The equivalence of the log-Sobolev inequality and a mixing condition for unbounded spin systems on the lattice^{1}

Nobuo Yoshida^{2}

Abstract: We consider a ferromagnetic spin system with unbounded spin spaces on the ddimensional integer lattice (d ≥ 1). We prove the equivalence of the log-Sobolev inequality, Poincaré inequality, and the exponential decay of the spin-spin correlation, which was originally obtained by D. W. Stroock and B. Zegarlinski [23, 24] in the compact spin space setting.

Résumé: Nous considérons un système de spins ferromagnétique avec un espace de spins nonborné, sur le réseau Z^{d} (d ≥ 1). Nous montrons l’équivalence entre l’inégalité Sobolev logarithmique, l’inégalité de Poincaré et la décroissance exponentielle de la corrélation spin-spin, qui fut montrée initialement par D. W. Stroock and B. Zegarlinski [23, 24] dans le cas d’un espace de spins compact.

1Last modified May 19, 1999; printed out November 24, 2000. The title in French: Equivalence entre l’inégalité Sobolev logarithmique et une condition de mélange pour les systèmes à espaces de spins non-borné sur le réseau

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

1 Introduction

For lattice spin systems with the compact spin space, remarkable progresses have been made to understand the relation between the mixing properties of the Gibbs states and the speed at which the associated Glauber dynamics relaxes to equilibrium. In particular, the results obtained by D. W. Stroock and B. Zegarlinski [23, 24] are very impressive. They state that a mixing property (the Dobrushin-Shlosman mixing condition) and the rapid relaxation property of the Glauber dynamics are, if properly defined, not only related to each other but in fact equivalent (See also works of F. Martinelli and E. Olivieri [20, 21]). The Dobrushin-Shlosman mixing condition referred to above is known to be true for example when the underlying lattice is one dimensional, or when the interaction potential is weak enough.

In this article, we consider a ferromagnetic lattice spin system with unbounded spin space. We prove in Theorem 2.1 below that the mixing conditions for the finite volume Gibbs states as follows are equivalent (each statement being understood to be uniform in the volume and the boundary condition); (i): the uniform log-Sobolev inequality holds, (ii): the uniform Poincaré inequality holds, (iii): the spin-spin correlation decays exponentially. This equivalence can be seen as an extension of the results in [23, 24].

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 (1.7) below). 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}}.

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: 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})^{2} + ∑_{x∈Λ} (U(σ_{x}) − ∑_{y∉Λ} J_{x,y}σ_{x}ω_{y}) .  (1.6)

Here, the coupling constants J = (J_{x,y} ∈ R ; x, y ∈ Z^{d}) are such that

R(J) =^{def.} sup{d(x, y) ; J_{x,y} ≠ 0} < ∞,  (1.7)
∥J∥ =^{def.} sup_{x} ∑_{y} |J_{x,y}| < ∞,  (1.8)
J_{x,y} = J_{y,x} ≥ 0 if x ≠ y,  (1.9)

For the function U : R → R in (1.6), we consider the following conditions.

(U0) There exist V, W ∈ C^{∞}(R → R), m ∈ (0, ∞) 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 ∥_{∞} + ∥W′∥_{∞} ≤ C_{1.12},  (1.12)

where ∥W ∥_{∞} = sup_{s} |W (s)|.

(U1) For any m > 0, there exist V, W ∈ C^{∞}(R → R) and C_{1.12} ∈ (0, ∞) such that (1.10)— (1.12) hold.

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

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

where N ≥ 2, a_{1}, 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. The finite volume Gibbs state: For Λ ⊂⊂ Z^{d} and a boundary condition ω ∈ Ω, we define a Borel probability measure E^{Λ,ω} on R^{Λ} by;

E^{Λ,ω}(dσ_{Λ}) = \frac{exp −H^{Λ,ω}(σ)}{Z^{Λ,ω}} ∏_{x∈Λ} dσ_{x},  (1.14)

where Z^{Λ,ω} is the normalizing constant. The measure E^{Λ,ω} is called the finite volume Gibbs state.

The Vassershtein distance: We let P(R^{Λ}) denote the set of Borel probability measures on R^{Λ}. For a finite set Λ and ν, ν̃ ∈ P(R^{Λ}), we define

R^{Λ}(ν, ν̃) = inf {∑_{x∈Λ} ∫_{R^{Λ}×R^{Λ}}μ(dσdσ̃)|σ_{x} − σ̃_{x}| ; μ ∈ K(ν, ν̃)} .  (1.15)

where

K(ν, ν̃) = {μ ∈ P(R^{Λ} × R^{Λ}) ; μ(dσ × R^{Λ}) = ν, μ(R^{Λ} × dσ̃) = ν̃ }.  (1.16)

The function (ν, ν̃) ↦ R^{Λ}(ν, ν̃) is called the Vassershtein distance.

Expectation and Covariance. In what follows, the following common notations for the expectation and the covariance with respect to a probability measure m is used; for functions f and g,

m(f) = ∫ m(dσ)f(σ), m (f; g) = m(fg) − m(f) · m(g).  (1.17)

2 The result

We are now ready to present main results of this paper. The first result suggests that “all happy families are alike” [11].

Theorem 2.1 Consider the following conditions.

(LS) There is C_{2.1} ∈ (0, ∞) such that for all Λ ⊂⊂ Z^{d}, f ∈ C_{Λ} and ω ∈ Ω;

E^{Λ,ω} (f^{2} log \frac{f^{2}}{E^{Λ,ω}(f^{2})}) ≤ C_{2.1}E^{Λ,ω} (∑_{x∈Λ} |∇_{x}f|^{2}) .  (2.1)

(SG) There is C_{2.2} ∈ (0, ∞) such that for all Λ ⊂⊂ Z^{d}, f ∈ C_{Λ} and ω ∈ Ω;

E^{Λ,ω} (f; f) ≤ C_{2.2}E^{Λ,ω} (∑_{x∈Λ} |∇_{x}f|^{2}) .  (2.2)

(DS1) There is C_{2.3} ∈ (0, ∞) such that for all Λ ⊂⊂ Z^{d}, ω ∈ Ω and f, g ∈ C_{Λ},

|E^{Λ,ω}(f; g)| ≤ B_{2.3}(f, g) exp (−d(S_{f}, S_{g})/C_{2.3}) ,  (2.3)
where the coefficient B_{2.3}(f, g) depends only on |||f|||, |||g|||, |S_{f}| and |S_{g}|.

(DS2) There is C_{2.4} ∈ (0, ∞) such that for all Λ ⊂⊂ Z^{d}, f ∈ C_{Λ}, y ∉ Λ and ω, ω̃ ∈ Ω with ω̃ ≡ ω off y,

|E^{Λ,_{ω}}(f) − E^{Λ,ω}(f)| ≤ B_{2.4}(f)|ω̃_{y} − ω_{y}| exp (−d(S_{f}, y)/C_{2.4}) ,  (2.4)
where the coefficient B_{2.4}(f) depends only on |||f||| and |S_{f}|.

