershtein distance R_{ρ} on M_{1,ρ}(X) as follows;

R_{ρ}(μ_{1}, μ_{2}) = inf {∫_{X^{2}}μ(dx_{1}dx_{2})ρ(x_{1}, x_{2}) ; μ ∈ K(μ_{1}, μ_{2})} ,  (1.40)

where

K(μ_{1}, μ_{2}) = {μ ∈ M_{1}(X^{2}) ; μ(dx_{1} × X) = μ_{1}, μ(X × dx_{2}) = μ_{2} }.  (1.41)

The case of X = R^{Λ} and ρ(σ^{1}, σ^{2}) = ∑_{x∈Λ} |σ_{x}^{1}−σ_{x}^{2}| is especially relevant to us. the Vassershtein distance in this case is denoted by R_{Λ}; for μ_{i} ∈ M_{1,ρ}(R^{Λ}) (i = 1, 2),

R_{Λ}(μ_{1}, μ_{2}) = inf {∫_{R^{Λ}×R^{Λ}}μ(dσ^{1}dσ^{2}) ∑_{x∈Λ} |σ_{x}^{1} − σ_{x}^{2}| ; μ ∈ K(μ_{1}, μ_{2})} .  (1.42)

2 Results

We have the following results for the lattice field described by the Hamiltonian (1.17). Recall that we have defined the measure E_{Λ,q}^{W,\overline{p}} by (1.37).

Theorem 2.1 Let F be either A or B(n_{0}) for arbitrarily fixed n_{0} > 0 (cf. (1.4) and (1.5)). Suppose that the following mixing condition holds; there exist positive constants B_{2.1} and C_{2.1} such that

sup_{q∈R^{Λ}} |E_{Λ,q}^{W,\overline{p}^{1}}(p_{z}) − E_{Λ,q}^{W,\overline{p}^{2}}(p_{z})|

≤ B_{2.1} (1 + ∑_{w∈Λ∩∂_{R}W}(|\overline{p}_{w}^{1}| + |\overline{p}_{w}^{2}|)) exp (−\frac{d(y, z)}{C_{2.1}})  (2.1)

whenever Λ ∈ F, W ⊂ Λ, y ∈ Λ\W and \overline{p}^{i} ∈ R^{Λ} (i = 1, 2) differs only at y. Then the log-Sobolev constants (cf.(1.32)) are uniformly bounded in the sense that;

sup{γ_{LS}(Λ) | Λ ∈ F} ≤ C_{2.2} < ∞,  (2.2)

where the constant C_{2.2} depends only on d, U, J, B_{2.1} and C_{2.1}. Therefore, the unique element μ in G_{t} (cf. Remark 2.1) satisfies the log-Sobolev inequality;

μ (f^{2} log \frac{f^{2}}{μ(f^{2})}) ≤ C_{2.2}μ(|∇_{Λ}f|^{2})  (2.3)

for any Λ ⊂⊂ Z^{d} and f ∈ C_{Λ}.

Remark 2.1 The mixing condition (2.1) implies the uniqueness of the tempared DLRstate. In fact, as we will see in Lemma 3.3 below, (2.1) implies the following;

|E^{Λ,ω}(σ_{x}; σ_{y})| ≤ C_{2.4} exp (−\frac{d(x, y)}{C_{2.4}}) ,  (2.4)

for any Λ ∈ F, ω ∈ R^{Λc} and x, y ∈ Λ, where C_{2.4} = C_{2.4}(U, J) ∈ (0, ∞). Furthermore, it is not difficult to prove by standard arguments that (2.4) implies the uniqueness of the tempared DLR-state (For example, Theorem IV.3, Proposition IV.4 and Proposition V.1 in [BH82] can be used to show that “pure phases” μ_{±} coincide.).

Remark 2.2 We will see that the mixing condition (2.1) holds if sup_{x} ∑_{y:y≠x} J_{x,y} is small enough (cf. Theorem 2.4). It is also known that if d = 1, then (2.1) holds regardless of the value of sup_{x} ∑_{y:y≠x} J_{x,y}. This can be seen from [Z96, Lemma 4.5].

Remark 2.3 As we experience in models with compact spin spaces, it may well be the case that for some J, U and h, a mixing condition like (2.1) does not hold for all Λ, but its restriction to “nice” Λ’s (typically to cubes or to “fat” enough boxes) does. This is why we introduced the class B(n_{0}).

Remark 2.4 Theorem 2.1 above is strongly motivated by [Z96, Theorem 5.1]. There, the potential function U is assumed only to satisfy (1.10) – (1.13) for some m > 0. [Z96, Theorem 5.1] says that if there is C_{2.4} = C_{2.4}(U, J, h) ∈ (0, ∞) such that (2.4) holds for large enough cubes Λ ⊂ Z^{d}, ω ∈ R^{Λc} and x, y ∈ Λ, then there is γ ∈ (0, ∞) for which the log-Sobolev inequality (1.32) holds for ω ≡ 0 and for large enough cubes Λ ⊂ Z^{d}. Therefore, as compared with [Z96, Theorem 5.1], Theorem 2.1 in this paper says more or less that a stronger results follows from stronger assumptions.

Remark 2.5 Let us briefly remark that the uniform bound (2.2) of the log-Sobolev constants implies “exponential convergence to the equibrium” of the associated stochastic dynamics (cf. [Y98] [Z96, Theorem 3.1]).

We also present the following weaker result:

Theorem 2.2 Suppose that the same mixing condition as in Theorem 2.2 holds. Then the inverse spectral gaps (cf.(1.30)) are uniformly bounded in the sense that;

sup{γ_{SG}(Λ) | Λ ∈ F} ≤ C_{2.5} < ∞,  (2.5)

where the constant C_{2.5} depends only on d, U, J, B_{2.1} and C_{2.1}. Therefore, the unique element μ in G_{t} (cf. Remark 2.1) satisfies

μ (f; f) ≤ C_{2.5}μ(|∇_{Λ}f|^{2})  (2.6)

for any Λ ⊂⊂ Z^{d} and f ∈ C_{Λ}.

This statement about the spectral gap follows easily from Theorem 2.1, since the inverse spectral gap (cf.(1.30)) is always bounded from above by a constant multiple of the log-Sobolev constant. In this paper, however, we prove Theorem 2.2 in advance of Theorem 2.1 and use it as a step to show Theorem 2.1.

We provide the following “constructive criterion” of the mixing condition (2.1). Recall that for Λ ⊂⊂ Z^{d}, we have defined the Vassershtein distance R_{Λ} by (1.42).

Proposition 2.3 Let Λ ⊂⊂ Z^{d} be arbitrary. Suppose that there exist V ⊂⊂ Z^{d}, ε_{2.8} ∈ (0, 1) and a matrix K = (K_{x,y} ≥ 0 : x, y ∈ Z^{d}) such that

K_{x,y} = 0 if y − x ∉ V ∪ ∂_{R}V,  (2.7)
∥K∥ =^{def.} sup_{y} ∑_{x} K_{x,y} ≤ ε_{2.8}|V |,  (2.8)
R_{W ∩(x+V)} (E_{Λ,q}^{W ∩(x+V),\overline{p}^{1}}, E_{Λ,q}^{W ∩(x+V),\overline{p}^{2}}) ≤ ∑_{y} K_{x,y}|\overline{p}_{y}^{1} − \overline{p}_{y}^{2}|  (2.9)

for all W ⊂ Λ, x ∈ Z^{d} and \overline{p}^{i} ∈ R^{Λ} (i = 1, 2). Then there exist constants B_{2.1} and C_{2.1} which depend only on d, R, V and ε_{2.8} such that (2.1) holds whenever W ⊂ Λ, y ∈ Λ\W and \overline{p}^{i} ∈ R^{Λ} (i = 1, 2) differs only at y.

Remark 2.6 In Proposition 2.3, we did not assume that V and ε_{2.8} are independent of the choice of Λ. However, in application of the proposition we have in mind (cf. Theorem 2.2, Theorem 2.1), it is important to find V and ε_{2.8} independently of the choice of Λ, since we need (2.1) with constants B_{2.1} and C_{2.1} not depending on Λ.

Remark 2.7 Conditions (2.7) –(2.9) in Proposition 2.3 are reminiscient of [DSh85, Theorem 2.1]. Note however we have put stronger assumptions to get stronger conclusion than that of above mentioned result.

By combining Theorem 2.1 and Proposition 2.3, we can prove the following main result in this article:

Theorem 2.4 There is β ∈ (0, ∞) such that if sup_{x} ∑_{y:y≠x} J_{x,y} ≤ β, then the following hold.

(a) There exists ε_{2.8} < 1 which depends only on d, U, J such that conditions (2.7) –(2.9) with V = {0} and with some K = (K_{x,y} ≥ 0 : x, y ∈ Z^{d}) are satisfied for all Λ ⊂⊂ Z^{d}, W ⊂ Λ, x ∈ Z^{d} and \overline{p}^{i} ∈ R^{Λ} (i = 1, 2).

(b) There exist constants B_{2.1} and C_{2.1} which depend only on d, U, J such that (2.1) holds whenever Λ ⊂⊂ Z^{d}, W ⊂ Λ, y ∈ Λ\W and \overline{p}^{i} ∈ R^{Λ} (i = 1, 2) differs only at y.

(c) The uniform bound on the log-Sobolev constants (2.2) with F = A holds with the constant C_{2.2} depending only on d, U J and thus, the inequality (2.3) for the unique μ ∈ G_{t} holds.

Remark 2.8 Part(b) of Theorem 2.4 implies that if sup_{x} ∑_{y:y≠x} J_{x,y} is small enough, then mixing condition (2.1), and thus (2.4) holds (cf. Lemma 3.3). In particular, we see that [Z96, Theorem 5.1] applies if sup_{x} ∑_{y:y≠x} J_{x,y} is small enough.

Remark 2.9 The assumptions (U1) and (U2) for the one body interaction U are mild enough to include examples like (1.16) which are often discussed in physical litratures. However, from the mathematical point of view, these assupmtions are not minimal to prove the part(c) of Theorem 2.4 with. Based on the result of B. Helffer [He97], T. Bodineau and B. Helffer improved the proof of part(c) of Theorem 2.4 in a very recent paper [BH98] and they succeeded in reducing the assumptions for U to, more or less, the minimal ones. In fact, they only assume (1.10)–(1.13) for some m > 0 and, (1.14) is not required.

3 Lemmas

In this section, we prove a couple of lemmas which will be used later. Here, we will use many ideas from [Z96, Section 4].

Lemma 3.1 For any λ > 0, there exists C_{3.1}(λ) = C_{3.1}(λ, U, J) ∈ (0, ∞) such that

E_{Λ,q}^{W,\overline{p}} exp (λ|p_{x}|^{2}) ≤ exp (C_{3.1}(λ) (1 + ∑_{y∈Λ∩∂_{R}W} |\overline{p}_{y}|^{2})) ,  (3.1)

whenever W ⊂ Λ ⊂⊂ Z^{d}, q ∈ R^{Λ} and \overline{p} ∈ R^{Λ}.

