Relaxed criteria of the Dobrushin-Shlosman mixing condition

Nobuo Yoshida

Division of Mathematics, Graduate School of Science, Kyoto University, Kyoto 606-01, Japan, e-mail:nobuo@kusm.kyoto-u.ac.jp

Abstract. An interacting particle system (Glauber dynamics), which evolves on a finite subset in the d-dimensional integer lattice is considered. It is known that a mixing property of the Gibbs state in the sense of Dobrushin and Shlosman ([DSh85]) is equivalent to several very strong estimates in terms of the Glauber dynamics ([SZ92b]). We show that similar, but seemingly much milder estimates are again equivalent to the Dobrushin-Shlosman mixing condition, hence to the original ones in [SZ92b]. This may be understood as the absence of intermediate speed of convergence to equilibrium.

Key words. relaxed criteria, the Dobrushin-Shlosman mixing condition, Glauber dynamics, convergence to equilibrium.

1 Introduction

A standard way to describe a mixing property of a Gibbs state is to estimate the difference of the expectations of a local observable with respect to finite volume Gibbs state with different boundary conditions. For example, the Dobrushin-Shlosman mixing condition which we will discuss about can roughly be ststed as follows; the difference mentioned above is exponentially small in the distance between the support of the observable and the sites at which the boundary conditions are different (cf. (2.16)). On the other hand, since a mixing property of a Gibbs state reflects to the rapid relaxation of the Glauber dynamics, the mixing property can be expressed in the following different ways in terms of the Glauber dynamics; (a) estimate of the logarithmic Sobolev constant, (b) estimate of the spectral gap, (c) estimate of the rate of convergence (the difference between the semigroup at time t and its equilibrium ). In [SZ92b], Stroock and Zegarlinski succeeded in rephrasing the Dobrushin-Shlosman mixing condition in three ways described above, which we will review as Theorem 3.1 below.

The purpose of this article is to present relaxed criteria of the Dobrushin-Shlosman mixing condition of the type (a), (b) and (c), which are respectively condition (2a), (2b) and (2c) in Theorem 3.2, our main result. Technically, the derivation of the original Dobrushin-Shlosman mixing condition from these relaxed criteria is based on the fact that the exponential decay in the statement of the Dobrushin-Shlosman mixing condition is equivalent to a certain polynomial decay. This point is also made clear as condition (2d) and (2e) in Theorem 3.2. These relaxed criteria are potentially useful to check the Dobrushin-Shlosman mixing condition, which is often accepted as an assumption to do something with, but is not always easy to verify in practical applications. Since our result is based on the very old and well-known idea that “an expontial mixing follows from a certain polynomial mixing”, there are a lot of results with the same sprit in earlier litratures. The relation between these earlier results and ours will be discussed in the series of remarks following the statement of Theorem 3.2.

2 Basic definitions

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^{∞}-norm; |x| = max_{1≤i≤d} |x_{i}|. For a set Λ ⊂ Z^{d}, diamΛ and |Λ| stand respectively for its diameter and the number of the points it contains. We write Λ ⊂⊂ Z^{d} when 1 ≤ |Λ| < ∞ and define a family A by

A = {Λ; Λ ⊂⊂ Z^{d}}.  (2.1)

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}.  (2.2)

The value of r will eventually be chosen as an upper bound r_{0} of the range of the interaction we consider (cf.(2.5) below). For v ∈ Z^{d} and an integer m ≥ 1, we define a subset B_{v}(m) of A as the totality of Λ ⊂⊂ Z^{d} of the following form;

Λ = {x ∈ Z^{d}; v_{i} + ma_{i} ≤ x_{i} < v_{i} + mb_{i} for i = 1, ⋯, d},  (2.3)

where a_{i} and b_{i} are integers with a_{i} < b_{i}. In particular, the class B_{v}(m) consists of boxes with every side-length positive multiple of m. We define a subset C_{v}(m) of B_{v}(m) as the totality of cubes in B_{v}(m), i.e., the totality of Λ ⊂⊂ Z^{d} of the form (2.3) with b_{i} − a_{i} (1 ≤ i ≤ d) identical.

The configurations. We take a finite set S as a single spin space and λ will stand for the uniform distribution on S; λ(dt) = \frac{1}{|S|} ∑_{s∈S} δ_{s}(dt). Configuration spaces are defined as follows;

Ω_{Λ} = {σ = (σ_{x})_{x∈Λ} ; σ_{x} ∈ S}, Λ ⊂ Z^{d},
Ω = Ω_{Z^{d}}.

For Λ ⊂ Z^{d} and (σ, ω) ∈ Ω^{2}, σ_{Λ} · ω_{Λ^{c}} denotes the following configuration;

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

For f : Ω_{Λ} → R, we introduce the notations

∇_{X}f(σ) = ∫_{S^{X}} f(ξ_{X} · σ_{X^{c}})λ^{X}(dξ_{X}) − f(σ), X ⊂⊂ Z^{d},

∥f∥ = sup_{σ∈Ω_{Λ}} |f(σ)|,

|||f||| = ∑_{x∈Λ} ∥∇_{x}f∥,

osc_{x}(f) = sup_{(σ,σ′)∈Ω^{2}}{|f(σ) − f(σ′)|; σ ≡ σ′ off x},

Δ_{f} = {x ∈ Λ ; f is not a constant with respect to σ_{x}}.