(DS3) There is C_{2.5} ∈ (0, ∞) such that for all Λ ⊂⊂ Z^{d}, ω ∈ Ω and x, y ∈ Λ

|E^{Λ,ω}(σ_{x}; σ_{y})| ≤ C_{2.5} exp (−d(x, y)/C_{2.5}) .  (2.5)

(DS4) There is C_{2.6} ∈ (0, ∞) such that for all Λ ⊂⊂ Z^{d}, z ∈ Λ, y ∉ Λ and ω, ω̃ ∈ Ω with ω̃ ≡ ω off y,

|E^{Λ,_{ω}}(σ_{z}) − E^{Λ,ω}(σ_{z})| ≤ C_{2.6}|ω̃_{y} − ω_{y}| exp (−d(z, y)/C_{2.6}) ,  (2.6)

(CC) There exist V ⊂⊂ Z^{d}, K_{2.7} ∈ (0, ∞) and ε_{2.7} ∈ (0, 1) such that

K_{2.7}|∂_{R}V | ≤ ε_{2.7}|V |,  (2.7)
R^{Λ∩(x+V)} (E^{Λ∩(x+V),ω}, E^{Λ∩(x+V),_{ω}}) ≤ K_{2.7} ∑_{y∈∂_{R}(Λ∩(x+V))} |ω_{y} − ω̃_{y}|  (2.8)

for all Λ ⊂⊂ Z^{d}, x ∈ Z^{d} and ω, ω̃ ∈ Ω.

The following hold;

a) Suppose that condition (U0) is satisfied. Then conditions listed above are related as;

(LS) ⇒ (SG) ⇒ (DS1) ⇒ (DS2)  (2.9)
⇓  ⇓
(DS3) ⇒ (DS4) ⇔ (CC).  (2.10)

b) Suppose that condition (U1) is satisfied. Then, (DS4) implies (LS), and therefore, conditions (LS), (SG), (DS1)–(DS4) and (CC) are equivalent.

Remark 2.1 Equivalence of this kind was first proved in the compact spin setting by D. Stroock and B. Zegarlinski [23, 24]. The discrete spin case in which the single spin space is a finite set was further investigated in several other papers, e.g., [19, 20, 21]. For unbounded spin case, B. Zegarlinski claims in [29, Theorem 5.1] that (DS3) implies (LS) for any d ≥ 1 and in a certain restricted sense. Unfortunately, the proof of this result is not presented in [29]. After this paper was written, the author received an article [7] by T. Bodineau and B. Helffer, which contains a version of “ (SG) ⇒ (DS1)” (where Λ is restricted to a class of fat boxes) shown by a different technique.

Remark 2.2 In the compact spin setting, R. L. Dobrushin and S. Shlosman [10, 11] introduced a mixing condition called “complete analyticity”. Conditions (DS2)–(DS4) in Theorem 2.1 can be understood as the transposition of the complete analyticity in the unbounded spin setting: for example, condition (DS2) can be compared with condition (IIIc) in [11]. Condition (CC) is similar to the “constructive criterion” C_{V} in [9]. Note however that, unlike in C_{V}, (2.8) is imposed for Λ ∩ (x + V) with arbitrary Λ ⊂⊂ Z^{d}. It is easy to verify by the argument in [3, pages 202–203] or [27, page 38–39] that (CC) is satisfied if sup_{x} ∑_{y:y≠x} J_{x,y} is sufficiently

small.

Remark 2.3 Sufficient conditions for the mixing conditions listed in Theorem 2.1 are studied in some recent papers;

• A direct application of the Bakry-Emery criterion ([4]) for (LS) is discussed in [2]. This is possible when U is a convex function.

• In d = 1, B. Zegarlinski proved (LS) without imposing any other assumptions than (U0) [29, Theorem 4.1]. In this case, the Bakry-Emery criterion is no longer available in general. Instead, one takes advantage of a stronger version of (DS3) which comes from the dimensionality.

• High temperature, (i.e., sup_{x} ∑_{y:y≠x} J_{x,y} is sufficiently small): Log-Sobolev inequality (LS) under this condition was studied first in [27] with the function U satisfying assumption (U1) plus another technical condition. Then, alternative approaches were proposed by T. Bodineau, B. Helffer and M. Ledoux [6, 7, 16], which prove (LS) at high temperature under assumption (U0). An observation made at the end of Remark 2.2 above proves (LS) at high temperature under condition (U1). I. Gentil and C. Roberto [13] investigate spectral gap (SG) at high temperature with functions U which do not necessarily satisfy condition (U1).

Remark 2.4 Conditions in Theorem 2.1 are stated uniformly over all Λ ⊂⊂ Z^{d}. However, as is pointed out by F. Martinelli and E. Olivieri [20, 21] in the finite spin space setting, it is sometime reasonable to restrict one’s attention only to regular enough Λ’s such as fat enough boxes, in order to avoid some pathological phenomena caused by Λ’s with irregular shapes. In our case, it is clear from the proofs that Theorem 2.1 remains true if each “for all Λ ⊂⊂ Z^{d}” is replaced by “for all Λ ∈ B(n_{0})” , where B(n_{0}) is the set of generalized box with the minimal side-length n_{0} defined in [27].

3 Proof of Theorem 2.1

3.1 A Lemma on integrability

We begin by preparing integrability properties which are needed later.

Lemma 3.1 The following hold;

a) If condition (U0) is satisfied, then there are ε ∈ (0, ∞) and C_{3.1} ∈ (0, ∞) such that

E^{Λ,ω} exp (ε|σ_{z} − E^{Λ,ω}(σ_{z})|^{2}) ≤ C_{3.1}  (3.1)

for all z ∈ Λ ⊂⊂ Z^{d} and ω ∈ Ω.

b) If condition (U1) is satisfied, then for any n = 1, 2, …, there exists C_{3.2}(n) = C_{3.2}(U, J, n) ∈ (0, ∞) such that

E^{Λ,ω} (f^{2} log \frac{f^{2}}{E^{Λ,ω}(f^{2})}) ≤ C_{3.2}(|S_{f}|)E^{Λ,ω} (∑_{x∈Λ} |∇_{x}f|^{2}) .  (3.2)

for Λ ⊂⊂ Z^{d}, ω ∈ Ω and f ∈ C_{Λ}.

Proof: The proof of part (a), due to M. Sugiura [25], is presented in [28]. Part (b) follows from the proof of [27, Lemmas 3.1, 3.2]. □

Remark 3.1 It is known that (3.2) implies (3.1). See [1].

