Exponential relaxation of Glauber dynamics with some special boundary conditions

Roberto H. Schonmann^{1} and Nobuo Yoshida ^{2,3}

1 Mathematics Department, UCLA, Los Angeles, CA 90024, U.S.A., e-mail: rhs@math.ucla.edu 2 Division of Mathematics, Graduate School of Science, Kyoto University, Kyoto 606-01, Japan, e-mail:nobuo@kusm.kyoto-u.ac.jp 3 Mathematics Department, 2-101, MIT, Cambridge, MA 02139, USA. e-mail:nobuo@math.mit.edu

This paper is dedicated to the memory of Professor Roland Dobrushin

Key words and phrases: Exponential convergence, mixing conditions, attractivity, Ising model, Basuev regime, boundary conditions, complete analyticity.

Abstract. We consider attractive finite-range Glauber dynamics and show that if a certain mixing condition is satisfied, then the system evolving on arbitrary subsets of the lattice, with appropriate boundary conditions, converges to equilibrium exponentially fast, in the uniform sense, uniformly over the subsets of the lattice. This result applies, for instance, to the ferromagnetic nearest neighbor Ising model in the so called “Basuev region”, where complete analyticity is expected to fail.

Technically the result in this paper is an extension of a result of Martinelli and Olivieri, who proved that under a weaker form of mixing the infinite system approaches equilibrium exponentially fast.

Conceptually this paper may be seen as a step towards developing and exploiting a restricted notion of complete analyticity in which the boundary conditions, rather than the shapes of the regions under consideration are being restricted.

1 Introduction

In [MO94a], Section 3, Martinelli and Olivieri solved an important and old problem, by showing that for attractive spin flip systems (subject to some natural and mild technical conditions) the following holds. Exponential decay of the influence from the boundary for the invariant measures of the system restricted to cubic boxes implies exponential convergence to equilibrium for the infinite system, i.e., exponential ergodicity. In the case of spin flip systems which are Glauber dynamics associated to some interaction, the invariant measure for the system restricted to such a box is a finite-volume Gibbs measure, and the hypothesis in the theorem quoted above corresponds to a sort of mixing condition for the associated Gibbs measures. In this sense the result says that good equilibrium mixing properties imply fast approach to equilibrium for the class of Glauber dynamics under consideration. (A precise statement of this result is postponed to Section 3, after the pertinent notation is introduced.) The fact that the result recalled above refers only to the infinite lattice dynamics and not to the corresponding dynamics restricted to subsets of the lattice, with boundary conditions frozen outside, may seem surprising and disappointing at first sight. Nevertheless this feature of the result is not simply a technical weakness of the approach used, but a matter of reality. Even for the nearest neighbor ferromagnetic Ising model in dimension d ≥ 3 it is believed that at low temperatures and under appropriate values of the external magnetic field, “boundary phase transitions” can occur and prevent the dynamics from approaching equilibrium exponentially fast uniformly over all cubic boxes, for carefully chosen boundary conditions (in which some spins are frozen as −1, and others as +1). Following the tradition, we will refer to the part of the phase diagram where this phenomenon is expected to occur as the “Basuev region”. (This name derives from the fact that according to oral tradition, transmitted to one of us by S. Shlosman, Basuev first proposed the existence of such a phenomenon in a Moskow seminar.) A similar phenomenon is rigorously known to occur for some non-ferromagnetic systems known as Czech models (see [Shl86]), and for SOS approximations to the ferromagnetic Ising model (see [DM94] [CM96a] and [CM96b]).

1Partially supported by the American N. S. F. under the grant DMS 9400644

In this paper we address the following question: would it be possible to extend the result above of [MO94a], to include exponential approach to equilibrium for finite subsets of the lattice, uniformly in the size of the subset, provided that only “non-tricky” boundary conditions are considered? Indeed, it is very natural to ask, for instance, whether for the Ising model such a result holds when the boundary condition corresponds to having all spins +1, or all spins −1. We provide a result of this type, which is stated as Theorem 3.2. The mixing condition that is assumed to hold in the hypothesis of our theorem is stronger than the one in the Martinelli-Olivieri theorem, and involves the special type of boundary condition being considered. The good news is that in several cases of interest, one can check that this mixing condition does hold, and hence conclude that exponential convergence to equilibrium, in the sense discussed above, takes place. These successful cases are summarized in Theorem 4.1 and include the Basuev region.