The function spaces C and C_{Λ} (Λ ⊂ Z^{d}) are defined respectively by

C = {f : Ω → R ; |Δ_{f}| < ∞ },
C_{Λ} = {f : Ω → R ; Δ_{f} ⊂ Λ }.

The interaction and finite volume Gibbs states. A family Φ = {Φ_{X} ∈ C_{X} : X ⊂⊂ Z^{d}} is called a bounded, finite range interaction if it satisfies the following

(Φ-1) There exists M_{0} < ∞ such that

∥Φ∥ := sup_{x∈Z^{d}} ∑_{X:X∋x} ∥Φ_{X}∥ ≤ M_{0}.  (2.4)

(Φ-2) There exists r_{0} < ∞ such that

r(Φ) := sup{diam(X); Φ_{X} ≢ 0} ≤ r_{0}.  (2.5)

∥Φ∥ in (2.4) and r(Φ) in (2.5) are called the norm and the range of the interaction, respectively.

From here on, we fix a bounded, finite range interaction Φ. For each Λ ⊂⊂ Z^{d}, we define the Hamiltonian H_{Λ} ∈ C by

H_{Λ} = ∑_{X:X∩Λ≠ϕ} Φ_{X}.  (2.6)

For each Λ ⊂⊂ Z^{d} and ω ∈ Ω, we define the finite volume Gibbs state μ^{Λ,ω} as the probability measure on Ω_{Λ}, in which each configuration σ_{Λ} ∈ Ω_{Λ} appears with probability

μ^{Λ,ω}(\{σ_{Λ}\}) = \frac{exp −H_{Λ}(σ_{Λ} · ω_{Λ^{c}})}{Z^{Λ,ω}},  (2.7)

where Z^{Λ,ω} is the normalizing constant.

The stochastic dynamics. We introduce now for the model above, the time evolution called Glauber dynamics. We define for each x ∈ Z^{d} an operator A_{x} : C → C by A_{x}f(σ) = ∑_{s∈S} c_{x}(σ, s)(f(σ ·^{x} s) − f(σ)), where σ ·^{x} s is a configuration obtained from σ by replacing σ_{x} by s and the coefficient c_{x}(σ, s), which is called flip rate is required to satisfy the following.

(R-1) Boundedness : There exist positive constants \underline{c} and \overline{c} such that

\underline{c} ≤ c_{x}(σ, s) ≤ \overline{c}, for all x ∈ Z^{d}, σ ∈ Ω and s ∈ S.

(R-2) Finite range: There exists r_{1} ≥ 0 such that

γ(x, y) = 0 if |x − y| > r_{1},  (2.8)

where

γ(x, y) = sup_{(σ,σ′)∈Ω^{2}} {∑_{s∈S} |c_{x}(σ, s) − c_{x}(σ′, s)|; σ ≡ σ′ off y} .  (2.9)

(R-3) Detailed Balance Condition: For all x ∈ Z^{d}, σ ∈ Ω and (s, s′) ∈ S^{2},

μ^{{x},σ}({s})c_{x}(σ ·^{x} s, s′) = μ^{{x},σ}({s′})c_{x}(σ ·^{x} s′, s),  (2.10)

Remark 2.1 This is a very technical remark and is relevant only when one would like to identify a numerical constant M_{1} which will appear in Theorem 3.2, Lemma 4.1 and Lemma 4.2. We set

M = sup_{x} ∑_{y:y≠x} γ(x, y),
M_{1} = ∑_{y∈Z^{d}} γ_{1}(y)
= (2r_{1} + 1)^{d}|S|(\overline{c} − \underline{c}),  (2.11)

where γ(·, ·) is defined by (2.9) and

γ_{1}(y) = \{ \begin{matrix}|S|(\overline{c} − \underline{c}) & if |y| ≤ r_{1} \\ 0 & otherwise. \end{matrix}  (2.12)

The function γ(·, ·), together with the constant M plays fundamental role in analyzing a particle system (cf. [Lig85, Chapter I]). The function γ_{1}(·) is introduced as a shift-invariant object which bound γ(·, ·) from above. Note that γ(x, y) ≤ γ_{1}(x − y) and hence M ≤ M_{1}.

For fixed Λ ⊂⊂ Z^{d} and ω ∈ Ω, we define an operator A^{Λ,ω} : C_{Λ} → C_{Λ} and the associated semigroup (T_{t}^{Λ,ω})_{t≥0} by

A^{Λ,ω}f(σ) = ∑_{x∈Λ} A_{x}f(σ_{Λ} · ω_{Λ^{c}}), f ∈ C_{Λ},

T_{t}^{Λ,ω} = exp tA^{Λ,ω}.

We then see from (2.10) that for all {f, g} ⊂ C_{Λ}

E^{Λ,ω}(f, g) =^{def} −μ^{Λ,ω}(fA^{Λ,ω}g)

= \frac{1}{2} ∑_{x∈Λ} ∑_{s∈S} ∫ μ^{Λ,ω}(dσ)c_{x}(σ, s)[f(σ ·^{x} s) − f(σ)][g(σ ·^{x} s) − g(σ)].  (2.13)

We have defined A^{Λ,ω} and T_{t}^{Λ,ω} as operators acting on C_{Λ}. But we will extend their domain of definition to C by applying them to “ σ_{Λ} ↦ f(σ_{Λ} · ω_{Λ})” .