3.2 Proof of (SG) ⟹ (DS1)

We prove that (SG) implies (DS1) by the “semi-group method ” which is standard in the compact spin space setting, e.g., [17, Chapter I, Theorem 4.20] and [23, Lemmas 1.8 and 2.5 ]. Here, we follow the argument in [17]. We will also borrow techniques from [23], especially in the proof of Lemma 3.2 below.

We introduce now for the model we are considering, 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}} (3.3)
P = the Wiener measure on Θ.  (3.4)

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}  (3.5)

The existence and the uniqueness of the solution to (3.5) is well-known even for Λ = Z^{d} as well as for Λ ⊂⊂ Z^{d} (cf. [8]).

Our proof of (SG) ⟹ (DS1) is based on the “finite speed propagation property” of the dynamics in the following form.

Lemma 3.2 For any ε ∈ (0, ∞), there exists C_{3.6} ∈ (0, ∞) such that

P (f (σ_{t}^{Λ,ω}); g (σ_{t}^{Λ,ω})) ≤ C_{3.6} ∑_{y,z∈Z^{d}} ∥∇_{y}f∥∥∇_{z}g∥ exp (C_{3.6}t − εd(y, z))  (3.6)

for all Λ ⊂⊂ Z^{d}, ω ∈ Ω and f, g ∈ C_{Λ}.

The proof of this lemma will be given in section 3.3.

Remark 3.2 In the compact spin setting, (3.6) is well known, e.g., [17, Chapter I, Proposition 4.18]. A different version of finite speed propagation property for the unbounded spin setting can be found in [29, Section 1].

Proof of (SG) ⟹ (DS1): The proof is based on the idea as follows. If we have (SG), then, the distribution of the random variable σ_{t}^{Λ,ω} is close to E^{Λ,ω} for large, but finite time t. Suppose on the other hand that the supports S_{f} and S_{g} of functions f, g ∈ C_{Λ} are disjoint. Then, (3.6) says that the correlation of random variables f (σ_{t}^{Λ,ω}) and g (σ_{t}^{Λ,ω}) at any finite time t are exponentially decaying in the distance of S_{f} and S_{g}.

We now transpose the idea above into estimates. For f ∈ C_{Λ}, set \overline{f}(σ) = f(σ) − E^{Λ,ω}f and P_{t}f(σ) = Pf(σ^{Λ,σ_{Λ}·ω_{Λ^{c}}}). We then have that for f, g ∈ C_{Λ},

E^{Λ,ω}(f; g) = J_{1} + J_{2} + J_{3},  (3.7)

where

J_{1} = E^{Λ,ω}(\overline{f}\overline{g}) − E^{Λ,ω}P_{t}(\overline{f}\overline{g}),
J_{2} = E^{Λ,ω}P_{t}(\overline{f}\overline{g}) − E^{Λ,ω} (P_{t}\overline{f} · P_{t}\overline{g}) ,
J_{3} = E^{Λ,ω} (P_{t}\overline{f} · P_{t}\overline{g}) .

We see from Lemma 3.2 that

|J_{2}| ≤ C_{3.6}|||f||| |||g||| exp (C_{3.6}t − εd(S_{f}, S_{g})) .  (3.8)

By part (a) of Lemma 3.1, E^{Λ,ω}(\overline{f}^{2})^{1/2} ≤ E^{Λ,ω}(\overline{f}^{4})^{1/4} ≤ |||f|||. We therefore have by (SG) that

|J_{1}| ≤ E^{Λ,ω} |E^{Λ,ω}(\overline{f}\overline{g}) − P_{t}(\overline{f}\overline{g})|
≤ exp(−t/C_{2.2})E^{Λ,ω} (|E^{Λ,ω}(\overline{f}\overline{g}) − \overline{f}\overline{g}|^{2})^{1/2}
≤ exp(−t/C_{2.2}) (E^{Λ,ω}(\overline{f}^{2})^{1/2}E^{Λ,ω}(\overline{g}^{2})^{1/2} + E^{Λ,ω}(\overline{f}^{4})^{1/4}E^{Λ,ω}(\overline{g}^{4})^{1/4})
≤ 2 exp(−t/C_{2.2})|||f||| |||g|||,  (3.9)
|J_{3}| ≤ E^{Λ,ω} (|P_{t}\overline{g}|^{2})^{1/2} E^{Λ,ω} (|P_{t}\overline{g}|^{2})^{1/2}
≤ exp(−2t/C_{2.2})E^{Λ,ω} (|\overline{f}|^{2})^{1/2} E^{Λ,ω} (|\overline{g}|^{2})^{1/2}
≤ exp(−2t/C_{2.2})|||f||| |||g|||.  (3.10)

Plugging (3.8), (3.9) and (3.10) into (3.7) and choosing t = εd(S_{f}, S_{g})/(2C_{3.6}), we obtain (DS1). □

3.3 Proof of Lemma 3.2

Lemma 3.3 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)) < ∞}  (3.11)
l(−ε) = {α = (α_{x} ≥ 0)_{x∈Z^{d}} | ∥α∥_{−ε} =^{def.} ∑_{x∈Z^{d}} α_{x} exp −εd(0, x) < ∞}  (3.12)

α ∗ β = ( ∑_{y∈Z^{d}} α_{x−y}β_{y})_{x∈Z^{d}} , for α ∈ l(ε) and β ∈ l(−ε).  (3.13)

Then the following hold; 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) < ∞,  (3.14)
f_{t,x} ≤ β_{x} + ∫ _{0}^{t}ds(α ∗ f_{s})_{x}  (3.15)

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)  (3.16)

for all x ∈ Z^{d} and t ≥ 0.

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, ….  (3.17)

where (α∗)^{n}β = \underbrace{α ∗ ⋯ ∗ α∗}_{n} β. It is easy to prove that

∥(α∗)^{n}β∥_{−ε} ≤ ∥α∥_{ε}^{n}∥β∥_{−ε}, n = 1, 2, ….  (3.18)

We now introduce the shift transformation;

τ_{y} : (α_{x})_{x∈Z^{d}} ↦ (α_{x+y})_{x∈Z^{d}}, y ∈ Z^{d}.  (3.19)

Then, α ∗ (τ_{y}β) = τ_{y}(α ∗ β) and hence (α∗)^{n}(τ_{y}β) = τ_{y}((α∗)^{n}β). From this and (3.18), 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 (3.17).