Proof: To prove (3.1), we begin by proving that

E_{Λ,q}^{x,\overline{p}} exp (λ|p_{x}|^{2}) ≤ E_{Λ,q}^{x,\overline{p}} exp (λ|p_{x}|^{2})|_{q≡0}  (3.2)

Since the left-hand-side of (3.2) depends on q only through q_{x}, it is enough to prove that

\frac{∂}{∂q_{x}} E_{Λ,q}^{x,\overline{p}} exp (λ|p_{x}|^{2}) \{ \begin{matrix}≤ 0 & if q_{x} ≥ 0, \\ ≥ 0 & if q_{x} ≤ 0. \end{matrix}  (3.3)

We have that

\frac{∂}{∂q_{x}} E_{Λ,q}^{x,\overline{p}} exp (λ|p_{x}|^{2}) = −E_{Λ,q}^{x,\overline{p}} (\frac{∂U}{∂q_{x}} (p_{x}, q_{x}); exp (λ|p_{x}|^{2}))
= − \frac{1}{2}∫ E_{Λ,q}^{x,\overline{p}}(dp_{x}^{1})∫ E_{Λ,q}^{x,\overline{p}}(dp_{x}^{2}) (\frac{∂U}{∂q_{x}} (p_{x}^{1}, q_{x}) − \frac{∂U}{∂q_{x}} (p_{x}^{2}, q_{x}))
· (exp (λ|p_{x}^{1}|^{2}) − exp (λ|p_{x}^{2}|^{2})) .  (3.4)

Suppose now that q_{x} ≥ 0,then |p_{x}| ↦ \frac{∂U}{∂q_{x}} is is non-decreasing by (1.14). This implies that the integrand in (3.4) is non-negative for all (p_{x}^{1}, p_{x}^{2}). Similarly, the integrand in (3.4) is non-positive for all (p_{x}^{1}, p_{x}^{2}), if q_{x} ≤ 0. Therefore, we have proved ( 3.3).

We next find a constant C_{3.5}(λ) ∈ (0, ∞) and a sequence r = (r_{x} ≥ 0) ∈ S, which depend only on λ, J and U such that

∑_{x} r_{x} < 1,
E_{Λ,q}^{x,\overline{p}} exp (λ|p_{x}|^{2})|_{q≡0} ≤ exp (C_{3.5}(λ) + λ ∑_{y} r_{x−y}|\overline{p}_{y}|^{2})  (3.5)

whenever x ∈ Λ ⊂⊂ Z^{d} and \overline{p} ∈ R^{Λ}. This can be done in the same way as in the proof of [BH82, Theorem III.2], where similar estimate for the integral of exp (λ|p_{x}|) is obtained. Here, we have to consider the integral of exp (λ|p_{x}|^{2}). However, it is not diffucult to generalize the computations in [BH82, Lemmas III.5 and III.6] to cover our case (by taking m > 0 in (1.10)–(1.13) large enough, depending on λ). Once (3.2) and (3.5) are established, one can proceed as in the proof of [BH82, Proposition III.1] to obtain (3.1). □

Lemma 3.2 For any n = 1, 2, …, there exists C_{3.6}(n) = C_{3.6}(U, J, n) ∈ (0, ∞) such that

E^{Λ,ω} (f^{2} log \frac{f^{2}}{E^{Λ,ω}(f^{2})}) ≤ C_{3.6}(|Δ|)E^{Λ,ω}(|∇_{Λ}f|^{2})  (3.6)

whenever Δ, Λ ⊂⊂ Z^{d} and f ∈ C satisfy S_{f} ∩ Λ ⊂ Δ ⊂ Λ.

To prove Lemma 3.2, we need another lemma.

Lemma 3.3 Suppose that the mixing condition (2.1) holds. Then,

|E^{Λ,ω}(σ_{x}; σ_{y})| ≤ C_{3.7} exp (−\frac{d(x, y)}{C_{2.1}}) ,  (3.7)

for any Λ ∈ F, ω ∈ R^{Λc} and x, y ∈ Λ, where C_{3.7} = C_{3.7}(U, J, B_{2.1}) ∈ (0, ∞). In general (without the mixing condition (2.1)), the following is true;

sup{|E^{Λ,ω}(σ_{x}; σ_{x})| ; Λ ⊂⊂ Z^{d}, x ∈ Λ, h ∈ R^{Λ}, ω ∈ R^{Λc}} ≤ C_{3.8} < ∞,  (3.8)

where C_{3.8} = C_{3.8}(U, J) ∈ (0, ∞).

Proof of Lemma 3.3: To prove the Lemma, recall that we have defined measures E_{Λ,q}, E_{Λ,+} and E_{Λ,q}^{Λ\Δ} respectively by (1.33), (1.35) and (1.37). Note that

E^{Λ}(σ_{x}; σ_{y}) = \frac{1}{2}∫ E^{Λ} ⊗ E^{Λ}(dσ^{1}dσ^{2}) (σ_{x}^{1} − σ_{x}^{2}) (σ_{y}^{1} − σ_{y}^{2})
= ∫ E_{Λ,+}(dq)∫ E_{Λ,q}(dp)p_{x}p_{y}
= ∫ E_{Λ,+}(dq)∫ E_{Λ,q}(dp)p_{x}E_{Λ,q}^{Λ\x,p}(p_{y})  (3.9)

and that we have from (2.1) that

|E_{Λ,q}^{Λ\x,p}(p_{y})| ≤ 2B_{2.1} (1 + |p_{x}|) exp (−\frac{d(x, y)}{C_{2.1}}) .

Plugging this into (3.9), we have

|E^{Λ,ω}(σ_{x}; σ_{y})| ≤ 2B_{2.1} exp (−\frac{d(x, y)}{C_{2.1}}) ∫ E_{Λ,+}(dq)∫ E_{Λ,q}(dp)|p_{x}| (1 + |p_{x}|) .  (3.10)

By (3.1), the integral in the right-hand-side of (3.10) is bounded by some constant which depends only on J and U. Therefore we get (3.7). The same argument as above, but without the use of (2.1), proves (3.8). □

Proof of Lemma 3.2: Let us set E_{Δ}^{Λ,ω}(dσ_{Δ}) = E^{Λ,ω}(R^{Λ\Δ} × dσ_{Δ}). Since

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

−\frac{1}{2} ∑_{x,y∈Λ\Δ} J_{x,y}σ_{x}σ_{y} + ∑_{x∈Λ\Δ} (U(σ_{x}) − h_{x}σ_{x}) − ∑_{x∈Λ\Δ} ∑_{y∉Λ} J_{x,y}σ_{x}ω_{y}

− ∑_{x∈Λ\Δ} ∑_{y∈Δ} J_{x,y}σ_{x}σ_{y},  (3.11)

the measure E_{Δ}^{Λ,ω}(dσ_{Δ}) has the following expression;

E_{Δ}^{Λ,ω}(dσ_{Δ}) = \frac{exp (− ∑_{x∈Δ} (U(σ_{x}) − h_{x}^{Λ,ω}σ_{x})) ZJ^{Λ\Δ,σ_{Δ}}νJ^{Δ}(dσ_{Δ})}{∫ νJ^{Δ}(dσ_{Δ}) exp (− ∑_{x∈Δ} (U(σ_{x}) − h_{x}^{Λ,ω}σ_{x})) ZJ^{Λ\Δ,σ_{Δ}}} ,  (3.12)

where

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

νJ^{Δ}(dσ_{Δ}) = exp (\frac{1}{2} ∑_{x,y∈Δ} J_{x,y}σ_{x}σ_{y}) ∏_{x∈Δ} dσ_{x},

ZJ^{Λ\Δ,σ_{Δ}} = ∫ νJ^{Λ\Δ}(dσ_{Λ\Δ}) exp (− ∑_{x∈Λ\Δ} (U(σ_{x}) − h_{x}σ_{x}) + ∑_{x∈Λ\Δ} ∑_{y∈Δ∪Λc} J_{x,y}σ_{x}(σ_{Δ} · ω_{Λc})_{y}) .

We would like to deform the above measure into another measure \overline{E}_{Δ}^{Λ,ω}(dσ_{Δ}) to which we can apply the Bakry-Emery Γ_{2}-criterion of the log-Sobolev inequality ([BE85, Corollaire 2]). To this end, we decompose U into V and W in a manner described in (1.10)–(1.13), where the parameter m > 0 is specified later, depending on |Δ|. We then have by (1.13) that

exp (−2C_{1.12}) ≤ \frac{dE_{Δ}^{Λ,ω}}{d\overline{E}_{Δ}^{Λ,ω}} (σ_{Δ}) ≤ exp (2C_{1.12}) ,  (3.13)

where

\overline{E}_{Δ}^{Λ,ω}(dσ_{Δ}) = \frac{exp (− ∑_{x∈Δ} (V (σ_{x}) − h_{x}^{Λ,ω}σ_{x})) ZJ^{Λ\Δ,σ_{Δ}}νJ^{Δ}(dσ_{Δ})}{∫ νJ^{Δ}(dσ_{Δ}) exp (− ∑_{x∈Δ} (V (σ_{x}) − h_{x}^{Λ,ω}σ_{x})) ZJ^{Λ\Δ,σ_{Δ}}} ,  (3.14)

Let us next prove that

∑_{x,y∈Δ} |ξ_{x}ξ_{y} \frac{∂^{2}}{∂σ_{x}∂σ_{y}} log ZJ^{Λ\Δ,σ_{Δ}} | ≤ C_{3.8}|Δ|∥J∥^{2} ∑_{x∈Δ} |ξ_{x}|^{2} ,  (3.15)

for any (ξ_{x})_{x∈Δ} ∈ R^{Δ}. In fact,

∑_{x,y∈Δ} |ξ_{x}ξ_{y} \frac{∂^{2}}{∂σ_{x}∂σ_{y}} log ZJ^{Λ\Δ,σ_{Δ}} | = ∑_{x,y∈Δ} |ξ_{x}ξ_{y} ∑_{z,w∈Λ\Δ} J_{x,z}J_{y,w}E^{Λ\Δ,σ_{Δ}·ω_{Λc}}(σ_{z}; σ_{w})|

≤ C_{3.8}∥J∥^{2} ( ∑_{x∈Δ} |ξ_{x}|)^{2}
≤ C_{3.8}∥J∥^{2}|Δ| ∑_{x∈Δ} |ξ_{x}|^{2}.