The spectral gap, the logarithmic Sobolev constant, and the Dobrushin-Shlosman mixing condition. We now introduce a couple of quantities and a notion which plays crucial roles to analyze the large time behaviour of the Glauber dynamics. We define inverse spectral gap γ_{SG}(Λ, ω) as the smallest γ for which the following inequality is true for all f ∈ C;

μ^{Λ,ω}(|f − μ^{Λ,ω}f|^{2}) ≤ γE^{Λ,ω}(f, f).  (2.14)

Similarly, we define the logarithmic Sobolev constant γ_{LS}(Λ, ω) as the smallest γ for which the following inequality is true for all f ∈ C;

μ^{Λ,ω} (f^{2} log \frac{f^{2}}{μ^{Λ,ω}(f^{2})}) ≤ 2γE^{Λ,ω}(f, f).  (2.15)

For F ⊂ A, an interaction Φ is said to satisfy the Dobrushin-Shlosman mixing condition over F, if there exist constant C_{i} ∈ (0, ∞) (i = 1, 2) such that for all Λ ∈ F, y ∉ Λ and f ∈ C_{Λ},

osc_{y}(μ^{Λ,·}f) ≤ C_{1}∥f∥ exp −\frac{d(y, Δ_{f})}{C_{2}}.  (2.16)

In the sequel, we will refer to the above condition as DSM(F).

It is convenient to introduce the following definition. We call a family {c(f) > 0}_{f∈C} admissible coefficient if it satisfies;

sup_{f}{c(f); ∥f∥ + diamΔ_{f} ≤ m} < ∞ for all m > 0.  (2.17)

What (2.17) requires is that c(f) should have an upper bound in terms of ∥f∥ and diamΔ_{f}, which are independent of where Δ_{f} is. Typically, const.∥f∥ or const.|||f||| appears as the admissible coefficient c(f).

Remark 2.2 We could have worked in continuous spin setting, in which the discrete spin space S is replaced by a smooth, compact Riemanian manifold (in fact, C^{2} is enough). What we have defined above can easily be modified so that they make sense in continuous spin setting and the all results as well as their proofs in this article remain valid up to the value of constants and boring technicalities.

3 The result

First we recall the following result:

Theorem 3.1 ([SZ92b]) Let F be either A or B_{v}(m) for arbitrarily fixed v ∈ Z^{d} and m ≥ 1 (cf.(2.1),(2.3)). Then, each of the following conditions is equivalent to the DSM (F) (cf.(2.16));

(1a) The logarithmic Sobolev constant(cf. (2.15)) satisfies;

sup_{Λ∈F,ω∈Ω} γ_{LS}(Λ, ω) < ∞.  (3.1)

(1b) The inverse spectral gap(cf. (2.14)) satisfies;

sup_{Λ∈F,ω∈Ω} γ_{SG}(Λ, ω) < ∞.  (3.2)

(1c) There exist a constant C > 0 and an admissble coefficient {c(f) > 0}_{f∈C} (cf. (2.17)) such that for all f ∈ C and t > 0;

sup_{Λ∈F,ω∈Ω} ∥T_{t}^{Λ,ω}f − μ^{Λ,ω}f∥ ≤ c(f) exp −\frac{t}{C} .  (3.3)

Remark 3.1 In the original statement of this theorem ([SZ92b, Theorem 1.8, part(c)]), only the family A is under consideration. The statements for B_{v}(m) is implicit in the argument in that paper. The point is that both A and B_{v}(m) is closed under intersection and contains arbitrarily large cubes. The DSM condition restricted to boxes is easier to check than DSM(A) is. In fact, DSM(A) is known to be true for the Ising ferromagnet with β < β_{c}/2 or |h| > 2d ([Hig], See also [DSh85, (2.32)] for the latter case), while the validity of DSM(B_{v}(1)) extends to |h| > d − 1. As is mentioned in [HY96, Remark 2.3], there is also a difference between DSM(B_{v}(1)) and DSM(B_{v}(m)) with a large m if d ≥ 3. The DSM condition based on cubes, together with an analogous statement to Theorem 3.1 in that context, is studied extensively in [MO94a], [MO94b], [MOS94] and [SS95]. In this connection, it should be mentioned that [MO94a, Theorem 4.1] contains the following nice observation; if DSM(∪_{v∈Z^{d}} ∪_{m≥m_{0}} C_{v}(m)) holds for some m_{0} ≥ 1, then DSM(∪_{v∈Z^{d}} ∪_{m≥m_{1}} B_{v}(m)) holds for large enough m_{1} ≥ 1.

Next, we state the main result of this paper.

Theorem 3.2 Let F be either A or B_{v}(m) for arbitrarily fixed v ∈ Z^{d} and m ≥ 1 (cf.(2.1),(2.3)). Then each of the following conditions is equivalent to DSM(F) (cf. (2.16));

(2a) The logarithmic Sobolev constant (cf. (2.15)) satisfies;

\overline{lim_{D→∞}} sup_{Λ∈F,ω∈Ω} \{\frac{γ_{LS}(Λ, ω)}{ψ(diamΛ)}; diamΛ ≥ D \} < \frac{1}{10(d − 1)r_{1}M_{1}} ,  (3.4)

where ψ(t) = t/ log t. The constants r_{1} and M_{1} are given respectively by (2.8) and (2.11).