To prove (3.16), we iterate (3.15) (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),  (3.20)

where we have used (3.17) to proceed to the second line. Since the second term on the righthand-side of (3.20) vanishes as n ↗ ∞, we obtain (3.16). □

Proof of Lemma 3.2: We fix Λ ⊂⊂ Z^{d} and ω ∈ R^{Λ^{c}} throughout this proof. Let us use the following notations; for f ∈ C^{2}(R^{Λ}),

∇_{x,y}f = ∇_{x}∇_{y}f,  (3.21)
A_{x}f = ∇_{x,x}f − ∇_{x}H^{Λ,ω}∇_{x}f,  (3.22)
Af = ∑_{x∈Λ} A_{x}f.  (3.23)

Step1;

\frac{1}{2}A_{x} (|∇_{y}f|^{2}) − ∇_{y}A_{x}f · ∇_{y}f ≥ \{ \begin{matrix}−C_{3.24}|∇_{x}f∇_{y}f|, & if d(x, y) ≤ R, \\ 0 & otherwise, \end{matrix}  (3.24)

where C_{3.24} = sup_{x,y} J_{x,y} + max{0, − inf_{s} U′′(s)}. In fact, it is easy to see that the LHS of (3.24) is equal to

|∇_{x,y}f|^{2} + ∇_{x,y}H^{Λ,ω}∇_{x}f∇_{y}f.  (3.25)

Since ∇_{x,y}H^{Λ,ω} = −J_{x,y} + δ_{x,y}U′′(σ_{x}), (3.25) is bounded from below by the RHS of (3.24).

Step2; Take f ∈ C_{c}^{∞}(R^{Λ}) : the set of smooth functions on R^{Λ} which vanish outside a compact set and define P_{t}f(σ) = Pf(σ^{Λ,σ_{Λ}·ω_{Λ^{c}}}). Note that P_{t}f(σ) has an integral kernel P_{t}(σ, τ) which is smooth in (t, σ, τ) ∈ (0, ∞) × R^{Λ} × R^{Λ} and solves \frac{∂}{∂t}P_{t}(·, τ) = \frac{1}{2}AP_{t}(·, τ). Therefore, P_{t}f(σ) is smooth in (t, σ) ∈ (0, ∞) × R^{Λ} and solves \frac{∂}{∂t}P_{t}f = \frac{1}{2}AP_{t}f. Let us prove that ∇_{x}P_{t}f(σ) is bounded in σ for any x ∈ Λ. This can be seen as follows. For a bounded smooth function F = (F_{x})_{x∈Λ} : R^{Λ} → R^{Λ}, define

A⃗F = (AF_{x} − ∑_{y∈Λ} ∇_{x,y}H^{Λ,ω}F_{y})_{x∈Λ}
P⃗_{t}F (σ) = P (M_{t}F (σ_{t}^{Λ,σ_{Λ}·ω_{Λ^{c}}} ))

where M_{t} = (M_{t;x,y})_{x,y∈Λ} is a matrix defined as the solution to the following integral equation;

M_{t;x,y} = δ_{x,y} − \frac{1}{2} ∑_{y∈Λ} ∫ _{0}^{t}dsM_{s;x,z}∇_{z,y}H^{Λ,ω}(σ_{s}^{Λ,σ_{Λ}·ω_{Λ^{c}}}).  (3.26)

We then have by standard arguments (See the proof of [26, Proposition 2.3] for example.) that

• (P⃗_{t})_{t≥0} extends to a strongly continuous semigroup on the Hilbert space L^{2}(E^{Λ,ω}) ⊗ R^{Λ},

• A⃗ extends to the generator of the semigroup,

• |M_{t}ξ| ≤ |ξ| exp(tC_{3.24}/2), for all ξ ∈ R^{Λ} and t ≥ 0, where | · | on each hand side stands for the Euclidean norm on R^{Λ}.

These imply that

\frac{∂}{∂t} P⃗_{t}F = \frac{1}{2} A⃗P⃗_{t}F = \frac{1}{2} P⃗_{t}A⃗F,  (3.27)
|P⃗_{t}F (σ)| ≤ P_{t}(|F |)(σ) exp(tC_{3.24}/2).  (3.28)

On the other hand, it is easy to see that

∇Af = A⃗∇f for f ∈ C_{c}^{∞}(R^{Λ}),  (3.29)

where ∇f = (∇_{x}f)_{x∈Λ}. It follows from (3.27), (3.28) and (3.29) that

|∇P_{t}f(σ)| = |P⃗_{t}∇f(σ)|
≤ P_{t}(|∇f|)(σ) exp(tC_{3.24}/2),

which implies that ∇_{x}P_{t}f(σ) are bounded in σ for any x ∈ Λ.

Step3; Define F_{t} = (F_{x,t})_{x∈Z^{d}} by

F_{x,t} = ∥∇_{x}P_{t}f∥, if x ∈ Λ,
F_{x,t} = 0. if x ∉ Λ.

Let us prove that for any ε ∈ (0, ∞), there is C_{3.30} ∈ (0, ∞) such that

F_{x,t} ≤ C_{3.30} ∑_{y∈Z^{d}} F_{y,0} exp (C_{3.30}t − 2εd(x, y)) .  (3.30)

To prove (3.30) by applying Lemma 3.3, we will check that

sup_{0≤s≤t} ∑_{y∈Z^{d}} F_{y,s} exp −2εd(x, y) < ∞,  (3.31)
F_{x,t} ≤ F_{x,0} + ∫ _{0}^{t}ds(α ∗ F_{s})_{x}  (3.32)

for all t ≥ 0 and x ∈ Z^{d}, where α_{x} = \frac{1}{2}C_{3.24}1_{\{d(0,x)≤R\}}. (3.31) is obvious. To prove (3.32), define φ(s) = P_{t−s} (|∇_{x}P_{s}f|^{2}), 0 ≤ s ≤ t, for any fixed t. We then have by (3.24) that

φ′(s) = −\frac{1}{2}P_{t−s}A (|∇_{x}P_{s}f|^{2}) + 2P_{t−s} (∇_{x}P_{s}f∇_{x} \frac{A}{2}P_{s}f)
≤ C_{3.24} ∑_{y:d(x,y)≤R} P_{t−s} (|∇_{x}P_{s}f||∇_{y}P_{s}f|)
≤ C_{3.24} ∑_{y:d(x,y)≤R} F_{x,s}F_{y,s}
= 2(α ∗ F_{s})_{x}F_{x,s},

from which we conclude

F_{x,t}^{2} = φ(t)
≤ φ(0) + 2∫ _{0}^{t}(α ∗ F_{s})_{x}F_{x,s}ds
≤ F_{x,0}^{2} + 2∫ _{0}^{t}(α ∗ F_{s})_{x}F_{x,s}ds.  (3.33)

By solving (3.33) as an integral inequality with respect to F_{x,t}, we obtain (3.32).

Step4; Take f, g ∈ C_{c}^{∞}(R^{Λ}) and define G_{x,t} for t ≥ 0 and x ∈ Z^{d} similarly as F_{x,t};

G_{x,t} = ∥∇_{x}P_{t}g∥, if x ∈ Λ,
G_{x,t} = 0 if x ∉ Λ.