At this point, we take m = C_{3.8}∥J∥^{2}|Δ| + 1 to have that

∑_{x,y∈Δ} ξ_{x}ξ_{y} \frac{∂^{2}}{∂σ_{x}∂σ_{y}} ( ∑_{x∈Δ} (V (σ_{x}) − h_{x}^{Λ,ω}σ_{x}) − log ZJ^{Λ\Δ,σ_{Δ}}) ≥ ∑_{x∈Δ} |ξ_{x}|^{2}.  (3.16)

This implies that Γ_{2} in the sense of [BE85], computed with respect to the measure \overline{E}^{Λ,ω} is bounded from below by 1 and thus that we have

\overline{E}_{Δ}^{Λ,ω} (f^{2} log \frac{f^{2}}{\overline{E}_{Δ}^{Λ,ω}(f^{2})}) ≤ 2\overline{E}_{Δ}^{Λ,ω}(|∇_{Λ}f|^{2}).  (3.17)

whenever f ∈ C and Λ ⊂⊂ Z^{d} satisfy S_{f} ∩ Λ ⊂ Δ ⊂ Λ. By (3.13), (3.17) and a standard comparison argument ([HS87, Lemma 5.1]), we get (3.6) with

C_{3.6} = 2 exp (4C_{1.12}) .  (3.18)

□

The following lemma is technically the most important step in our proof of Theorem 2.2 and Theorem 2.1;

Lemma 3.4 Suppose that mixing condition (2.1) is true. Then, for any f ∈ C, Λ ∈ F, x ∈ Λ and Δ ⊂ Λ such that S_{f} ∩ Λ ⊂ Δ ⊂ Λ,

|E^{Λ}(f; σ_{x})| ≤ C_{3.19}|Δ|^{2} exp (−\frac{d(x, Δ)}{C_{2.1}}) E^{Λ}(f; f)^{1/2},  (3.19)
|E^{Λ}(f^{2}; σ_{x})| ≤ C_{3.19}|Δ|^{2} exp (−\frac{d(x, Δ)}{C_{2.1}})

·E^{Λ}(f^{2})^{1/2} (E^{Λ}(f; f)^{1/2} + E^{Λ} (f^{2} log \frac{f^{2}}{E^{Λ}(f^{2})})^{1/2}) ,  (3.20)

where C_{3.19} = C_{3.19}(B_{2.1}, C_{3.1}(1)) ∈ (0, ∞). Furthermore, for y ∉ Λ,

|∇_{y}E^{Λ}(f)| ≤ |E^{Λ} (∇_{y}f) | + ∥J∥ sup_{x∈Λ_{d(x,y)≤R}} |E^{Λ}(f; σ_{x})|  (3.21)

≤ |E^{Λ} (∇_{y}f) | + C_{3.19}∥J∥|Δ|^{2} exp (−\frac{d(x, Δ)}{C_{2.1}}) E^{Λ}(f; f)^{1/2}, (3.22)
|∇_{y}\sqrt{E^{Λ}(f^{2})}| ≤ E^{Λ} (|∇_{y}f|^{2})^{1/2} + \frac{1}{2}E^{Λ}(f^{2})^{−1/2}∥J∥ sup_{x∈Λ_{d(x,y)≤R}} |E^{Λ}(f^{2}; σ_{x})|  (3.23)

≤ E^{Λ} (|∇_{y}f|^{2})^{1/2} + C_{3.19}∥J∥|Δ|^{2} exp (−\frac{d(y, Δ)}{C_{2.1}})

· (E^{Λ}(f; f)^{1/2} + E^{Λ} (f^{2} log \frac{f^{2}}{E^{Λ}(f^{2})})^{1/2}) .  (3.24)

Remark 3.1 We will use (3.19),(3.21) and (3.22) to prove Theorem 2.2. On the other hand, (3.20),(3.23) and (3.24) will be used in the proof of Theorem 2.1, where the term E^{Λ}(f; f) on the right-hand-side of (3.20) and (3.24) will eventually be bounded by C_{2.5}E^{Λ} (|∇_{Λ}f|^{2}) by using Theorem 2.2.

Remark 3.2 As will become clear from the proof, if we do not assume any mixing condition, we have (3.19)–(3.24) without the factor exp (−\frac{d(y,Δ)}{C_{2.1}}).

Proof of Lemma 3.4: Let us begin by proving (3.21) and (3.23). In fact, it is easy to see that

|∇_{y}E^{Λ}(f)| = |E^{Λ} (∇_{y}f) + ∑_{x∈Λ} J_{x,y}E^{Λ}(f; σ_{x})|
≤ |E^{Λ} (∇_{y}f) | + ∥J∥ sup_{x∈Λ_{d(x,y)≤R}} |E^{Λ}(f; σ_{x})|

and that

|∇_{y}\sqrt{E^{Λ}(f^{2})}| = |\frac{1}{2}E^{Λ}(f^{2})^{−1/2}∇_{y}E^{Λ}(f^{2})|
= |\frac{1}{2}E^{Λ}(f^{2})^{−1/2} (2E^{Λ} (f∇_{y}f) + ∑_{x∈Λ} J_{x,y}E^{Λ}(f^{2}; σ_{x}))|

≤ \frac{1}{2}E^{Λ}(f^{2})^{−1/2} (2E^{Λ} (f^{2})^{1/2} E^{Λ} (|∇_{y}f|^{2})^{1/2} + ∥J∥ sup_{x∈Λ_{d(x,y)≤R}} |E^{Λ}(f^{2}; σ_{x})|)

= E^{Λ} (|∇_{y}f|^{2})^{1/2} + \frac{1}{2}E^{Λ}(f^{2})^{−1/2}∥J∥ sup_{x∈Λ_{d(x,y)≤R}} |E^{Λ}(f^{2}; σ_{x})| .

By (3.21) and (3.23), the proof of (3.22) and (3.24) comes down to that of (3.19) and (3.20).

To prove (3.19), recall that we have defined mesures E_{Λ,q}, E_{Λ,q}^{Λ\Δ} and E_{Λ,+} respectively by (1.33) and (1.37) and (1.35). The correlation E^{Λ}(f; σ_{x}) can be expressed in terms of these measures as follows;

E^{Λ}(f; σ_{x})
= \frac{1}{2}∫ E^{Λ} ⊗ E^{Λ}(dσ^{1}dσ^{2}) (σ_{x}^{1} − σ_{x}^{2}) (f(σ^{1}) − f(σ^{2}))
= \frac{1}{\sqrt{2}}∫ E_{Λ,+}(dq)∫ E_{Λ,q}(dp) (f (\frac{q + p}{\sqrt{2}}) − f (\frac{q − p}{\sqrt{2}})) p_{x}
= \frac{1}{\sqrt{2}}∫ E_{Λ,+}(dq)∫ E_{Λ,q}(dp) (f (\frac{q + p}{\sqrt{2}}) − f (\frac{q − p}{\sqrt{2}})) E_{Λ,q}^{Λ\Δ,p}(p_{x})  (3.25)

On the other hand, we have from (2.1) that

|E_{Λ,q}^{Λ\Δ,p}(p_{x})| ≤ 2B_{2.1}|Δ| (1 + ∑_{z∈Δ} |p_{z}|) exp (−\frac{d(x, Δ)}{C_{2.1}})

Plugging this into (3.25), we see that

|E^{Λ}(f; σ_{x})| ≤ \sqrt{2}B_{2.1}|Δ| exp (−\frac{d(x, Δ)}{C_{2.1}}) (I_{1}(f) + ∑_{z∈Δ} I_{2}(f, z)) ,  (3.26)

where

I_{1}(f) = ∫ E_{Λ,+}(dq)∫ E_{Λ,q}(dp) |f (\frac{q + p}{\sqrt{2}}) − f (\frac{q − p}{\sqrt{2}})|
= ∫ E^{Λ} ⊗ E^{Λ}(dσ^{1}dσ^{2}) |f(σ^{1}) − f(σ^{2})| ,  (3.27)
I_{2}(f, z) = ∫ E_{Λ,+}(dq)∫ E_{Λ,q}(dp) |f (\frac{q + p}{\sqrt{2}}) − f (\frac{q − p}{\sqrt{2}})| |p_{z}|
= \frac{1}{\sqrt{2}}∫ E^{Λ} ⊗ E^{Λ}(dσ^{1}dσ^{2}) |(f(σ^{1}) − f(σ^{2})) (σ_{z}^{1} − σ_{z}^{2})|  (3.28)

We first observe that

I_{2}(f, z) ≤ \frac{1}{\sqrt{2}}I_{3}(f)^{1/2}I_{4}(f, z)^{1/2},  (3.29)

where

I_{3}(f) = ∫ E^{Λ} ⊗ E^{Λ}(dσ^{1}dσ^{2}) |f(σ^{1}) − f(σ^{2})|^{2} ,
I_{4}(f, z) = ∫ E^{Λ} ⊗ E^{Λ}(dσ^{1}dσ^{2}) |σ_{z}^{1} − σ_{z}^{2}|^{2} ,

The integrals I_{1}(f) and I_{3}(f) can be estimated as follows;

I_{1}(f) ≤ I_{3}(f)^{1/2} ≤ 2 (E^{Λ} |f − E^{Λ}f|^{2})^{1/2}
= 2E^{Λ}(f; f)^{1/2}  (3.30)

On the other hand, it follows from (3.1) and Jensen inequality that

I_{4}(f, z) ≤ C_{3.31}  (3.31)

for some C_{3.31} = C_{3.31}(C_{3.1}(1)) ∈ (0, ∞). Putting (3.29),(3.30) and (3.31) together, we obtain

I_{1}(f) + ∑_{z∈Δ} I_{2}(f, z) ≤ I_{1}(f) + \frac{1}{\sqrt{2}}I_{3}(f)^{1/2} ∑_{z∈Δ} I_{4}(f, z)^{1/2}
≤ 2E^{Λ}(f; f)^{1/2}(1 + C_{3.31}^{1/2}|Δ|)

which, in conjunction with (3.26), implies (3.19).

The proof of (3.20) is similar to that of (3.19). We see from (3.26) that

|E^{Λ}(f^{2}; σ_{x})| ≤ \sqrt{2}B_{2.1}|Δ| exp (−\frac{d(x, Δ)}{C_{2.1}}) (I_{1}(f^{2}) + ∑_{z∈Δ} I_{2}(f^{2}, z)) .  (3.32)

We first observe that

I_{1}(f^{2}) = ∫ E^{Λ} ⊗ E^{Λ}(dσ^{1}dσ^{2}) |f(σ^{1})^{2} − E^{Λ}(f)^{2} − (f(σ^{2})^{2} − E^{Λ}(f)^{2})|
≤ 2E^{Λ}(f; f).  (3.33)

Next, we have by Schwartz inequality that

I_{2}(f^{2}, z) ≤ \frac{1}{\sqrt{2}}I_{3}(f)^{1/2}I_{5}(f, z)^{1/2},  (3.34)

where

I_{5}(f, z) = ∫ E^{Λ} ⊗ E^{Λ}(dσ^{1}dσ^{2}) |(f(σ^{1}) + f(σ^{2}))^{2} (σ_{z}^{1} − σ_{z}^{2})^{2}| .