(2b) The inverse spectral gap (cf. (2.14)) satisfies;

\overline{lim_{D→∞}} sup_{Λ∈F,ω∈Ω} \{\frac{γ_{SG}(Λ, ω)}{ψ(diamΛ)}; diamΛ ≥ D \} < \frac{1}{10(d − 1)r_{1}M_{1}} ,  (3.5)

where ψ(t) = t/ log t The constants r_{1} and M_{1} are given respectively by (2.8) and (2.11).

(2c) There exists an admissible coefficient {c(f) > 0}_{f∈C} (cf. (2.17)) and a non-increasing function ϕ : (0, ∞) → (0, ∞) such that ϕ(t) = o(t^{−2(d−1)}) as t → ∞ and such that

sup_{Λ∈F,ω∈Ω} μ^{Λ,ω}|T_{t}^{Λ,ω}f − μ^{Λ,ω}f| ≤ c(f)ϕ(t), for all f ∈ C.  (3.6)

(2d) There exist C_{1} ∈ (2, ∞), an admissible coefficient {c(f) > 0}_{f∈C} and a non-increasing function ϕ : (0, ∞) → (0, ∞) such that ϕ(t) = o(t^{−2(d−1)}) as t → ∞ and such that

osc_{y}(μ^{Λ,·}f) ≤ c(f)ϕ(d(y, Δ_{f}))  (3.7)
for all Λ ∈ F, f ∈ C_{Λ} and y ∈ ∂_{r_{0}}Λ which satisfy diamΛ ≤ C_{1}d(y, Δ_{f}).

(2e) There exists a non-increasing function ϕ : (0, ∞) → (0, ∞) such that ϕ(t) = o(t^{−(d−1)})

as t → ∞ and such that

osc_{y}(μ^{Λ,·}f) ≤ ∥f∥ϕ(d(y, Δ_{f}))  (3.8)

for all Λ ∈ F, f ∈ C_{Λ} and y ∈ ∂_{r_{0}}Λ.

Remark 3.2 It follows from Theorem 3.2 that the condition (2a) or even (2b) implies (1c) in Theorem 3.1. This is reminiscent of [HS87, Theorem 5.3]. Although (2a) and (2b) require control over boxes, not only over cubes, the conclusion (1c) is quite rewarding as compared with that of the above quoted reference. The bound 1/{10(d−1)r_{1}M_{1}} on the right-hand-sides of (3.4), (3.5) is technical and is in no way sharp.

Remark 3.3 It follows from Theorem 3.2 is that (2c) implies (1c), which roughly says the following: if the rate of convergence of a Glauber dynamics is bounded by a certain negative power of the time t, then the convergence is necessarily exponentially fast in t. As is mentioned in the introduction, this is not the first time a result of this kind is obtained. For example, it is shown in [SZ92a, Theorem 3.6] that L^{∞}-convergence as fast as t^{−(2d+ε)} implies exponentially fast L^{∞}-convergence. Also, it is mentioned in [Lar95, Remarque 4.3] that L^{2}-convergence as fast as t^{−(2d+ε)} implies exponentially fast L^{2}-convergence. On the other hand, our result ((2c) ⇒(1c)) says that L^{1}-convergence faster than t^{−2(d−1)} in the sense of (2c) is enough for exponentially fast L^{∞}-convergence, which improves two results mentioned above. It should also be mentioned that in the case S = {−1, +1} and the flip rate is attractive, (1c) is also implied by the following condition;

lim_{t→∞} t^{d} sup_{Λ∈F,ω∈Ω} sup_{x∈Λ} {(T_{t}^{Λ,ω}σ_{x})(+) − (T_{t}^{Λ,ω}σ_{x})(−)} = 0,  (3.9)

where (±) stand for configurations with all spins equal to ±1. This follows from the proof of [AH87, Theorem 4], with a slight adjustment in order to circumvent the lack of shift-invariance. In d = 2, condition (2c) is milder than (3.9)(cf.[AH87, Lemma 2.1]).

Remark 3.4 Condition (2e) says that the exponential decay in the definition of DSM(F) is equivalent to a certain polynomial decay. Technically, condition (2d) plays a role of a junction between a statement in terms of dynamics, like (2b) or (2c) and that in terms of equilibrium, like (2e) (cf. Lemma 4.2 and Lemma 4.3). Let us compare condition (2d) with a similar condition in [DSh87]. In [DSh87], it is proved that DSM(F) has the following relaxed criterion, which is called condition (IIIb) in that paper; there exists a non-increasing function ϕ : (0, ∞) → (0, ∞) such that ϕ(t) = o(t^{−2(d−1)}) as t → ∞ and such that

osc_{y}(μ^{Λ,·}f) ≤ ∥f∥ ∑_{x∈Δ_{f}} ϕ(|x − y|)  (3.10)

for all Λ ∈ F, f ∈ C_{Λ} and y ∈ ∂_{r_{0}}Λ. As can easily be seen, condition (2d) relaxes condition (IIIb) even further.