Let us prove that

∑_{x∈Z^{d}} F_{x,t}G_{x,t} ≤ C_{3.34} ∑_{y,z∈Z^{d}} ∥∇_{y}f∥∥∇_{z}f∥ exp (C_{3.34}t − εd(y, z)) ,  (3.34)

where C_{3.34} ∈ (0, ∞) is independent of Λ and ω. Set γ_{x} = exp (−2εd(0, x)). It is easy to see that for any x ∈ Z^{d},

exp (εd(0, x)) γ_{x} ≤ ∥γ ∗ γ∥_{ε} ≤ ∥γ∥_{ε}^{2}.  (3.35)

Therefore, we have for any y, z ∈ Z^{d} that

∑_{x∈Z^{d}} γ_{x−y}γ_{x−z} = γ ∗ γ(y − z) ≤ ∥γ∥_{ε}^{2} exp (−εd(y, z)) .  (3.36)

We now conclude (3.34) from (3.30) and (3.36) as follows;

∑_{x∈Z^{d}} F_{x,t}G_{x,t} ≤ C_{3.30}^{2} exp (2C_{3.30}t) ∑_{x∈Z^{d}} ∑_{y,z∈Z^{d}} ∥∇_{y}f∥∥∇_{z}f∥γ_{x−y}γ_{x−z}
≤ C_{3.30}^{2}∥γ∥_{ε}^{2} exp (2C_{3.30}t) ∑_{y,z∈Z^{d}} ∥∇_{y}f∥∥∇_{z}f∥ exp (−εd(y, z)) .

Step5; Conclusion of the proof. For f, g ∈ C_{c}^{∞}(R^{Λ}), we define ψ(s) = P_{s} (P_{t−s}fP_{t−s}g), 0 ≤ s ≤ t for any fixed t. We then have that

ψ′(s) = P_{s} (\frac{1}{2}A (P_{t−s}fP_{t−s}g) − \frac{1}{2}AP_{t−s}fP_{t−s}g − \frac{1}{2}P_{t−s}fAP_{t−s}g)
= P_{s} (∑_{x∈Λ} ∇_{x}P_{t−s}f∇_{x}P_{t−s}g)
≤ ∑_{x∈Λ} F_{x,t−s}G_{x,t−s}

and therefore that

P (f (σ_{t}^{Λ,ω}); g (σ_{t}^{Λ,ω})) = P_{t} (fg) − P_{t} (f) P_{t} (g)
= ψ(t) − ψ(0)
≤ ∫ _{0}^{t} ∑_{x∈Λ} F_{x,t−s}G_{x,t−s}ds
= ∫ _{0}^{t} ∑_{x∈Λ} F_{x,s}G_{x,s}ds.  (3.37)

Plugging (3.34) into (3.37), we arrive at (3.6) for f, g ∈ C_{c}(R^{Λ}). (3.6) for f, g ∈ C_{Λ} can be obtained by considering a suitable approximation. □

3.4 Proof of (DS4) ⇒ (CC)

We will prove that

R^{Λ} (E^{Λ,ω}, E^{Λ,_{ω}}) ≤ K_{3.38} ∑_{y∈∂_{R}Λ} |ω_{y} − ω̃_{y}|,  (3.38)

for all Λ ⊂⊂ Z^{d} and ω, ω̃ ∈ Ω, where K_{3.38} = C_{2.6} ∑_{z∈Z^{d}} exp(−|z|/C_{2.6}). Condition (CC) follows from (3.38) by taking V as a large enough cube and applying (3.38) to Λ ∩ (x + V) instead of Λ. The proof of (3.38) reduces to the case ω ≡ ω̃ off a site y ∈ ∂_{R}Λ. The reduction is a simple argument which uses the triangle inequality for the Vassershtein distance and an interpolating sequence ω_{0}, …, ω_{n} (n = |∂_{R}Λ|) of boundary conditions such that ω_{0} = ω, ω_{n} = ω̃ and ω_{j−1} ≡ ω_{j} off a site in ∂_{R}Λ (j = 1, …, n).

We now assume that ω ≡ ω̃ off a site y ∈ ∂_{R}Λ and that ω_{y} ≥ ω̃_{y} without loss. We then have by the FKG inequality that

E^{Λ,ω} (∩_{x∈Λ} {σ : σ_{x} ≥ τ_{x}}) ≥ E^{Λ,_{ω}} (∩_{x∈Λ} {σ : σ_{x} ≥ τ_{x}})

for all (τ_{x})_{x∈Λ} ∈ R^{Λ}. From this and Strassen’s theorem on the existence of an order preserving coupling [18, page 129, (4.2)], there exists an measure E^{Λ,ω,_{ω}} ∈ K(E^{Λ,ω}, E^{Λ,_{ω}}) such that

E^{Λ,ω,_{ω}} (∩_{x∈Λ} {(σ, σ̃) : σ_{x} ≥ σ̃_{x}}) = 1.  (3.39)

We have by (3.39) and (2.6) that

R^{Λ} (E^{Λ,ω}, E^{Λ,_{ω}}) ≤ ∑_{z∈Λ} E^{Λ,ω,_{ω}}|σ_{z} − σ̃_{z}|
= ∑_{z∈Λ} (E^{Λ,ω}(σ_{z}) − E^{Λ,_{ω}}(σ_{z}))
≤ C_{2.6}|ω_{y} − ω̃_{y}| ∑_{z∈Λ} exp(−|z − y|/C_{2.6})
≤ K_{3.38}|ω_{y} − ω̃_{y}|,

which proves (3.38). □

3.5 Proof of (CC)⇒ (DS4)

We follow the the argument in [27, pages 33–38] which originally is adapted from [10, pages 353–356]. We begin with lemmas.

Lemma 3.4 Suppose that Λ ⊂⊂ Z^{d}, y ∉ Λ and ω, ω̃ ∈ Ω are such that ω ≡ ω̃ off y. Then, there exists a measure E^{Λ,ω,_{ω}} ∈ K(E^{Λ,ω}, E^{Λ,_{ω}}) such that if we define f_{z}^{Λ,ω,_{ω}} by

f_{z}^{Λ,ω,_{ω}} = ∫ E^{Λ,ω,_{ω}}(dσdσ̃)|σ_{z} − σ̃_{z}|,  (3.40)

then

∑_{z∈Λ} f_{z}^{Λ,ω,_{ω}} = R^{Λ}(E^{Λ,ω}, E^{Λ,_{ω}}),  (3.41)
sup_{z∈Λ} f_{z}^{Λ,ω,_{ω}} ≤ C_{3.42}|ω_{y} − ω̃_{y}|,  (3.42)
∑_{z∈Λ∩(x+V)} f_{z}^{Λ,ω,_{ω}} ≤ K_{2.7} ∑_{z∈Λ∩(x+∂_{R}V)} f_{z}^{Λ,ω,_{ω}} + K_{2.7}|ω_{y} − ω̃_{y}|1_{{y∈x+V }}.  (3.43)

where the constant C_{3.42} depends only on d, J and U.