Let us note that ab ≤ exp(a) + b log b for a, b ≥ 0 and that

∫ E^{Λ} ⊗ E^{Λ}(dσ^{1}dσ^{2}) exp (|σ_{z}^{1} − σ_{z}^{2}|^{2}) = 2∫ E_{Λ,+}(dq)∫ E_{Λ,q}(dp) exp (|p_{z}|^{2})
≤ C_{3.35}  (3.35)

by (3.1), where C_{3.35} = C_{3.35}(C_{3.1}(1)). We then have that

I_{5}(f, z) ≤ 2∫ E^{Λ} ⊗ E^{Λ}(dσ^{1}dσ^{2}) |f(σ^{1})^{2} (σ_{z}^{1} − σ_{z}^{2})^{2}|
= 2E^{Λ}(f^{2})∫ E^{Λ} ⊗ E^{Λ}(dσ^{1}dσ^{2}) |(σ_{z}^{1} − σ_{z}^{2})^{2} \frac{f(σ^{1})^{2}}{E^{Λ}(f^{2})}|
≤ 2C_{3.35}E^{Λ}(f^{2}) + 2E^{Λ} (f^{2} log \frac{f^{2}}{E^{Λ}(f^{2})}) .  (3.36)

Putting (3.30), (3.33), (3.34) and (3.36) together, we obtain

I_{1}(f^{2}) + ∑_{z∈Δ} I_{2}(f^{2}, z)

≤ I_{1}(f^{2}) + \frac{1}{\sqrt{2}}I_{3}(f)^{1/2} ∑_{z∈Δ} I_{5}(f, z)^{1/2}
≤ E^{Λ}(f; f) + \frac{2}{\sqrt{2}}E^{Λ}(f; f)^{1/2}

·\sqrt{2}|Δ| (\sqrt{C_{3.35}}E^{Λ}(f^{2})^{1/2} + E^{Λ} (f^{2} log \frac{f^{2}}{E^{Λ}(f^{2})})^{1/2})

≤ C_{3.37}|Δ|E^{Λ}(f^{2})^{1/2} (E^{Λ}(f; f)^{1/2} + E^{Λ} (f^{2} log \frac{f^{2}}{E^{Λ}(f^{2})})^{1/2})  (3.37)

which, in conjunction with (3.32), implies (3.20). □

4 Proof of Theorem 2.2

In this section, we prove Theorem 2.2 by using lemmas presented in the previous section. We first consider the case F = A. What we want to prove is equivalent to that

sup_{n≥1} γ_{n} < ∞,  (4.1)

where

γ_{n} = sup{γ_{SG}(Λ) | |Λ| ≤ n}.  (4.2)

To prove (4.1), it is enough to find some constants C_{4.3} and N_{4.3} which depend only on d, R, U, J, B_{2.1} and C_{2.1} such that

γ_{2n} ≤ \frac{4}{5}γ_{n} + C_{4.3}, for n ≥ N_{4.3}.  (4.3)

To this end, we take arbitrary Λ ⊂⊂ Z^{d} with |Λ| ≤ 2n and 0 < f ∈ C_{Λ}. We then choose Λ_{0} ⊂ Λ such that max{|Λ_{0}|, |Λ\Λ_{0}|} ≤ n and define

Λ_{j} = Λ_{0} ∪ {x_{1}, …, x_{j}}, j = 1, 2, …, m,  (4.4)
f_{j} = E^{Λ_{j}}f,  (4.5)
Λ_{j,k} = \{x ∈ Λ_{j} ; d(x, x_{j+1}) < \frac{k}{2} \} , k = 0, 1, 2, …,  (4.6)
f_{j,k} = f, if k ≥ 0 and Λ_{j,k} = ϕ,
f_{j,k} = E^{Λ_{j,k}}f, if k ≥ 1 and Λ_{j,k} ≠ ϕ,  (4.7)

where {x_{j}}_{j=1}^{m} is an enumeration of Λ\Λ_{0}. We will prove (4.3) after a series of lemmas.

Lemma 4.1

E^{Λ} (f; f) ≤ γ_{n}E^{Λ} (|∇_{Λ_{0}}f|^{2}) + C_{4.8}E^{Λ} (|∇_{Λ\Λ_{0}}f|^{2}) + C_{4.8} ∑_{j=0}^{m−1} E^{Λ}Q_{j}(f),  (4.8)

where C_{4.8} = C_{4.8}(C_{3.6}(1), ∥J∥) ∈ (0, ∞) and

Q_{j}(f) = sup {E^{Λ_{j}}(f; σ_{x})^{2} ; x ∈ Λ_{j}, d(x, x_{j+1}) ≤ R} .  (4.9)

Proof: We first divide the left-hand-side of (4.8) into two terms;

E^{Λ} (f; f) = E^{Λ} (f^{2} − f_{0}^{2}) + E^{Λ} (f_{0}^{2} − f_{m}^{2}) .  (4.10)

The first term on the right-hand-side can be estimated as follows;

E^{Λ} (f^{2} − f_{0}^{2}) = E^{Λ}E^{Λ_{0}} (f^{2} − f_{0}^{2})
≤ γ_{n}E^{Λ}E^{Λ_{0}} (|∇_{Λ_{0}}f|^{2})
= γ_{n}E^{Λ} (|∇_{Λ_{0}}f|^{2}) .  (4.11)

As for the second term, we have

E^{Λ} (f_{0}^{2} − f_{m}^{2}) = ∑_{j=0}^{m−1} E^{Λ} (f_{j}^{2} − f_{j+1}^{2})
= ∑_{j=0}^{m−1} E^{Λ}E^{Λ_{j+1}} (f_{j}^{2} − f_{j+1}^{2})  (4.12)

Note that S_{f_{j}} ∩ Λ_{j+1} = {x_{j+1}} and hence by (3.6) and (3.21) that

E^{Λ_{j+1}} (f_{j}^{2} − f_{j+1}^{2}) ≤ C_{3.6}(1)E^{Λ_{j+1}} (|∇_{x_{j+1}}f_{j}|^{2})
≤ C_{4.13}E^{Λ_{j+1}} (|∇_{x_{j+1}}f|^{2}) + C_{4.13}E^{Λ_{j+1}}Q_{j}(f),  (4.13)

where C_{4.13} = C_{4.13}(C_{3.6}(1), ∥J∥). It follows from (4.12) and (4.13) that

E^{Λ} (f_{0}^{2} − f_{m}^{2}) ≤ C_{4.13}E^{Λ} (|∇_{Λ\Λ_{0}}f|^{2}) + C_{4.13} ∑_{j=0}^{m−1} E^{Λ}Q_{j}(f).  (4.14)

By (4.10),(4.11) and (4.14), we conclude (4.8). □

Lemma 4.2

Q_{j}(f) ≤ C_{4.15} ∑_{k=0}^{∞} exp (−\frac{k}{2C_{2.1}} ) E^{Λ_{j}} (E^{Λ_{j,k+1}}(f; f)^{1/2})^{2} ,  (4.15)

where C_{4.15} = C_{4.15}(d, R, C_{2.1}, C_{3.19}) ∈ (0, ∞). Proof: Suppose that x ∈ Λ_{j} and d(x, x_{j+1}) ≤ R. We then have that

E^{Λ_{j}}(f; σ_{x})^{2} = E^{Λ_{j}} ((f − f_{j})σ_{x})^{2}
= (∑_{k=0}^{∞} E^{Λ_{j}} ((f_{j,k} − f_{j,k+1})σ_{x}))^{2}
= (∑_{k=0}^{∞} E^{Λ_{j}}E^{Λ_{j,k+1}}(f_{j,k}; σ_{x}))^{2}
≤ C_{4.16} ∑_{k=0}^{∞}(k + 1)^{2} (E^{Λ_{j}}E^{Λ_{j,k+1}}(f_{j,k}; σ_{x}))^{2}  (4.16)

Since Λ_{j,k+1} ∩ S_{f_{j,k}} ⊂ Λ_{j,k+1}\Λ_{j,k} and d(x, Λ_{j,k+1}\Λ_{j,k}) ≥ \frac{k}{2} − R, it follows from (3.19) that

|E^{Λ_{j,k+1}}(f_{j,k}; σ_{x})| ≤ C_{4.17}(k + 1)^{2d} exp (−\frac{k}{2C_{2.1}} ) E^{Λ_{j,k+1}}(f_{j,k}; f_{j,k})^{1/2}  (4.17)

where C_{4.17} = C_{4.17}(R, C_{3.19}). We see from Jensen inequality that

E^{Λ_{j,k+1}}(f_{j,k}; f_{j,k}) = E^{Λ_{j,k+1}} (f_{j,k}^{2} − f_{j,k+1}^{2})
≤ E^{Λ_{j,k+1}} (f^{2} − f_{j,k+1}^{2})
= E^{Λ_{j,k+1}}(f; f)  (4.18)

Putting (4.17) and (4.18) together, we have that

E^{Λ_{j}} (E^{Λ_{j,k+1}}(f_{j,k}; σ_{x}))^{2}
≤ C_{4.17}^{2}(k + 1)^{4d} exp (−\frac{k}{C_{2.1}} ) E^{Λ_{j}} (E^{Λ_{j,k+1}}(f_{j,k}; f_{j,k})^{1/2})^{2}
≤ C_{4.17}^{2}(k + 1)^{4d} exp (−\frac{k}{C_{2.1}} ) E^{Λ_{j}} (E^{Λ_{j,k+1}}(f; f)^{1/2})^{2}

Plugging this into (4.16), we arrive at (4.15). □

Remark 4.1 This remark will become relevant when we turn to the proof of Theorem 2.2 for the case F = B(n_{0}). From Remark 3.2 and the proof of Lemma 4.1 we presented above, we see that (4.15) without the factor exp (−\frac{k}{2C_{2.1}} ) in the right-hand-side summation is true when we do not assume any mixing condition. On the other hand, as will be seen from the way (4.15) is used later (cf. (4.23) in the proof of Lemma 4.3), the factor exp (−\frac{k}{2C_{2.1}} ) in the right-hand-side summation of (4.15) are used only for sufficiently large k’s. It is thus sufficient for us to require (3.19)–(3.22) to be valid only for Λ = Λ_{j,k} with sufficiently large k’s.

Lemma 4.3 For k_{0} = 1, 2, …, ⌊n^{1/d}⌋ − 1,

E^{Λ} (f; f) ≤ γ_{n}E^{Λ} (|∇_{Λ_{0}}f|^{2}) + γ_{n}C_{4.19} exp (−\frac{k_{0}}{3C_{2.1}} ) E^{Λ} (|∇_{Λ}f|^{2})
+D_{4.19}E^{Λ} (|∇_{Λ}f|^{2}) + C_{4.19} exp (−\frac{n^{1/d}}{3C_{2.1}} ) E^{Λ} (f; f)  (4.19)

where C_{4.19} = C_{4.19}(d, U, J, B_{2.1}, C_{2.1}) and D_{4.19} = D_{4.19}(k_{0}, d, U, J, B_{2.1}, C_{2.1})

Proof: We see from (4.15) that

∑_{j=0}^{m−1} E^{Λ}Q_{j}(f) ≤ C_{4.15} ∑_{k=0}^{∞} exp (−\frac{k}{2C_{2.1}} ) ∑_{j=0}^{m−1} E^{Λ}E^{Λ_{j,k+1}}(f; f).  (4.20)

Let us first note that E^{Λ}E^{Λ_{j,k+1}}(f; f) has the following two upper bounds;

γ_{(k+1)^{d}}E^{Λ} (|∇_{Λ_{j,k+1}}f|^{2}) , E^{Λ} (f; f) .  (4.21)

The first bound in (4.21) come from (4.2) and the Markov property. In fact,

E^{Λ}E^{Λ_{j,k+1}}(f; f) ≤ γ_{(k+1)^{d}}E^{Λ}E^{Λ_{j,k+1}} (|∇_{Λ_{j,k+1}}f|^{2})
= γ_{(k+1)^{d}}E^{Λ} (|∇_{Λ_{j,k+1}}f|^{2}) .

The second bound E^{Λ} (f; f) in (4.21) can be seen as follows;

E^{Λ}E^{Λ_{j,k+1}}(f; f) = E^{Λ}E^{Λ_{j,k+1}}(f^{2}) − E^{Λ}(f_{j,k+1}^{2})
≤ E^{Λ}(f^{2}) − E^{Λ}(f)^{2}

where we have used Jensen inequality and Markov property.