4 Proof of Theorem 3.2

Lemma 4.1 Let Λ ⊂⊂ Z^{d} and ω ∈ Ω be fixed. Suppose that ϕ(t) is a non-increasing function such that lim_{t→∞} ϕ(t) = 0 and that

μ^{Λ,ω}|T_{t}^{Λ,ω}f − μ^{Λ,ω}f| ≤ c(f)ϕ(t) for all f ∈ C and t > 0,  (4.1)

with some coefficient c(f). Then, there exists C ∈ (0, ∞) which depends only on the flip rate such that for all {f, g} ⊂ C

μ^{Λ,ω}(f; g) =^{def.} μ^{Λ,ω}(fg) − μ^{Λ,ω}fμ^{Λ,ω}g

≤ c(f, g)ϕ (\frac{d(Δ_{f}, Δ_{g})}{5M_{1}r_{1}}) + C|||f||| |||g||| exp −\frac{d(Δ_{f}, Δ_{g})}{5r_{1}},  (4.2)

where c(f, g) = c(fg) + ∥f∥c(g) + c(f)∥g∥. The constants r_{1} and M_{1} are given respectively by (2.8) and (2.11).

Proof: We first bound the left-hand-side of (4.2) by three terms;

μ^{Λ,ω}|T_{t}^{Λ,ω}(fg) − T_{t}^{Λ,ω}f · T_{t}^{Λ,ω}g| ≤ μ^{Λ,ω}|T_{t}^{Λ,ω}(fg) − T_{s}^{Λ,ω}(fg)|  (4.3)
+ μ^{Λ,ω}|T_{s}^{Λ,ω}(fg) − T_{s}^{Λ,ω}f · T_{s}^{Λ,ω}g|  (4.4)
+ μ^{Λ,ω}|T_{s}^{Λ,ω}f · T_{s}^{Λ,ω}g − T_{t}^{Λ,ω}f · T_{t}^{Λ,ω}g|.  (4.5)

The condition (4.1) is available to bound the first and the third terms;

(4.3) ≤ μ^{Λ,ω}|T_{t}^{Λ,ω}(fg) − μ^{Λ,ω}(fg)| + μ^{Λ,ω}|T_{s}^{Λ,ω}(fg) − μ^{Λ,ω}(fg)|
≤ c(fg)(ϕ(t) + ϕ(s)).  (4.6)

Similarly,

(4.5) ≤ (∥f∥c(g) + c(f)∥g∥)(ϕ(t) + ϕ(s)).  (4.7)

We now apply Proposition 4.18 of [Lig85, page 40] to (4.4) to see that

(4.4) ≤ C|||f||| |||g||| exp(4M_{1}s − δρ)  (4.8)

where δ = 1/r_{1}, ρ = d(Δ_{f}, Δ_{g}) and C > 0 is a constant which depnds only on the flip rate (cf. Remark 4.1 below). At this point, we take s = δρ/5M_{1}. We then have by (4.6), (4.7) and (4.8) that

μ^{Λ,ω}|T_{t}^{Λ,ω}(fg) − T_{t}^{Λ,ω}f · T_{t}^{Λ,ω}g|

≤ (c(fg) + ∥f∥c(g) + c(f)∥g∥) (ϕ(t) + ϕ(δρ/5M_{1}))

+C|||f||| |||g||| exp −\frac{δρ}{5}

which proves (4.2) by letting t → ∞.

q.e.d.

Remark 4.1 Since (T_{t}^{Λ,ω})_{t≥0} is not a shift-invariant particle system, (4.8) does not come from a direct application of [Lig85, Proposition 4.18] but from a modification of the proof of that proposition, to be precise. The modification is simply to replace γ(x, y) by its upper bound γ_{1}(x − y) (cf. Remark 2.1). The choice δ = 1/r_{1} is possible by the following reason. Going back to the proof of the proposition, δ has been chosen so that

∑_{x∈Z^{d}} γ_{1}(x) exp(δ|x|) ≤ 2M_{1}.  (4.9)

Since the left-hand-side of (4.9) is not greater than exp(δr_{1})M_{1}, our choice δ = 1/r_{1} is more than enough.

Lemma 4.2 Let C_{0} ∈ [0, ∞), C_{1} ∈ (0, ∞) and F ⊂ A be arbitrary. Suppose that there exists C_{2} ∈ (0, ∞) such that

\overline{lim_{D→∞}} sup_{Λ∈F,ω∈Ω} \{\frac{γ_{SG}(Λ, ω)}{ψ(diamΛ)}; diamΛ ≥ D \} < C_{2},  (4.10)

where ψ(t) = t/ log t. Then there exists C_{3} ∈ (0, ∞) such that

osc_{y}(μ^{Λ,·}f) ≤ C_{3}|||f|||d(y, Δ_{f})^{−1/(5C_{1}C_{2}M_{1}r_{1})}  (4.11)

for all Λ ∈ F, f ∈ C_{Λ} and y ∈ ∂_{r_{0}}Λ which satisfy

diamΛ ≤ C_{1}d(y, Δ_{f}) + C_{0}.  (4.12)

The constant r_{1} and M_{1} are given by (2.8) and (2.11), respectively.