Proof: We may assume ω_{y} ≥ ω̃_{y} without loss. for all (τ_{x})_{x∈Λ} ∈ R^{Λ}. Therefore, there exists an measure E^{Λ,ω,_{ω}} ∈ K(E^{Λ,ω}, E^{Λ,_{ω}}) such that (3.39) holds. Therefore, if we define f_{z}^{Λ,ω,_{ω}} by (3.40), we then have for any z ∈ Λ and any μ ∈ K(E^{Λ,ω}, E^{Λ,_{ω}}) that

f_{z}^{Λ,ω,_{ω}} = ∫ E^{Λ,ω,ω̃}(dσdσ̃)(σ_{z} − σ̃_{z})
= E^{Λ,ω}(σ_{z}) − E^{Λ,ω̃}(σ_{z})
= ∫ μ(dσdσ̃)(σ_{z} − σ̃_{z})
≤ ∫ μ(dσdσ̃)|σ_{z} − σ̃_{z}|,

and hence that

∑_{z∈Λ} f_{z}^{Λ,ω,_{ω}} ≤ ∑_{z∈Λ} ∫ μ(dσdσ̃)|σ_{z} − σ̃_{z}|,

which implies (3.41).

We prove (3.42) as follows. We repeat the argument in the proof of (DS1) ⟹ (DS2) to see that

f_{z}^{Λ,ω,_{ω}} = E^{Λ,ω}(σ_{z}) − E^{Λ,ω̃}(σ_{z})
≤ (ω̃_{y} − ω_{y}) sup_{ζ∈Ω} ∑_{x∈Λ} J_{x,y}E^{Λ,ζ}(σ_{z}; σ_{x}).  (3.44)

This estimate proves (3.42), since we have (3.1).

We now turn to the proof of (3.43). We begin by taking a measurable map (ξ, ξ̃) ↦ Ê^{x}(·|ξ, ξ̃) from R^{Λ} × R^{Λ} to P(R^{Λ∩(x+V)} × R^{Λ∩(x+V)}) such that

R^{Λ∩(x+V)} (E^{Λ∩(x+V),ξ}, E^{Λ∩(x+V),_{ξ}}) = ∫ Ê^{x}(dσdσ̃|ξ, ξ̃) ∑_{z∈Λ∩(x+V)} |σ_{z} − σ̃_{z}|,  (3.45)

for all (ξ, ξ̃). The possibility of this measurable selection can be shown as an application of [22, Thorem 12.1.10]. (Use also Lemma 12.1.7 in that book to check that the set of measures which attain R^{Λ∩(x+V)} (E^{Λ∩(x+V),ξ}, E^{Λ∩(x+V),_{ξ}}) is measurable as a set-valued function of (ξ, ξ̃)).

We now define a measure \overline{E}^{x,ω,_{ω}} ∈ P(R^{Λ} × R^{Λ}) by

\overline{E}^{x,ω,_{ω}}(A × B) = ∫_{A}E^{Λ,ω,_{ω}}(dξdξ̃)∫_{B}Ê^{x}(dσdσ̃|ξ, ξ̃),

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

\overline{E}^{x,ω,_{ω}} ∈ K(E^{Λ,ω}, E^{Λ,_{ω}}),  (3.46)
\overline{E}^{x,ω,_{ω}} = E^{Λ,ω,_{ω}} on R^{Λ\(x+V)} × R^{Λ\(x+V)}.  (3.47)

To see (3.43), it is sufficient to prove that

∑_{z∈Λ∩(x+V)} f_{z}^{Λ,ω,_{ω}} ≤ ∑_{z∈Λ∩(x+V)} \overline{f}_{z}^{x,ω,_{ω}},  (3.48)
∑_{z∈Λ∩(x+V)} \overline{f}_{z}^{x,ω,_{ω}} ≤ K_{2.7} ∑_{z∈Λ∩(x+∂_{R}V)} f_{z}^{Λ,ω,_{ω}} + K_{2.7}|ω_{y} − ω̃_{y}|1_{\{y∈x+V \}}  (3.49)

where

\overline{f}_{z}^{x,ω,_{ω}} = ∫ \overline{E}^{x,ω,_{ω}}(dσdσ̃)|σ_{z} − σ̃_{z}|.  (3.50)

The first inequality (3.48) can be seen as follows. Since (3.47) implies that f_{z}^{Λ,ω,_{ω}} = \overline{f}_{z}^{x,ω,_{ω}} for z ∉ W ∩ (x + V), we have from this, (3.41) and (3.46) that

∑_{z∈Λ∩(x+V)} (f_{z}^{Λ,ω,_{ω}} − \overline{f}_{z}^{x,ω,_{ω}}) = ∑_{z∈Λ} (f_{z}^{Λ,ω,_{ω}} − \overline{f}_{z}^{x,ω,_{ω}})
≤ ∑_{z∈Λ} f_{z}^{Λ,ω,_{ω}} − R^{Λ}(E^{Λ,ω}, E^{Λ,_{ω}})
= 0.

To prove the second inequality (3.49), we will use (3.45) and (2.8) as follows;

∑_{z∈Λ∩(x+V)} \overline{f}_{z}^{x,ω,_{ω}} = ∫ E^{Λ,ω,_{ω}}(dξdξ̃)∫ Ê^{x}(dσdσ̃|ξ, ξ̃) ∑_{z∈Λ∩(x+V)} |σ_{z} − σ̃_{z}|
≤ K_{2.7}∫ E^{Λ,ω,_{ω}}(dξdξ̃) ∑_{z∈∂_{R}(Λ∩(x+V))} |ξ_{z} − ξ̃_{z}|
= K_{2.7} ∑_{zΛ∩(x+∂_{R}V)} f_{z}^{Λ,ω,_{ω}} + K_{2.7}|ω_{y} − ω̃_{y}|1_{{y∈x+V }}.

This completes the proof of Lemma 3.4. □

Lemma 3.5 For any A ⊂ Λ, L ≥ 1 and ω, ω̃ ∈ Ω with ω ≡ ω̃ off y,

∑_{z∈Λ} f_{z}^{Λ,ω,_{ω}} exp (−d(z, A)/C_{3.51}) ≤ B_{3.51} ∑_{{z∈Λ \atop d(z,y)≤L+D_{3.51}}} f_{z}^{Λ,ω,_{ω}} exp (−d(z, A)/C_{3.51})

+ B_{3.51}|ω_{y} − ω̃_{y}| ∑_{{z;d(z,y)>L \atop z+V ∋y}} exp (−d(z, A)/C_{3.51}) , (3.51)

where f_{z}^{Λ,ω,_{ω}} is defined by (3.40), D_{3.51} = diam(V ∪ ∂_{R}V), B_{3.51} = B_{3.51}(R, V, ε_{2.7}) and C_{3.51} =

C_{3.51}(R, V, ε_{2.7}).