We will divide the summation in k on the right-hand-side of (4.20) in three parts as follows;

∑_{k=0}^{∞} = ∑_{k=0}^{k_{0}−1} + ∑_{k=k_{0}}^{k_{1}−1} + ∑_{k=k_{1}}^{∞},  (4.22)

where k_{1} = ⌊n^{1/d}⌋ − 1. We use the first bound in (4.21) to estimate the first and the second summations in (4.22) as follows;

∑_{k=0}^{k_{0}−1} ≤ ∑_{k=0}^{k_{0}−1} exp (−\frac{k}{2C_{2.1}} ) ∑_{j=0}^{m−1} γ_{(k+1)^{d}}E^{Λ} (|∇_{Λ_{j,k+1}}f|^{2})

≤ ∑_{x∈Λ} E^{Λ} (|∇_{x}f|^{2}) ∑_{k=0}^{k_{0}−1} exp (−\frac{k}{2C_{2.1}} ) γ_{(k+1)^{d}} ∑_{j:Λ_{j,k+1}∋x} 1
≤ C_{4.23}(k_{0})E^{Λ} (|∇_{Λ}f|^{2}) .  (4.23)
∑_{k=k_{0}}^{k_{1}−1} ≤ ∑_{k=k_{0}}^{k_{1}−1} exp (−\frac{k}{2C_{2.1}} ) ∑_{j=0}^{m−1} γ_{n}E^{Λ} (|∇_{Λ_{j,k+1}}f|^{2})

≤ γ_{n}C_{4.24} exp (−\frac{k_{0}}{3C_{2.1}} ) E^{Λ} (|∇_{Λ}f|^{2}) .  (4.24)

Note that γ_{(k+1)^{d}} ≤ C_{3.6}(k_{0}^{d}) for k ≤ k_{0}−1 by Lemma 3.2 and hence that we can make C_{4.23}(k_{0}) depend only on k_{0}, d, U, J, B_{2.1} and C_{2.1}. To estimate the second summation in (4.22), we make use of the second bound in (4.21);

∑_{k=k_{1}}^{∞} ≤ ∑_{k=k_{1}}^{∞} exp (−\frac{k}{2C_{2.1}} ) ∑_{j=0}^{m−1} E^{Λ} (f; f)

≤ C_{4.25} exp (−\frac{n^{1/d}}{3C_{2.1}} ) E^{Λ} (f; f)  (4.25)

Now, (4.19) can be seen from (4.8), (4.20) and (4.23)–(4.25). □

Proof of (4.3): With Lemma 4.3 in hand, (4.3) can be proved as follows. By exchanging the role of Λ_{0} and Λ\Λ_{0}, we have that for k_{0} ≤ ⌊n^{1/d}⌋ − 1,

E^{Λ} (f; f)
≤ γ_{n}E^{Λ} (|∇_{Λ\Λ_{0}}f|^{2}) + γ_{n}C_{4.19} exp (−\frac{k_{0}}{3C_{2.1}} ) E^{Λ} (|∇_{Λ}f|^{2})
+D_{4.19}E^{Λ} (|∇_{Λ}f|^{2}) + C_{4.19} exp (−\frac{n^{1/d}}{3C_{2.1}} ) E^{Λ} (f; f) ,  (4.26)

and hence by averaging (4.19) and (4.26) that

E^{Λ} (f; f)
≤ \frac{γ_{n}}{2} E^{Λ} (|∇_{Λ}f|^{2}) + γ_{n}C_{4.19} exp (−\frac{k_{0}}{3C_{2.1}} ) E^{Λ} (|∇_{Λ}f|^{2})
+D_{4.19}E^{Λ} (|∇_{Λ}f|^{2}) + C_{4.19} exp (−\frac{n^{1/d}}{3C_{2.1}} ) E^{Λ} (f; f) .  (4.27)

Since our choice of Λ and 0 < f ∈ C_{Λ} was arbitrary as long as |Λ| ≤ 2n, we see from (4.27) that

γ_{2n} ≤ \frac{(\frac{1}{2} + C_{4.19} exp (−\frac{k_{0}}{3C_{2.1}} )) γ_{n} + D_{4.19}}{1 − C_{4.19} exp (−\frac{n^{1/d}}{3C_{2.1}} )} .  (4.28)

At this point, we choose k_{0} such that \frac{1}{2} + C_{4.19} exp (−\frac{k_{0}}{3C_{2.1}} ) < \frac{3}{5}. We then have (4.3) with

C_{4.3} = \frac{4}{3}D_{4.19}, whenever 1 − C_{4.19} exp (−\frac{n^{1/d}}{3C_{2.1}} ) ≥ \frac{3}{4}. □

Proof of Theorem 2.2 for F = B(n_{0}): We modify the proof for the case F = A as follows. The goal is equivalent to that;

sup_{n_{0}≤n_{i}<∞} γ(n_{1}, …, n_{d}) < ∞,  (4.29)

where

γ(n_{1}, …, n_{d}) = sup \{γ_{SG}(Λ) ; \begin{matrix}Λ is a generalized box with size (m_{1}, …, m_{d}), \\ n_{0} ≤ m_{i} ≤ n_{i}, 1 ≤ i ≤ d \end{matrix} \} . (4.30)

To prove (4.29), it is enough to find some constants C_{4.31} and N_{4.31} which depend only on d, R, U, J, n_{0},B_{2.1} and C_{2.1} such that

γ(2n_{1}, n_{2}, …, n_{d}) ≤ \frac{4}{5}γ(n_{1}, n_{2}, …, n_{d}) + C_{4.31},  (4.31)

for n_{i} ≥ N_{4.31}. To this end, we take arbitrary Λ ∈ B(n_{0}) with the size at most (2n_{1}, n_{2}, …, n_{d}) and 0 < f ∈ C_{Λ}. We then decompose Λ into Λ_{0} ∈ B(n_{0}) and Λ\Λ_{0} ∈ B(n_{0}) such that both of them are of the size at most (n_{1}, n_{2}, …, n_{d}). We can now choose an enumeration {x_{j}}_{j=1}^{m} of Λ\Λ_{0} so that Λ_{j} ∈ B(n_{0}) for all j = 1, …, m, where Λ_{j} is defined by (4.4). Note also that Λ_{j,k} ∈ B(n_{0}) for k > 2n_{0}, where Λ_{j,k} is defined by (4.6). We thus see that the proof of (4.3) for the case F = A works almost without change (Recall that we need to apply (3.19)–(3.22) to the set Λ_{j,k} with k > 2n_{0}, cf. Remark 4.1). □

5 Proof of Theorem 2.1

In this section, we prove Theorem 2.1 by using Theorem 2.2 as well as lemmas presented in Section 3. The basic strategy is the same as that in the proof of Theorem 2.2. We first consider the case F = A. What we want to prove is equivalent to that

sup_{n≥1} γ_{n} < ∞,  (5.1)

where

γ_{n} = sup{γ_{LS}(Λ) | |Λ| ≤ n}.  (5.2)

To prove (5.1), it is enough to find some constants C_{5.3} and N_{5.3} which depend only on d, R, U, J, B_{2.1} and C_{2.1} such that

γ_{2n} ≤ \frac{4}{5}γ_{n} + C_{5.3}, for n ≥ N_{5.3}.  (5.3)

To this end, we take arbitrary Λ ⊂⊂ Z^{d} with |Λ| ≤ 2n and 0 < f ∈ C_{Λ}. We then choose Λ_{0} ⊂ Λ such that max{|Λ_{0}|, |Λ\Λ_{0}|} ≤ n and define

Λ_{j} = Λ_{0} ∪ {x_{1}, …, x_{j}}, j = 1, 2, …, m,  (5.4)
f_{j} = \sqrt{E^{Λ_{j}}(f^{2})},  (5.5)
Λ_{j,k} = \{x ∈ Λ_{j} ; d(x, x_{j+1}) < \frac{k}{2} \} , k = 0, 1, 2, …,  (5.6)
f_{j,k} = f, if k ≥ 0 and Λ_{j,k} = ϕ,
f_{j,k} = \sqrt{E^{Λ_{j,k}}(f^{2})}, if k ≥ 1 and Λ_{j,k} ≠ ϕ,  (5.7)

where {x_{j}}_{j=1}^{m} is an enumeration of Λ\Λ_{0}.We will prove (5.3) after a series of lemmas.

Lemma 5.1

E^{Λ} (f^{2} log \frac{f^{2}}{E^{Λ} (f^{2})}) ≤ γ_{n}E^{Λ} (|∇_{Λ_{0}}f|^{2}) + C_{5.8}E^{Λ} (|∇_{Λ\Λ_{0}}f|^{2}) + C_{5.8} ∑_{j=0}^{m−1} E^{Λ} (\frac{R_{j}(f)}{f_{j}^{2}}) , (5.8)

where C_{5.8} = C_{5.8}(C_{3.6}(1), ∥J∥) ∈ (0, ∞) and

R_{j}(f) = sup {E^{Λ_{j}}(f^{2}; σ_{x})^{2} ; x ∈ Λ_{j}, d(x, x_{j+1}) ≤ R} .  (5.9)

Proof: We first divide the left-hand-side of (5.8) into two terms;

E^{Λ} (f^{2} log \frac{f^{2}}{E^{Λ} (f^{2})}) = E^{Λ} (f^{2} log \frac{f^{2}}{f_{0}^{2}} ) + E^{Λ} (f^{2} log \frac{f_{0}^{2}}{f_{m}^{2}} ) .  (5.10)

The first term on the right-hand-side can be estimated as follows;

E^{Λ} (f^{2} log \frac{f^{2}}{f_{0}^{2}} ) = E^{Λ}E^{Λ_{0}} (f^{2} log \frac{f^{2}}{f_{0}^{2}} )
≤ γ_{n}E^{Λ}E^{Λ_{0}} (|∇_{Λ_{0}}f|^{2})
= γ_{n}E^{Λ} (|∇_{Λ_{0}}f|^{2}) .  (5.11)

As for the second term, we have

E^{Λ} (f^{2} log \frac{f_{0}^{2}}{f_{m}^{2}} ) = ∑_{j=0}^{m−1} E^{Λ} (f_{j}^{2} log f_{j}^{2} − f_{j+1}^{2} log f_{j+1}^{2})

= ∑_{j=0}^{m−1} E^{Λ} (f_{j}^{2} log f_{j}^{2} − E^{Λ_{j+1}}(f_{j}^{2}) log f_{j+1}^{2})

= ∑_{j=0}^{m−1} E^{Λ}E^{Λ_{j+1}} (f_{j}^{2} log \frac{f_{j}^{2}}{f_{j+1}^{2}} ) .  (5.12)

Note that S_{f_{j}} ∩ Λ_{j+1} = {x_{j+1}} and hence by (3.6) and (3.23) that

E^{Λ_{j+1}} (f_{j}^{2} log \frac{f_{j}^{2}}{f_{j+1}^{2}} ) ≤ C_{3.6}(1)E^{Λ_{j+1}} (|∇_{x_{j+1}}f_{j}|^{2})
≤ C_{5.13}E^{Λ_{j+1}} (|∇_{x_{j+1}}f|^{2}) + C_{5.13} \frac{R_{j}(f)}{f_{j}^{2}},  (5.13)

where where C_{5.13} = C_{5.13}(C_{3.6}(1), ∥J∥) ∈ (0, ∞). It follows from (5.12) and (5.13) that