Proof: By (4.10), there exist θ ∈ (0, 1) and D ≥ 1 such that

\frac{sup_{ω∈Ω} γ_{SG}(Λ, ω)}{C_{2}ψ(diamΛ)} ≤ θ ≤ \frac{C_{1}D}{C_{1}D + C_{0}} ,  (4.13)

for any Λ ∈ F with diamΛ ≥ D. To prove (4.11), we may assume ρ := d(y, Δ_{f}) ≥ D + r_{0}, since the left-hand-side of (4.11) is not greater than |||f||| in any case. Let (ω, ω′) ∈ Ω^{2} be such that ω′ ≡ ω off y. We set g = \frac{dμ^{Λ,ω′}}{dμ^{Λ,ω}} . Note that

∥g∥ ≤ exp(4∥Φ∥) and Δ_{g} ⊂ Λ ∩ ∂_{r_{0}}{y}.

Using these and Lemma 4.1, we have

|(μ^{Λ,ω′} − μ^{Λ,ω})f| = |μ^{Λ,ω}(g; f)|
≤ c_{1}|||f||| exp −\frac{ρ − r_{0}}{5M_{1}r_{1}γ_{SG}(Λ, ω)}  (4.14)
+ c_{2}|||f||| exp −\frac{ρ − r_{0}}{5r_{1}} .  (4.15)

Here and in what follows, c_{i} (i = 1, 2, ⋯) stand for constants, which are independent of Λ, f and y. Since diamΛ ≥ ρ − r_{0} ≥ D, we can take advantage of (4.13) and (4.12) to bound (4.14) from above as follows ;

(4.14) ≤ c_{3}|||f||| exp (−\frac{ρ}{5M_{1}r_{1}θC_{2}(C_{1}ρ + C_{0})} log(C_{1}ρ + C_{0}))
≤ c_{3}|||f||| exp (−\frac{1}{5C_{1}C_{2}M_{1}r_{1}} log(C_{1}ρ + C_{0}))
≤ c_{3}|||f|||ρ^{−1/(5C_{1}C_{2}M_{1}r_{1})}.  (4.16)

Similarly, we have that

(4.15) ≤ c_{4}|||f||| exp −\frac{ρ}{5r_{1}}
≤ c_{5}|||f|||ρ^{−1/(5C_{1}C_{2}M_{1}r_{1})}.  (4.17)

Putnting (4.16) and (4.17) together, we conclude (4.11).

q.e.d.

Lemma 4.3 Let C_{1} ∈ (2, ∞) and n > 1 be such that \frac{n(4n−3)}{2(n−1)^{2}} ≤ C_{1} and F = B_{v}(m) for arbitrarily fixed v ∈ Z^{d} and m ≥ 1. Suppose that there exist an admissible coefficient {c(f) > 0}_{f∈C} and a non-increasing function ϕ : (0, ∞) → (0, ∞) such that

osc_{y}(μ^{Λ,·}f) ≤ c(f)ϕ(d(y, Δ_{f}))  (4.18)

for all Λ ∈ F, f ∈ C_{Λ} and y ∈ ∂_{r_{0}}Λ which satisfy diamΛ ≤ C_{1}d(y, Δ_{f}). Then there exists C_{2} ∈ (0, ∞), such that

osc_{y}(μ^{Λ,·}f) ≤ C_{2}∥f∥d(y, Δ_{f})^{d−1}ϕ(d(y, Δ_{f}) − 2m − r_{0})  (4.19)

for all Λ ∈ F, f ∈ C_{Λ} and y ∈ ∂_{r_{0}}Λ which satisfy d(y, Δ_{f}) ≥ n(2m + r_{0}).

For F = A, the same statement with m = 1 is true.

Proof: Take Λ ∈ F, f ∈ C_{Λ} and y ∈ ∂_{r_{0}}Λ which satisfy ρ := d(y, Δ_{f}) ≥ n(2m + r_{0}). We are going to apply our assumption to Λ ∩ Γ, rather than Λ itself, where Γ is an element of C_{v}(m) we now define. We define Γ_{k} ∈ C_{v}(m) (k = 1, 2, ⋯) by

Γ_{k} = {x : d(x, Γ_{0}) ≤ km},

where Γ_{0} is an element in C_{v}(m) such that diamΓ_{0} = m and y ∈ Γ_{0}. We set Γ = Γ_{K}, where K = max{k ≥ 1 : Γ_{k} ∩ Δ_{f} = ϕ}. We then have by definition that

Km ≤ ρ ≤ (K + 2)m  (4.20)

and hence that

K ≥ \frac{ρ}{m} − 2
≥ 2(n − 1).  (4.21)

For fixed (ω, ω′) ∈ Ω^{2} with ω ≡ ω′ off y, we set g = \frac{dμ^{Λ,ω′}}{dμ^{Λ,ω}}. Note that

∥g∥ ≤ exp(4∥Φ∥), Δ_{g} ⊂ Λ ∩ ∂_{r_{0}}{y}  (4.22)

and that for z ∈ Λ ∩ ∂_{r_{0}}Γ,

diam(Γ ∩ Λ) ≤ C_{1}d(z, Δ_{g}).  (4.23)

The first two observations are standard. The third one can be seen as follows. We have by (4.22) and (4.20) that

d(z, Δ_{g}) ≥ |z − y| − r_{0}
≥ Km − r_{0}
≥ (\frac{ρ}{m} − 2) m − r_{0}
= ρ − 2m − r_{0}
≥ \frac{n − 1}{n}ρ.  (4.24)

By (4.20), (4.21) and (4.24), we get

diam(Γ ∩ Λ) ≤ (2K + 1)m
≤ (2 + \frac{1}{K} ) ρ
≤ (2 + \frac{1}{2(n − 1)}) \frac{n}{n − 1}d(z, Δ_{g})
≤ C_{1}d(z, Δ_{g}).