Proof: We choose C_{3.51} = C_{3.51}(R, V, ε_{2.7}) so large that

C_{3.52} =^{def.} exp (−D_{3.51}/C_{3.51}) − ε_{2.7} exp (D_{3.51}/C_{3.51}) > 0.  (3.52)

We set

e_{x} = exp (−d(x, A)/C_{3.51}) , C_{3.53} = exp(D_{3.51}/C_{3.51}).  (3.53)

We then define l_{z}^{0} = ∑_{x:x+V ∋z} e_{x}, l_{z}^{1} = K_{2.7} ∑_{x:x+∂_{R}V ∋z} e_{x}, 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∈Λ} f_{z}^{Λ,ω,_{ω}}l_{z} ≤ ∑_{z∈Λ} f_{z}^{Λ,ω,_{ω}}r_{z} + K_{2.7}|ω_{y} − ω̃_{y}| ∑_{{x:d(x,y)≥L \atop x+V ∋y}} e_{z}.  (3.54)

We have by (3.43) that

∑_{x:d(x,y)>L} e_{x} ∑_{z∈Λ∩(x+V)} f_{z}^{Λ,ω,_{ω}} ≤ K_{2.7} ∑_{x:d(x,y)>L} e_{x} ∑_{z∈Λ∩(x+∂_{R}V)} f_{z}^{Λ,ω,_{ω}}
+K_{2.7}|ω_{y} − ω̃_{y}| ∑_{{x:d(x,y)≥L \atop x+V ∋y}} e_{z}.  (3.55)

Since

∑_{x:d(x,y)>L} e_{x} ∑_{z∈Λ∩(x+V)} f_{z}^{Λ,ω,_{ω}} = ∑_{z∈Λ} f_{z}^{Λ,ω,_{ω}} ∑_{x:d(x,y)>L_{x+V ∋z}} e_{x}

= ∑_{z∈Λ} f_{z}^{Λ,ω,_{ω}} (l_{z}^{0} − r_{z}) ,
K_{2.7} ∑_{x:d(x,y)>L} e_{x} ∑_{z∈Λ∩(x+∂_{R}V)} f_{z}^{Λ,ω,_{ω}} ≤ ∑_{z∈Λ} f_{z}^{Λ,ω,_{ω}}l_{z}^{1},

it follows from (3.55) that

∑_{z∈Λ} f_{z}^{Λ,ω,_{ω}} (l_{z}^{0} − r_{z}) ≤ ∑_{z∈Λ} f_{z}^{Λ,ω,_{ω}}l_{z}^{1} + K_{2.7}|ω_{y} − ω̃_{y}| ∑_{{x:d(x,y)≥L \atop x+V ∋y}} e_{x},

which is equivalent to (3.54).

Let us next prove that

r_{z} ≤ C_{3.56}e_{z},  (3.56)
r_{z} = 0 if d(z, y) > L + D_{3.51},  (3.57)
l_{z} ≥ C_{3.58}e_{z},  (3.58)

where C_{3.56}, C_{3.58} ∈ (0, ∞) depend only on R, V and ε_{2.7}. To verify (3.56) and (3.57), note first that an easy to prove fact that

C_{3.53}^{−1} ≤ e_{x}/e_{z} ≤ C_{3.53} if d(x, z) ≤ D_{3.51}.  (3.59)

We thus see that

r_{z} ≤ C_{3.53}e_{z} ∑_{{x:x+V ∋z \atop d(x,y)≤L}} 1,

which proves (3.56) and (3.57).

On the other hand, it follows from (3.59) and (2.7) that

l_{z} ≥ e_{z} {C_{3.53}^{−1} ∑_{x:x+V ∋z} 1 − K_{2.7}C_{3.53} ∑_{x:x+∂_{R}V ∋z} 1}
≥ e_{z} {C_{3.53}^{−1}|V | − C_{3.53}ε_{2.7}|V |}
= C_{3.52}|V |e_{z},  (3.60)

which proves (3.58). By plugging (3.56), (3.57) and (3.58) into (3.54), we obtain (3.51). □

3.6 Proof of (DS4) ⇒ (LS)

To prove that (DS4) implies (LS), we will use the method of S.-L. Lu and H.-T. Yau [19], in which (SG) and (LS) in compact spin settings are obtained from a certain mixing condition. The following lemma plays a key role in carrying out the strategy of the above mentioned paper in the unbounded spin setting. The proof we present here is based mainly on that of [27, Lemma 3.4]. An idea in a recent paper [6] by T. Bodineau and B. Helffer is also used.

Lemma 3.6 Suppose that conditions (U0) and (DS4) are satisfied. Then, the following hold;

a) There exists C_{3.61} ∈ (0, ∞) such that

|E^{Λ,ω}(f; σ_{x})| ≤ C_{3.61}|Δ| exp (−d(x, Δ)/C_{2.6}) E^{Λ,ω}(f; f)^{1/2},  (3.61)

whenever f ∈ C, Λ ⊂⊂ Z^{d}, x ∈ Λ and Δ ⊂ Λ are such that S_{f} ∩ Λ ⊂ Δ ⊂ Λ.

b) There exists C_{3.62} ∈ (0, ∞) such that

|E^{Λ,ω}(f^{2}; σ_{x})| ≤ C_{3.62}|Δ| exp (−d(x, Δ)/C_{2.6})
·E^{Λ,ω}(f^{2})^{1/2} (E^{Λ,ω}(f; f)^{1/2} + E^{Λ,ω} (f^{2} log \frac{f^{2}}{E^{Λ,ω}(f^{2})})^{1/2})(3.62)

whenever f ∈ C, Λ ⊂⊂ Z^{d}, x ∈ Λ and Δ ⊂ Λ are such that S_{f} ∩ Λ ⊂ Δ ⊂ Λ.

Proof: Set E^{Λ,ξ,_{ξ}} = E^{Λ,ξ} ⊗ E^{Λ,_{ξ}} ∈ P(R^{Λ} × R^{Λ}). We have that

E^{Λ,ω}(f; σ_{x}) = \frac{1}{2}∫ E^{Λ,ω,ω}(dσdσ̃) (f(σ) − f(σ̃)) (σ_{x} − σ̃_{x})
= \frac{1}{2}∫ E^{Λ,ω,ω}(dξdξ̃) (f(ξ) − f(ξ̃)) ∫ E^{Λ\Δ,ξ,_{ξ}}(dσdσ̃) (σ_{x} − σ̃_{x})
= \frac{1}{2}∫ E^{Λ,ω,ω}(dξdξ̃) (f(ξ) − f(ξ̃)) (E^{Λ\Δ,ξ}(σ_{x}) − E^{Λ\Δ,_{ξ}}(σ_{x})) (3.63)

where the second equality can be verified by applying Markov property to E^{Λ,ω}(dσ) and E^{Λ,ω}(dσ̃) separatedly. Since ξ ≡ ξ̃ ≡ ω outside Λ in (3.63), we see from (DS4) that

|E^{Λ\Δ,ξ}(σ_{x}) − E^{Λ\Δ,_{ξ}}(σ_{x})| ≤ C_{2.6} ∑_{y∈Δ} |ξ_{y} − ξ̃_{y}| exp (−d(x, Δ)/C_{2.6})  (3.64)