E^{Λ} (f^{2} log \frac{f_{0}^{2}}{f_{m}^{2}} ) ≤ C_{5.13}E^{Λ} (|∇_{Λ\Λ_{0}}f|^{2}) + C_{5.13} ∑_{j=0}^{m−1} E^{Λ} (\frac{R_{j}(f)}{f_{j}^{2}}) .  (5.14)

By (5.10),(5.11) and (5.14), we conclude (5.8). □

Lemma 5.2

\frac{R_{j}(f)}{f_{j}^{2}} ≤ C_{5.15} ∑_{k=0}^{∞} exp (−\frac{k}{2C_{2.1}} ) E^{Λ_{j}} (|∇_{Λ_{j,k+1}}f|^{2} + f^{2} log \frac{f^{2}}{f_{j,k}^{2}} + f^{2} log \frac{f^{2}}{f_{j,k+1}^{2}} ) , (5.15)

where C_{5.15} = C_{5.15}(d, R, C_{2.1}, C_{2.5}, C_{3.19}) ∈ (0, ∞). Proof: Suppose that x ∈ Λ_{j} and d(x, x_{j+1}) ≤ R. We then have that

E^{Λ_{j}}(f^{2}; σ_{x})^{2} = E^{Λ_{j}} ((f^{2} − f_{j}^{2})σ_{x})^{2}
= (∑_{k=0}^{∞} E^{Λ_{j}} ((f_{j,k}^{2} − f_{j,k+1}^{2})σ_{x}))^{2}
= (∑_{k=0}^{∞} E^{Λ_{j}}E^{Λ_{j,k+1}}(f_{j,k}^{2}; σ_{x}))^{2}
≤ C_{5.16} ∑_{k=0}^{∞}(k + 1)^{2}E^{Λ_{j}} (E^{Λ_{j,k+1}}(f_{j,k}^{2}; σ_{x}))^{2}  (5.16)

Since Λ_{j,k+1} ∩ S_{f_{j,k}} ⊂ Λ_{j,k+1}\Λ_{j,k} and d(x, Λ_{j,k+1}\Λ_{j,k}) ≥ \frac{k}{2} − R, it follows from (3.20) that

|E^{Λ_{j,k+1}}(f_{j,k}^{2}; σ_{x})| ≤ C_{5.17}(k + 1)^{2d} exp (−\frac{k}{2C_{2.1}} ) f_{j,k+1} (I_{1}^{1/2} + I_{2}^{1/2}) ,  (5.17)

where C_{5.17} = C_{5.17}(R, C_{3.19}),

I_{1} = E^{Λ_{j,k+1}} (f_{j,k}; f_{j,k}) and I_{2} = E^{Λ_{j,k+1}} (f_{j,k}^{2} log \frac{f_{j,k}^{2}}{f_{j,k+1}^{2}} ) .

I_{1} can be estimated as follows;

I_{1} ≤ C_{2.5}E^{Λ_{j,k+1}} (|∇_{Λ_{j,k+1}}f_{j,k}|^{2})  (5.18)
≤ 2C_{2.5}E^{Λ_{j,k+1}} (|∇_{Λ_{j,k+1}}f|^{2})
+C_{5.19}(k + 1)^{5d} (E^{Λ_{j,k+1}}(f; f) + E^{Λ_{j,k+1}} (f^{2} log \frac{f^{2}}{f_{j,k}^{2}} ))  (5.19)
≤ C_{5.20}(k + 1)^{5d} (E^{Λ_{j,k+1}} (|∇_{Λ_{j,k+1}}f|^{2}) + E^{Λ_{j,k+1}} (f^{2} log \frac{f^{2}}{f_{j,k}^{2}} )) .  (5.20)

Here, we have used (2.5) in both (5.18) and (5.20), whereas (5.19) is an application of (3.24). On the other hand, we see from Jensen inequality that

I_{2} ≤ E^{Λ_{j,k+1}} (f^{2} log \frac{f^{2}}{f_{j,k+1}^{2}} ) .  (5.21)

Putting (5.17), (5.20) and (5.21) together, we have that

E^{Λ_{j}} (E^{Λ_{j,k+1}}(f_{j,k}^{2}; σ_{x}))^{2}
≤ C_{5.17}^{2}(k + 1)^{4d} exp (−\frac{k}{C_{2.1}} ) E^{Λ_{j}} (f_{j,k+1} (I_{1}^{1/2} + I_{2}^{1/2}))^{2}
≤ 2C_{5.17}^{2}(k + 1)^{4d} exp (−\frac{k}{C_{2.1}} ) f_{j}^{2}E^{Λ_{j}}(I_{1} + I_{2})
≤ C_{5.22}(k + 1)^{9d} exp (−\frac{k}{C_{2.1}} )
·f_{j}^{2}E^{Λ_{j}} (|∇_{Λ_{j,k+1}}f|^{2} + f^{2} log \frac{f^{2}}{f_{j,k}^{2}} + f^{2} log \frac{f^{2}}{f_{j,k+1}^{2}} ) .  (5.22)

Plugging this into (5.16), we arrive at the following bound;

\frac{E^{Λ_{j}}(f^{2}; σ_{x})^{2}}{f_{j}^{2}} ≤ C_{5.23} ∑_{k=0}^{∞}(k + 1)^{9d+2} exp (−\frac{2k}{C_{2.1}} )
·E^{Λ_{j}} (|∇_{Λ_{j,k+1}}f|^{2} + f^{2} log \frac{f^{2}}{f_{j,k}^{2}} + f^{2} log \frac{f^{2}}{f_{j,k+1}^{2}} ) ,  (5.23)

which proves (5.15). □

Remark 5.1 This remark will become relevant when we turn to the proof of Theorem 2.1 for the case F = B(n_{0}). From Remark 3.2 and the proof of Lemma 5.1 we presented above, we see that (5.15) without the factor exp (−\frac{k}{C_{2.1}} ) in the right-hand-side summation is true when we do not assume any mixing condition. On the other hand, as will be seen from the way (5.15) is used later (cf. (5.28) in the proof of Lemma 5.3), the factor exp (−\frac{k}{C_{2.1}} ) in the right-hand-side summation of (5.15) are used only for sufficiently large k’s. It is thus sufficient for us to require (3.19)–(3.22) to be valid only for Λ = Λ_{j,k} with sufficiently large k’s.

Lemma 5.3 For k_{0} = 1, 2, …, ⌊n^{1/d}⌋ − 1,

E^{Λ} (f^{2} log \frac{f^{2}}{E^{Λ}(f^{2})})

≤ γ_{n}E^{Λ} (|∇_{Λ_{0}}f|^{2}) + γ_{n}C_{5.24} exp (−\frac{k_{0}}{3C_{2.1}} ) E^{Λ} (|∇_{Λ}f|^{2})
+D_{5.24}E^{Λ} (|∇_{Λ}f|^{2}) + C_{5.24} exp (−\frac{n^{1/d}}{3C_{2.1}} ) E^{Λ} (f^{2} log \frac{f^{2}}{E^{Λ}(f^{2})})  (5.24)

where C_{5.24} = C_{5.24}(d, U, J, B_{2.1}, C_{2.1}) and D_{5.24} = D_{5.24}(k_{0}, d, U, J, B_{2.1}, C_{2.1}).

Proof: We see from (5.15) that

∑_{j=0}^{m−1} E^{Λ} (\frac{R_{j}(f)}{f_{j}^{2}}) ≤ C_{5.15} ∑_{k=0}^{∞} exp (−\frac{k}{2C_{2.1}} )

· ∑_{j=0}^{m−1} E^{Λ} (|∇_{Λ_{j,k+1}}f|^{2} + f^{2} log \frac{f^{2}}{f_{j,k}^{2}} + f^{2} log \frac{f^{2}}{f_{j,k+1}^{2}} ) . (5.25)

Let us first note that E^{Λ} (f^{2} log \frac{f^{2}}{f_{j,k}^{2}} + f^{2} log \frac{f^{2}}{f_{j,k+1}^{2}} ) has the following two upper bounds;

2γ_{(k+1)^{d}}E^{Λ} (|∇_{Λ_{j,k+1}}f|^{2}) , 2E^{Λ} (f^{2} log \frac{f^{2}}{E^{Λ}(f^{2})}) .  (5.26)

The first bound in (5.26) comes from (5.2) and the Markov property. For example,

E^{Λ} (f^{2} log \frac{f^{2}}{f_{j,k}^{2}} ) ≤ E^{Λ}E^{Λ_{j,k}} (f^{2} log \frac{f^{2}}{f_{j,k}^{2}} )
≤ γ_{k^{d}}E^{Λ}E^{Λ_{j,k}} (|∇_{Λ_{j,k}}f|^{2})
= γ_{k^{d}}E^{Λ} (|∇_{Λ_{j,k}}f|^{2}) .

The second bound E^{Λ} (f^{2} log \frac{f^{2}}{E^{Λ}(f^{2})}) in (5.26) can be seen as follows;

E^{Λ} (f^{2} log \frac{f^{2}}{f_{j,k}^{2}} ) = E^{Λ} (f^{2} log(f^{2})) − E^{Λ} (f^{2} log(f_{j,k}^{2}))
= E^{Λ} (f^{2} log(f^{2})) − E^{Λ} (f_{j,k}^{2} log(f_{j,k}^{2}))
≤ E^{Λ} (f^{2} log(f^{2})) − E^{Λ}(f^{2}) log E^{Λ}(f^{2}),

where we have used Jensen inequality and Markov property in the last line.