The proof of (4.19) comes down to the following estimate;

∥μ^{Γ∩Λ,·}(g − 1)∥ ≤ C_{2}ρ^{d−1}ϕ(ρ − 2m − r_{0}).  (4.25)

In fact,

|(μ^{Λ,ω′} − μ^{Λ,ω})f| = |μ^{Λ,ω}[f(g − 1)]|
≤ ∥f∥∥μ^{Γ∩Λ,·}(g − 1)∥.

On the other hand, (4.25) can be seen as follows;

|μ^{Γ∩Λ,ξ}(g − 1)| = |μ^{Λ,ω}(μ^{Γ∩Λ,ξ}g − μ^{Γ∩Λ,·}g)|
≤ sup_{(ξ,η)∈Ω^{2}} {|μ^{Γ∩Λ,ξ}g − μ^{Γ∩Λ,η}g|; ξ ≡ η outside Λ}
≤ |Λ ∩ ∂_{r_{0}}Γ| sup_{z∈Λ∩∂_{r_{0}}Γ} osc_{z}(μ^{Γ∩Λ,·}g)
≤ c_{1}ρ^{d−1}c(g) sup_{z∈Λ∩∂_{r_{0}}Γ} ϕ(d(z, Δ_{g}))
≤ c_{2}ρ^{d−1}ϕ(ρ − 2m − r_{0}).

The first equality comes from an identity: 1 = μ^{Λ,ω}g = μ^{Λ,ω}(μ^{Γ∩Λ·}g). The inequality in the forth line is an application of (4.18) to Γ∩Λ ∈ F, which is allowed by (4.23) under our present assumption. To proceed to the last line, we have used (4.22).

q.e.d.

Lemma 4.4 Let F be either A or B_{v}(m) for arbitrarily fixed v ∈ Z^{d} and m ≥ 1. Suppose that (2e) in Theorem 3.2 holds and set α_{x,y} = ϕ(|x − y|). Then,

lim_{L→∞} \frac{1}{L^{d}} ∑_{x∈[0,L)^{d}} ∑_{y∈∂_{r_{0}}[0,L)^{d}} α_{x,y} = 0  (4.26)

and for all X ∈ F and y ∉ X,

∥∇_{y}μ^{X,·}f − μ^{X,·}∇_{y}f∥ ≤ ∑_{x∈X} ∥∇_{x}f∥α_{x,y}.  (4.27)

Remark 4.2 A mixing property defined by the set of conditions (4.26) and (4.27) is essentially the same as the condition (GS2) in [Hol85] and the condition DSM(Y) in [SZ92b]. This condition turns out to be equivalent to DSM(F) in the end (cf. the proof of Theorem 3.2).

Proof: It is easy to see (4.26). In fact,

∑_{x∈[0,L)^{d}} ∑_{y∈∂_{r_{0}}[0,L)^{d}} α_{x,y} = ∑_{x∈[0,L)^{d}} ∑_{y∈∂_{r_{0}}[0,L)^{d}} ϕ(|x − y|)

≤ ∑_{m=1}^{L+r_{0}} ∑_{y∈∂_{r_{0}}[0,L)^{d}} ∑_{x:|x−y|=m} ϕ(m)

≤ c_{1}L^{d−1} ∑_{m=1}^{L+r_{0}} m^{d−1}ϕ(m),

and thus,

\frac{1}{L^{d}} ∑_{x∈[0,L)^{d}} ∑_{y∈∂_{r_{0}}[0,L)^{d}} α_{x,y} ≤ \frac{c_{1}}{L} ∑_{m=1}^{L+r_{0}} m^{d−1}ϕ(m)
⟶ 0, as L → ∞.

We next prove (4.27). Take X ∈ F and an alignment {x_{m}}_{m=1}^{n} of points in X. We set I_{m} = {x_{j}}_{j=1}^{m−1} and J_{m} = {x_{j}}_{j=m}^{n}. Note first that

∇_{X}f(σ) = ∑_{m=1}^{n} ∫ λ^{I_{m}}(dξ_{I_{m}})∇_{x_{m}}f(ξ_{I_{m}} · σ_{I_{m}}^{c}).

Using this expression, we have that

∇_{y}μ^{X,ω}f − μ^{X,ω}∇_{y}f
= ∫ λ(ds)(μ^{X,ω·s} − μ^{X,ω})(dσ_{X})f(σ_{X} · (ω · s)_{X^{c}})
= − ∫ λ(ds)(μ^{X,ω·s} − μ^{X,ω})(dσ_{X})∇_{X}f(σ_{X} · (ω · s)_{X^{c}})
= − ∑_{m=1}^{n} ∫ λ(ds) ∫ (μ^{X,ω·s} − μ^{X,ω})(dσ_{X})\overline{f_{m}}(σ_{X} · (ω · s)_{X^{c}}),  (4.28)

where ω · s stand for a configuration obtained from σ by replacing σ_{y} by s and

\overline{f_{m}}(σ) = ∫ λ^{I_{m}}(dξ_{I_{m}})∇_{x_{m}}f(ξ_{I_{m}} · σ_{I_{m}}^{c}).