Plugging this into (3.63), we see that

|E^{Λ,ω}(f; σ_{x})| ≤ C_{3.65} exp (−d(x, Δ)/C_{2.6}) ∑_{z∈Δ} I_{1}(f, z),  (3.65)

where

I_{1}(f, z) = ∫ E^{Λ,ω,ω}(dσdσ̃) |f(σ) − f(σ̃)| |σ_{z} − σ̃_{z}| .  (3.66)

We have by Schwarz inequality that

I_{1}(f, z) ≤ (∫ E^{Λ,ω,ω}(dσdσ̃) |f(σ) − f(σ̃)|^{2})^{1/2} (∫ E^{Λ,ω,ω}(dσdσ̃) |σ_{z} − σ̃_{z}|^{2})^{1/2}
= 2E^{Λ,ω}(f; f)^{1/2}E^{Λ,ω}(σ_{z}; σ_{z})^{1/2}.  (3.67)

Since it follows from (3.1) that E^{Λ,ω}(σ_{z}; σ_{z}) is bounded from above by a constant C which is independent of Λ, ω and z, we obtain

∑_{z∈Δ} I_{1}(f, z) ≤ 2C^{1/2}|Δ|E^{Λ,ω}(f; f)^{1/2}.  (3.68)

This, in conjunction with (3.65), implies (3.61).

The proof of (3.62) is similar. We see from (3.65) that

|E^{Λ,ω}(f^{2}; σ_{x})| ≤ C_{2.6} exp (−d(x, Δ)/C_{2.6}) ∑_{z∈Δ} I_{1}(f^{2}, z).  (3.69)

We have by Schwarz inequality that

I_{1}(f^{2}, z) ≤ \sqrt{2}E^{Λ,ω}(f; f)^{1/2}I_{2}(f, z)^{1/2},  (3.70)

where

I_{2}(f, z) = ∫ E^{Λ,ω,ω}(dσdσ̃) |f(σ) + f(σ̃)|^{2} |σ_{z} − σ̃_{z}|^{2} .

To estimate I_{2}(z), we will use an inequality: ab ≤ exp(a) + b log b for a, b ≥ 0 and the following fact which follows from (3.1);

sup_{Λ,ω,z} ∫ E^{Λ,ω,ω}(dσdσ̃) exp (ε |σ_{z} − σ̃_{z}|^{2}) < ∞.  (3.71)

We see that

I_{2}(f, z) ≤ 4∫ E^{Λ,ω,ω}(dσdσ̃)f(σ)^{2} |σ_{z} − σ̃_{z}|^{2}
= 4E^{Λ,ω}(f^{2})∫ E^{Λ,ω,ω}(dσdσ̃) |σ_{z} − σ̃_{z}|^{2} \frac{f(σ)^{2}}{E^{Λ,ω}(f^{2})}
≤ 4E^{Λ,ω}(f^{2})∫ E^{Λ,ω,ω}(dσdσ̃) exp (ε |σ_{z} − σ̃_{z}|^{2}) + (4/ε)E^{Λ,ω} (f^{2} log \frac{f^{2}}{εE^{Λ,ω}(f^{2})})
≤ C_{3.72}E^{Λ,ω}(f^{2}) + C_{3.72}E^{Λ,ω} (f^{2} log \frac{f^{2}}{E^{Λ,ω}(f^{2})}) ,  (3.72)

where C_{3.72} ∈ (0, ∞) is independent of Λ, ω, z and f. Putting (3.70) and (3.72) together, we obtain

∑_{z∈Δ} I_{1}(f^{2}, z)

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

≤ \sqrt{2}C_{3.72}^{1/2}|Δ|E^{Λ,ω}(f; f)^{1/2} (E^{Λ,ω}(f^{2})^{1/2} + E^{Λ,ω} (f^{2} log \frac{f^{2}}{E^{Λ,ω}(f^{2})})^{1/2})

≤ \sqrt{2}C_{3.72}^{1/2}|Δ|E^{Λ,ω}(f^{2})^{1/2} (E^{Λ,ω}(f; f)^{1/2} + E^{Λ,ω} (f^{2} log \frac{f^{2}}{E^{Λ,ω}(f^{2})})^{1/2})

which, in conjunction with (3.69), implies (3.62). □

Proof of (DS4) ⟹ (LS): The proof is divided into two steps;

(3.2) and (3.61) ⟹ (SG),  (3.73)
(3.2), (3.62) and (SG) ⟹ (LS).  (3.74)

Each of above steps can be carried out in the same way as in [27, Sections 4 and 5]. In the second step, (SG) is used to handle E^{Λ,ω}(f; f) on the RHS of (3.62). □

3.7 Completion of the proof of Theorem 2.1

(LS) ⟹ (SG); This is well known (cf. [12, Corollary 6.1.17]). (SG) ⟹ (DS1); This has already been shown in section 3.2. (DS1) ⟹ (DS2); Define ζ = ζ(θ) ∈ Ω (0 ≤ θ ≤ 1) by ζ_{x} = θω̃_{x} + (1 − θ)ω_{x} (x ∈ Z^{d}), which interpolates two boundary conditions ω and ω̃. We then have that

E^{Λ,_{ω}}(f) − E^{Λ,ω}(f) = ∫ _{0}^{1}dθ\frac{∂}{∂θ}E^{Λ,ζ}(f).  (3.75)

On the other hand, we have by direct computations that

\frac{∂}{∂θ}E^{Λ,ζ}(f) = −E^{Λ,ζ} (f; \frac{∂H^{Λ,ζ}}{∂θ} )
= (ω̃_{y} − ω_{y}) ∑_{x∈Λ} J_{x,y}E^{Λ,ζ}(f; σ_{x}).  (3.76)

We see from (3.76) and (DS1) that

|\frac{∂}{∂θ}E^{Λ,ζ}(f)| ≤ C_{3.77}|ω̃_{y} − ω_{y}| exp (−d(S_{f}, y)/C_{2.3}) .  (3.77)

Putting (3.75) and (3.77) together, we get (DS2). (DS1) ⟹ (DS3); Trivial. (DS2) ⟹ (DS4); Trivial. (DS3) ⟹ (DS4); This can be done by putting f(σ) = σ_{z} in the proof of (DS1) ⟹ (DS2). (DS4) ⟺ (CC): This has already been shown in sections 3.4, 3.5 (DS4) ⟹ (LS): This has already been shown in section 3.6. □

Acknowledgement: The author would like to thank T. Bodineau, B. Helffer and M. Ledoux for useful discussions and for communicating their works [6, 7, 16] prior to publication. The author also would like to thank the anonymous referee for many valuable comments.

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