We will divide the summation in k on the right-hand-side of (5.25) in three parts as follows;

∑_{k=0}^{∞} = ∑_{k=0}^{k_{0}−1} + ∑_{k=k_{0}}^{k_{1}−1} + ∑_{k=k_{1}}^{∞},  (5.27)

where k_{1} = ⌊n^{1/d}⌋ − 1. We use the first bound in (5.26) to estimate the first and the second summations in (5.27) as follows;

∑_{k=0}^{k_{0}−1} ≤ ∑_{k=0}^{k_{0}−1} exp (−\frac{k}{2C_{2.1}} ) ∑_{j=0}^{m−1} (1 + 2γ_{(k+1)^{d}}) E^{Λ} (|∇_{Λ_{j,k+1}}f|^{2})

≤ ∑_{x∈Λ} E^{Λ} (|∇_{x}f|^{2}) ∑_{k=0}^{k_{0}−1} exp (−\frac{k}{2C_{2.1}} ) (1 + 2γ_{(k+1)^{d}}) ∑_{j:Λ_{j,k+1}∋x} 1
≤ C_{5.28}(k_{0})E^{Λ} (|∇_{Λ}f|^{2}) .  (5.28)
∑_{k=k_{0}}^{k_{1}−1} ≤ ∑_{k=k_{0}}^{k_{1}−1} exp (−\frac{k}{2C_{2.1}} ) ∑_{j=0}^{m−1} (1 + 2γ_{n}) E^{Λ} (|∇_{Λ_{j,k+1}}f|^{2})

≤ (1 + 2γ_{n}) C_{5.29} exp (−\frac{k_{0}}{3C_{2.1}} ) E^{Λ} (|∇_{Λ}f|^{2}) .  (5.29)

Note that γ_{(k+1)^{d}} ≤ C_{3.6}(k_{0}^{d}) for k ≤ k_{0}−1 by Lemma 3.2 and hence that we can make C_{5.28}(k_{0}) depend only on k_{0}, d, U, J, B_{2.1} and C_{2.1}. To estimate the second summation in (5.27), we make use of the second bound in (5.26);

∑_{k=k_{1}}^{∞} ≤ ∑_{k=k_{1}}^{∞} exp (−\frac{k}{2C_{2.1}} ) ∑_{j=0}^{m−1} (E^{Λ} (|∇_{Λ_{j,k+1}}f|^{2}) + 2E^{Λ} (f^{2} log \frac{f^{2}}{E^{Λ}(f^{2})}))

≤ C_{5.30}E^{Λ} (|∇_{Λ}f|^{2}) + C_{5.30} exp (−\frac{n^{1/d}}{3C_{2.1}} ) E^{Λ} (f^{2} log \frac{f^{2}}{E^{Λ}(f^{2})})  (5.30)

Now, (5.24) can be seen from (5.8), (5.25) and (5.28)–(5.30). □

Proof of (5.3): With Lemma 5.3 in hand, (5.3) can be proved in the same way as we derived (4.3) from Lemma 4.3. □

6 Proof of Proposition 2.3 and Theorem 2.4

In this section, we prove Proposition 2.3 and Theorem 2.4. The proof of Proposition 2.3 is based on the two lemmas presented below.

Lemma 6.1 For W ⊂ Λ and for \overline{p}^{i} ∈ R^{Λ} (i = 1, 2), there exists a measure E^{W,\overline{p}^{1},\overline{p}^{2}} ∈ K(E_{Λ,q}^{W,\overline{p}^{1}}, E_{Λ,q}^{W,\overline{p}^{2}}) such that

∑_{z∈W ∩(x+V)} f_{z}^{W,\overline{p}^{1},\overline{p}^{2}} ≤ ∑_{z} K_{x,z}f_{z}^{W,\overline{p}^{1},\overline{p}^{2}},  (6.1)

where

f_{z}^{W,\overline{p}^{1},\overline{p}^{2}} = ∫ E^{W,\overline{p}^{1},\overline{p}^{2}}(dp^{1}dp^{2})|p_{z}^{1} − p_{z}^{2}|.  (6.2)

Proof: Let us take E^{W,\overline{p}^{1},\overline{p}^{2}} ∈ K(E_{Λ,q}^{W,\overline{p}^{1}}, E_{Λ,q}^{W,\overline{p}^{2}}) which attains the Vassershtein distance of E_{Λ,q}^{W,\overline{p}^{i}} (i = 1, 2), i.e.,

∑_{z∈W} f_{z}^{W,\overline{p}^{1},\overline{p}^{2}} = R_{W} (E_{Λ,q}^{W,\overline{p}^{1}}, E_{Λ,q}^{W,\overline{p}^{2}}).  (6.3)

The existence of such measure is guaranteed by the compactness of the set K(E_{Λ,q}^{W,\overline{p}^{1}}, E_{Λ,q}^{W,\overline{p}^{2}}) and the fact that the map μ ↦ ∫ μ(dp^{1}dp^{2})|p_{z}^{1} − p_{z}^{2}| from M_{1}(R^{W} × R^{W}) to [0, ∞) is lower semi-continuous.

We next take a measure

Ê^{x}(·|p̂^{1}, p̂^{2}) ∈ K(E_{Λ,q}^{W ∩(x+V),p^{1}}, E_{Λ,q}^{W ∩(x+V),p^{2}})

in such a way that it attains the Vassershtein distance of E_{Λ,q}^{W ∩(x+V),p^{i}} (i = 1, 2), i.e.,

R_{W ∩(x+V)} (E_{Λ,q}^{W ∩(x+V),p^{1}}, E_{Λ,q}^{W ∩(x+V),p^{2}}) = ∫ Ê^{x}(dp^{1}dp^{2}|p̂^{1}, p̂^{2}) ∑_{z∈W ∩(x+V)} |p_{z}^{1} − p_{z}^{2}|,  (6.4)

and that the map (p̂^{1}, p̂^{2}) ↦ Ê^{x}(·|p̂^{1}, p̂^{2}) from R^{Λ} × R^{Λ} to M_{1}(R^{W ∩(x+V)} × R^{W ∩(x+V)}) is measurable. The possibility of such measurable selection can be shown as an apllication of [SV79, Thorem 12.1.10]. (Use also Lemma 12.1.7 in that book to check that the set of minimizers of the Vassershtein distance of E_{Λ,q}^{W ∩(x+V),p^{i}} (i = 1, 2) is measurable as a set-valued function of (p̂^{1}, p̂^{2})).

We now define a measure Ẽ^{x,\overline{p}^{1},\overline{p}^{2}} ∈ M_{1}(R^{W} × R^{W}) by

Ẽ^{x,\overline{p}^{1},\overline{p}^{2}}(A × B) = ∫_{A}E^{W,\overline{p}^{1},\overline{p}^{2}}(dp̂^{1}dp̂^{2})∫_{B}Ê^{x,\overline{p}^{1},\overline{p}^{2}}(dp^{1}dp^{2}|p̂^{1}, p̂^{2}),

where A ⊂ R^{W \(x+V)} × R^{W \(x+V)} and B ⊂ R^{(x+V)∩W} × R^{(x+V)∩W}. It follows from the above definition that

Ẽ^{x,\overline{p}^{1},\overline{p}^{2}} ∈ K(E_{Λ,q}^{W,\overline{p}^{1}}, E_{Λ,q}^{W,\overline{p}^{2}}),  (6.5)
Ẽ^{x,\overline{p}^{1},\overline{p}^{2}} = E^{W,\overline{p}^{1},\overline{p}^{2}} on R^{W \(x+V)} × R^{W \(x+V)}.  (6.6)

To see (6.1), it is sufficient to prove that

∑_{z∈W ∩(x+V)} f_{z}^{W,\overline{p}^{1},\overline{p}^{2}} ≤ ∑_{z∈W ∩(x+V)} f̃_{z}^{x,\overline{p}^{1},\overline{p}^{2}},  (6.7)
∑_{z∈W ∩(x+V)} f̃_{z}^{x,\overline{p}^{1},\overline{p}^{2}} ≤ ∑_{z} K_{x,z}f_{z}^{W,\overline{p}^{1},\overline{p}^{2}},  (6.8)

where

f̃_{z}^{x,\overline{p}^{1},\overline{p}^{2}} = ∫ Ẽ^{x,\overline{p}^{1},\overline{p}^{2}}(dp^{1}dp^{2})|p_{z}^{1} − p_{z}^{2}|.  (6.9)

The first inequality (6.7) can be seen as follows. Since (6.6) implies that f_{z}^{W,\overline{p}^{1},\overline{p}^{2}} = f̃_{z}^{x,\overline{p}^{1},\overline{p}^{2}} for z ∉ W ∩ (x + V), we have from this, (6.3) and (6.5) that

∑_{z∈W ∩(x+V)} (f_{z}^{W,\overline{p}^{1},\overline{p}^{2}} − f̃_{z}^{x,\overline{p}^{1},\overline{p}^{2}}) = ∑_{z∈W} (f_{z}^{W,\overline{p}^{1},\overline{p}^{2}} − f̃_{z}^{x,\overline{p}^{1},\overline{p}^{2}})
≤ ∑_{z∈W} f_{z}^{W,\overline{p}^{1},\overline{p}^{2}} − R_{W} (E_{Λ,q}^{W,\overline{p}^{1}}, E_{Λ,q}^{W,\overline{p}^{2}})
= 0.

To prove the second inequality (6.8), we will use (6.4) and (2.8) as follows;

∑_{z∈W ∩(x+V)} f̃_{z}^{x,\overline{p}^{1},\overline{p}^{2}} = ∫ E^{W,\overline{p}^{1},\overline{p}^{2}}(dp̂^{1}dp̂^{2})∫ Ê^{x,\overline{p}^{1},\overline{p}^{2}}(dp^{1}dp^{2}|p̂^{1}, p̂^{2}) ∑_{z∈W ∩(x+V)} |p_{z}^{1} − p_{z}^{2}|
≤ ∫ E^{W,\overline{p}^{1},\overline{p}^{2}}(dp̂^{1}dp̂^{2}) ∑_{z} K_{x,z}|p̂_{z}^{1} − p̂_{z}^{2}|
= ∑_{z} K_{x,z}f_{z}^{W,\overline{p}^{1},\overline{p}^{2}}.

This completes the proof of Lemma 6.1. □

Lemma 6.2 For any A ⊂ W ⊂ Λ, L ≥ 1 and \overline{p}^{i} ∈ R^{Λ} (i = 1, 2) with \overline{p}^{1} ≡ \overline{p}^{2} off y

∑_{z∈W} f_{z}^{W,\overline{p}^{1},\overline{p}^{2}} exp (−\frac{d(z, A)}{C_{6.10}}) ≤ B_{6.10} ∑_{z∈W_{d(z,y)≤L+D_{6.10}}} f_{z}^{W,\overline{p}^{1},\overline{p}^{2}} exp (−\frac{d(z, A)}{C_{6.10}})

+ B_{6.10} ∑_{z;d(z,y)>L} K_{z,y}|\overline{p}_{y}^{1} − \overline{p}_{y}^{2}| exp (−\frac{d(z, A)}{C_{6.10}}) , (6.10)

where f_{z}^{W,\overline{p}^{1},\overline{p}^{2}} is defined by (6.2), D_{6.10} = diam(V ∪ ∂_{R}V), B_{6.10} = B_{6.10}(R, V, ε_{2.8}) and C_{6.10} = C_{6.10}(R, V, ε_{2.8}). In addition, by (2.7), the second term on the right-hand-side of (6.10) is zero when L ≥ D_{6.10}.