At this point, fix y and choose the alignment so that |x_{j} − y| is non-decreasing in j = 1, 2, · · ·. Since ∥\overline{f_{m}}∥ ≤ ∥∇_{x_{m}}f∥ and “σ_{X} ↦ \overline{f_{m}}(σ_{X} · (ω · s)_{X^{c}})” ∈ C_{J_{m}}, we have by (4.28) and (2e) that

∥∇_{y}μ^{X,·}f − μ^{X,·}∇_{y}f∥ ≤ ∑_{m=1}^{n} ∥∇_{x_{m}}f∥ϕ(d(y, J_{m}))
= ∑_{m=1}^{n} ∥∇_{x_{m}}f∥ϕ(|x_{m} − y|)
= ∑_{x∈X} ∥∇_{x}f∥α_{x,y}.

This proves (4.27).

q.e.d.

Proof of Theorem 3.2 In view of Theorem 3.1, all conditions in Theorem 3.2 are obviously necessary for the the Dobrushin-Shlosman mixing condition (2.16) to be true. To show that (2a), (2b),(2d) and (2e) are sufficient, we prove the the following sequence of implications;

(2a) ⇒ (2b) ⇒ (2d) ⇒ (2e) ⇒ (1b).  (4.29)

The condition (1b) is equivalent to the Dobrushin-Shlosman mixing condition by Theorem 3.1. The condition (2a) immediately implies (2b), since γ_{SG} ≤ γ_{LS} ([DS89, page 224, (6.1.7)]). Implication from (2b) to (2d) is a consequence of Lemma 4.2, and (2d) implies (2e) by Lemma 4.3. Now suppose that (2e) holds. This means that we may assume the conclusion of Lemma 4.4. Then, by the argument used in section 2 of [SZ92b], we obtain (1b) in Theorem 3.1. See the derivation of Corollary 2.8 in that paper. This completes the proof of (4.29). On the other hand, Lemma 4.1 says that (2c) implies (2d), and hence also (1b) by (4.29).

q.e.d.

Acknowledgement: This work originates in conversations with Yasunari Higuchi whom I would like to thank. I am also indebted to Roberto Schonmann for stimulating discussions, through which the improvement of the previous version of this paper was incubated. A question which Tetsuya Hattori asked me about the difference among DSM(F) for different F gave me another incentive to reconsider the framework of the older version. Finally, I am grateful for Dan Stroock for his interest in this work and for hospitality at MIT.

References

[AH87] Aizenmann, M., Holley, R. : Rapid convergence to equilibrium of stochastic Ising models in the Dobrushin-Shlosman regime, In: Percolation Theory and infinite particle systems, H. Kesten (ed.) IMA volumes in Math. and Appl., Springer pp.1–11,(1987).

[DS89] Deuschel, J. D. and Stroock, D. W. : “Large Deviations”, Academic Press, 1989.

[DSh85] Dobrushin, R. L., Shlosman, S. : Completely analytical Gibbs fields, in “Statistical Physics and Dynamical Systems” ed. by J. Fritz, A. Jaffe and D. Szasz, Birkhäuser (1985).

[DSh87] Dobrushin, R. L., Shlosman, S. : Completely analytical interactions: Constructive description, J. Stat. Phys., 46, 983–1014, (1987).

[Hig] Higuchi, Y.: A hand-written note in Japanese.

[HY96] Higuchi,Y., Yoshida, N.: Slow relaxation of stochastic Ising models with random and nonrandom boundary conditions, To appear in Proceeding of Taniguchi Symposium, (1996).

[Hol85] Holley, R. : Possible rate of convergence in finite range, attractive spin systems, Comtemp. Math. 41 215–234 (1985).

[HS87] Holley, R., Stroock, D. W. : Logarithmic Sobolev inequality and stochastic Ising models, J. Stat. Phys. 46, 1159-1194 (1987).

[Lar95] Laroche, E. : Hypercontractivité pour des systèm de spins de portée infinie, Prob. Th. Rel. Fields, 101, 89–132 (1995).

[Lig85] Liggett, T. M. :Interacting Particle Systems, Springer Verlag, Berlin-Heidelberg-Tokyo (1985).

[MO94a] Martinelli, F., Olivieri, E.: Approach to equilibrium of Glauber dynamics in the one phase region I: Attractive case, Commun, Math. Phys. 161, 447–486 (1994)

[MO94b] Martinelli, F., Olivieri, E.: Approach to equilibrium of Glauber dynamics in the one phase region II: General case, Commun, Math. Phys. 161, 487–514 (1994)

[MOS94] Martinelli, F., Olivieri, E., Schonmann, R. : For 2-D lattice spin systems weak mixing implies strong mixing, Commun. Math. Phys. 165, 33–47 (1994).

[SS95] Schonmann, R. H. and Shlosman, S. B.: Complete analyticity for 2D Ising completed, Commun. Math. Phys. 170, 453–482 (1995).

[SZ92a] Stroock, D. W. and Zegarlinski, B. : The equivalence of the logarithmic Sobolev inequality and the Dobrushin-Shlosman mixing condition, Commun. Math. Phys. 144, 303-323 (1992).

[SZ92b] Stroock, D. W. and Zegarlinski, B. : The logarithmic Sobolev inequality for discrete spin systems on the lattice, Commun. Math. Phys. 149,175-193 (1992).