Proof: We set e_{x} = exp (−\frac{d(x,A)}{C_{6.10}}), where C_{6.10} = C_{6.10}(R, V, ε_{2.8}) is choosen so large that

C_{6.11} =^{def.} exp (−\frac{D_{6.10}}{C_{6.10}} ) − ε_{2.8} exp (\frac{D_{6.10}}{C_{6.10}} ) > 0.  (6.11)

We then define l_{z}^{0} = ∑_{x:x+V ∋z} e_{x}, l_{z}^{1} = ∑_{x} e_{x}K_{x,z}, l_{z} = l_{z}^{0} − l_{z}^{1} and r_{z} = ∑_{{x:x+V ∋z \atop d(x,y)≤L}} e_{x}. Let us

first prove that

∑_{z∈W} f_{z}^{W,\overline{p}^{1},\overline{p}^{2}}l_{z} ≤ ∑_{z∈W} f_{z}^{W,\overline{p}^{1},\overline{p}^{2}}r_{z} + ∑_{z:d(z,y)≥L} e_{z}K_{z,y}|\overline{p}_{y}^{1} − \overline{p}_{y}^{2}|.  (6.12)

We have by (6.1) that

∑_{x:d(x,y)>L} e_{x} ∑_{z∈W ∩(x+V)} f_{z}^{W,\overline{p}^{1},\overline{p}^{2}} ≤ ∑_{x:d(x,y)>L} e_{x} ∑_{z∈W} K_{x,z}f_{z}^{W,\overline{p}^{1},\overline{p}^{2}}
+ ∑_{x:d(x,y)>L} e_{x}K_{x,y}|\overline{p}_{y}^{1} − \overline{p}_{y}^{2}|.  (6.13)

Since

∑_{x:d(x,y)>L} e_{x} ∑_{z∈W ∩(x+V)} f_{z}^{W,\overline{p}^{1},\overline{p}^{2}} = ∑_{z∈W} f_{z}^{W,\overline{p}^{1},\overline{p}^{2}} ∑_{x:d(x,y)>L_{x+V ∋z}} e_{x}

= ∑_{z∈W} f_{z}^{W,\overline{p}^{1},\overline{p}^{2}} (l_{z}^{0} − r_{z}) ,
∑_{x:d(x,y)>L} e_{x} ∑_{z∈W} K_{x,z}f_{z}^{W,\overline{p}^{1},\overline{p}^{2}} ≤ ∑_{z∈W} f_{z}^{W,\overline{p}^{1},\overline{p}^{2}}l_{z}^{1},

it follows from (6.13) that

∑_{z∈W} f_{z}^{W,\overline{p}^{1},\overline{p}^{2}} (l_{z}^{0} − r_{z}) ≤ ∑_{z∈W} f_{z}^{W,\overline{p}^{1},\overline{p}^{2}}l_{z}^{1} + ∑_{x:d(x,y)>L} e_{x}K_{x,y}|\overline{p}_{y}^{1} − \overline{p}_{y}^{2}|,

which is equivalent to (6.12).

Let us next prove that

r_{z} ≤ C_{6.14}e_{z},  (6.14)
r_{z} = 0 if d(z, y) > L + D_{6.10},  (6.15)
l_{z} ≥ C_{6.16}e_{z},  (6.16)

where C_{6.14}, C_{6.16} ∈ (0, ∞) depend only on R, V and ε_{2.8}. To verify (6.14) and (6.15), note first that an easy to prove fact that

exp (−\frac{D_{6.10}}{C_{6.10}} ) ≤ \frac{e_{x}}{e_{z}} ≤ exp (\frac{D_{6.10}}{C_{6.10}} ) if d(x, z) ≤ D_{6.10}.  (6.17)

We thus see that

r_{z} ≤ exp (\frac{D_{6.10}}{C_{6.10}} ) e_{z} ∑_{{x:x+V ∋z \atop d(x,y)≤L}} 1,

which proves (6.14) and (6.15).

On the other hand, it follows from (6.17) and (2.8) that

l_{z} ≥ e_{z} \{exp (−\frac{D_{6.10}}{C_{6.10}} ) ∑_{x:x+V ∋z} 1 − exp (\frac{D_{6.10}}{C_{6.10}} ) ∑_{x∈Z^{d}} K_{x,z}\}
≥ e_{z} \{exp (−\frac{D_{6.10}}{C_{6.10}} ) |V | − exp (\frac{D_{6.10}}{C_{6.10}} ) ε_{2.8}|V |\}
= C_{6.11}|V |e_{z},  (6.18)

which proves (6.16).

By plugging (6.14), (6.15) and (6.16) into (6.12), we obtain (6.10). □

Proof of Proposition 2.3: Suppose that A ⊂ W ⊂ Λ. We denote by (E_{Λ,q}^{W,\overline{p}})_{A} the restriction

of E_{Λ,q}^{W,\overline{p}} to R^{A}. Let E^{W,\overline{p}^{1},\overline{p}^{2}} ∈ K (E_{Λ,q}^{W,\overline{p}^{1}}, E_{Λ,q}^{W,\overline{p}^{2}}) be the measure we have found in Lemma 6.1.

Note that the restriction of E^{W,\overline{p}^{1},\overline{p}^{2}} to R^{A} × R^{A} is an element of K ((E_{Λ,q}^{W,\overline{p}^{1}})_{A} , (E_{Λ,q}^{W,\overline{p}^{2}})_{A}). We thus have that

|∑_{z∈A} (E_{Λ,q}^{W,\overline{p}^{1}}(p_{z}) − E_{Λ,q}^{W,\overline{p}^{2}}(p_{z}))|
≤ R_{A} ((E_{Λ,q}^{W,\overline{p}^{1}})_{A}, (E_{Λ,q}^{W,\overline{p}^{2}})_{A})
≤ ∑_{z∈A} E^{W,\overline{p}^{1},\overline{p}^{2}} (|p_{z}^{1} − p_{z}^{2}|)
≤ ∑_{z∈W} E^{W,\overline{p}^{1},\overline{p}^{2}} (|p_{z}^{1} − p_{z}^{2}|) exp (−\frac{d(z, A)}{C_{6.10}})
≤ B_{6.10} ∑_{{z∈W \atop d(z,y)≤2D_{6.10}}} E^{W,\overline{p}^{1},\overline{p}^{2}} (|p_{z}^{1} − p_{z}^{2}|) exp (−\frac{d(z, A)}{C_{6.10}}) .  (6.19)

Here, in passage to the last line, we have used (6.10) with L = D_{6.10} (and thus without the second term on the right-hand-side of (6.10)). To proceed from (6.19), note that we have d(z, A) ≥ d(y, A) − 2D_{6.10} in the exponential in (6.19) and that by (3.1),

E^{W,\overline{p}^{1},\overline{p}^{2}} (|p_{z}^{1} − p_{z}^{2}|) ≤ E_{Λ,q}^{W,\overline{p}^{1}}(|p_{z}|) + E_{Λ,q}^{W,\overline{p}^{2}}(|p_{z}|)
≤ C_{3.1} (4 + ∑_{w∈Λ∩∂_{R}W} (|\overline{p}_{w}^{1}| + |\overline{p}_{w}^{2}|)) .  (6.20)

Plugging these into (6.19), we conclude that

|∑_{z∈A} (E_{Λ,q}^{W,\overline{p}^{1}}(p_{z}) − E_{Λ,q}^{W,\overline{p}^{2}}(p_{z}))| ≤ C_{6.21} (1 + ∑_{y∈Λ∩∂_{R}W} (|\overline{p}_{y}^{1}| + |\overline{p}_{y}^{2}|)) exp (−\frac{d(y, A)}{C_{6.10}}) . (6.21)

The mixing condition (2.1) can be obtained as a special case of A = {z}. □

Proof of Theorem 2.4: We will use an argument in the proof of [AKRT97, Theorem 4.1] which uses the log-Sobolev inequality to get the desired upper bound of the Vassershtein distance.

By Theorem 2.1 and Proposition 2.3, it is sufficient for us to prove part (a) of the theorem. It can be seen from the same computation as in the proof of [COPP78, Theorem 2.3] that

R_{x} (E_{Λ,q}^{x,\overline{p}^{1}}, E_{Λ,q}^{x,\overline{p}^{2}}) ≤ ∑_{y} K_{x,y}|\overline{p}_{y}^{1} − \overline{p}_{y}^{2}|  (6.22)

for all x ∈ W and \overline{p}^{i} ∈ R^{Λ} (i = 1, 2), where

K_{x,y} = \{ \begin{matrix}0, & if x = y, \\ J_{x,y} sup_{\overline{p}∈R^{Λ}} E_{Λ,q}^{x,\overline{p}}(p_{x}; p_{x}), & if x ≠ y. \end{matrix}  (6.23)

If we set V = {0} and define K = (K_{x,y} ≥ 0 : x, y ∈ Z^{d}) by (6.23), then we have (2.7) and (2.9). To see that (2.8) is satisfied if sup_{x} ∑_{y:y≠x} J_{x,y} is small enough, it is sufficient to prove that

E_{Λ,q}^{x,\overline{p}}(p_{x}; p_{x}) ≤ C_{6.24},  (6.24)

where C_{6.24} = C_{6.24}(U) ∈ (0, ∞). In fact, (6.22) and (6.24) imply that

sup_{y} ∑_{x} K_{x,y} ≤ C_{6.24} sup_{x} ∑_{y:y≠x} J_{x,y}

and therefore that (2.8) is true if sup_{x} ∑_{y:y≠x} J_{x,y} < min{1, C_{6.24}^{−1}}.

The proof of (6.24) can be given as an application of the log-Sobolev inequality to the measure E_{Λ,q}^{x,\overline{p}} as follows. We begin by decomposing U into V and W as in (1.10)–(1.13), where the parameter m > 0 is arbitrary. We then have by (1.38) that

H_{Λ,q}^{x,\overline{p}}(p_{x}) = U(p_{x}, q_{x}) − p_{x} ∑_{y∈Λ\x} J_{x,y}\overline{p}_{y}
= V(p_{x}, q_{x}) + W(p_{x}, q_{x}),  (6.25)

where U(p_{x}, q_{x}) = U (\frac{q_{x}+p_{x}}{\sqrt{2}} )+U (\frac{q_{x}−p_{x}}{\sqrt{2}} ), V(p_{x}, q_{x}) = V (\frac{q_{x}+p_{x}}{\sqrt{2}} )+V (\frac{q_{x}−p_{x}}{\sqrt{2}} )−p_{x} ∑_{y∈Λ\x} J_{x,y}\overline{p}_{y},

and W(p_{x}, q_{x}) = W (\frac{q_{x}+p_{x}}{\sqrt{2}} ) + W (\frac{q_{x}−p_{x}}{\sqrt{2}} ). Since \frac{∂^{2}}{∂p_{x}^{2}} V(p_{x}, q_{x}) ≥ m and |W(p_{x}, q_{x})| ≤ 2∥W ∥_{∞}, we see from the Bakry-Emery criterion together with a comparison argument ([HS87, Lemma 5.1], cf. proof of Lemma 3.2) that

E_{Λ,q}^{x,\overline{p}} (f^{2} log \frac{f^{2}}{E_{Λ,q}^{x,\overline{p}}(f^{2})}) ≤ γE_{Λ,q}^{x,\overline{p}} (|\frac{∂f}{∂p_{x}} |^{2})  (6.26)

for all f ∈ C_{{x}} with γ = 2 exp(16∥W ∥_{∞})/m. It is well known that (6.26) implies that

E_{Λ,q}^{x,\overline{p}} (f; f) ≤ \frac{γ}{2}E_{Λ,q}^{x,\overline{p}} (|\frac{∂f}{∂p_{x}} |^{2})  (6.27)

for all f ∈ C_{{x}} (See [DS89, Corollary 6.1.17]). Putting f(p_{x}) = p_{x} in (6.27), we get (6.24) with C_{6.24} = \frac{γ}{2}. □

Acknowledgement: I am especially indebted to several colleagues. One is M. Sugiura for discussions, from which I learned the argument in [LY93]. The others are T. Bodineau, B. Helffer, M. Ledoux and Y. Gentil for careful reading of the manuscript and for pointing out an error in the previous version. I am also very thankful T. Bodineau and B. Helffer for sending me their works [He97, BH98] prior to publication.

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