We conclude this introduction with some remarks. A major direction of recent research on the issues of mixing properties of Gibbs measures and fast convergence to equilibrium of the associated Glauber dynamics, has concerned the equivalence of the (very strong) type of mixing condition called Dobrushin-Shlosman complete analyticity (introduced in [DS85]) and exponential convergence uniformly over subsets of the lattice, with arbitrary boundary conditions. The major breakthrough in this direction was obtained in [SZ92]. Such results are very strong and impressive, but come with one drawback. In the way the Dobrushin-Shlosman complete analyticity condition was originally introduced, it referred to uniform properties over all the subsets of the lattice, with arbitrary boundary conditions. Even for the ferromagnetic Ising model in two dimensions, counterexamples to complete analyticity in this original sense can actually be obtained in parts of the phase diagram where the system is supposed to be very well behaved. This is done by taking sequences of subsets of the lattice which have boundaries of sizes comparable to their volume. It was realised in [MO94a] and [MO94b] that by restricting the class of subsets of the lattice to, say, only cubes (actually much larger classes can be considered) the pathologic counterexamples would on one hand disappear, while on the other hand, the relations with rapid convergence to equilibrium would be preserved, uniformly now only over the allowed subsets of the lattice, but still uniformly over all boundary conditions. One way to look at the present paper is as a first attempt to restore the relations between good equilibrium mixing properties and fast convergence of the dynamics uniformly over subsets of the lattice, by restricting the set of allowed boundary conditions, rather than the set of allowed subsets of the lattice. Indeed, the mixing condition (3.3), which appears in the hypothesis of Theorem 3.2 is a natural analogue of the complete analyticity mixing condition (in the ferromagnetic case) restricted to the boundary condition ω. The problem of obtaning stronger results than the one provided in Theorem 3.2 under such a mixing condition, e.g., a bound on the Logarithmic Sobolev constant, uniform on the subsets of the lattice, but restricted to the appropriate boundary condition, remains open. Such a result would imply, in particular, the finiteness of the logarithmic Sobolev constant for the unique Gibbs measure of the infinite ferromagnetic Ising model at low temperatures, for all non-null values of the external magnetic field, including hence the Basuev region. (Naturally, one can also contemplate the possible advantages of introducing a type of mixing in which both, the class of lattice subsets and the boundary conditions are restricted.)

It is natural to remind the reader at this point that in d = 2 the mixing condition in the hypothesis of Theorem 3.1 is actually known to imply the stronger type of mixing (complete analyticity restricted to squares), which, in particular implies exponential convergence to equilibrium uniformly over all squares, with arbitrary boundary conditions (see [MOS94]). Still, it seems to us that even in d = 2 the results in Theorem 3.2 and Theorem 4.1 are new, since they refer to arbitrary subsets of the lattice.

2 Notation and 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^{1}-metric; ∥x∥_{1} = ∑_{i=1}^{d} |x_{i}|. The number of points contained in a set Λ ⊂ Z^{d} is denoted by |Λ| and we write Λ ⊂⊂ Z^{d} when 1 ≤ |Λ| < ∞. The diameter of Λ will be denoted by diam(Λ). For x ∈ Z^{d} and l = 1, 2, ⋯, Q_{l}(x) denotes the cube with the center x and the radius l;

Q_{l}(x) = {y ∈ Z^{d} : max_{i} |x_{i} − y_{i}| ≤ l}.  (2.1)

The configurations. The set of all spin configurations of S = {−1, 1} on Λ ⊂ Z^{d} is denoted by S^{Λ};

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

As usual, S^{Λ} is endowed with the product topology inherited from the discrete topology on S. In S^{Λ}, the following partial order is introduced;

σ ≤ σ′ if σ_{x} ≤ σ′_{x} for all x ∈ Λ.

Clearly, the maximal and the minimum element in this partial order are + and −, which are respectively the configurations with all spins +1 and −1. For Λ ⊂⊂ Z^{d} and (σ, ω) ∈ S^{Z^{d}} ×S^{Z^{d}}, 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 : S^{Λ} → R and x ∈ Λ, we set ∇_{x}f(σ) = f(σ^{x}) − f(σ), where σ^{x} = (σ_{y}^{x})_{y∈Λ} is the configuration obtained from σ by flipping the spin at x;

σ_{y}^{x} = \{ \begin{matrix}−σ_{x} & if y = x, \\ σ_{y} & if y ≠ x. \end{matrix}

For f : S^{Λ} → R, we introduce the notation

Δ_{f} = {x ∈ Λ ; ∇_{x}f is not identically zero.},

∥f∥ = sup_{σ∈S^{Λ}} |f(σ)|,

and |||f||| = ∑_{x∈Λ} ∥∇_{x}f∥.

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

C = {f : S^{Z^{d}} → R ; |Δ_{f}| < ∞ },
and C_{Λ} = {f : S^{Z^{d}} → 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},  (2.2)

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

r(Φ) := sup{diam(X); Φ_{X} ≢ 0} ≤ r_{0}.  (2.3)

∥Φ∥ in (2.2) and r(Φ) in (2.3) 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≠ø} Φ_{X}.  (2.4)

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

μ^{Λ,ω}(\{σ_{Λ}\}) = \frac{exp −H_{Λ}(σ_{Λ} · ω_{Λ^{c}})}{Z^{Λ,ω}},  (2.5)

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

The infinite volume Gibbs state. A Borel probability measure μ is called an infinite volume Gibbs state with respect to the interaction Φ or simply a Gibbs state if it solves the following Dobrushin-Lanford-Ruelle equation;

μ(μ^{Λ,·}f) = μf, ∀Λ ⊂⊂ Z^{d}, ∀f ∈ C.  (2.6)

The set of Gibbs states with respect to Φ will be denoted by Gibbs(Φ).

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 = c_{x}∇_{x}f, where the flip rates c_{x} : S^{Z^{d}} → (0, ∞) are functions on which we assume the following.

(R-1) Boundedness : There exist positive constants \underline{c}(Φ) and \overline{c}(Φ), which depends only on the norm ∥Φ∥ of the interaction such that

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

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

∇_{y}c_{x} ≡ 0 if |x − y| > r_{1}.  (2.7)

(R-3) Detailed Balance Condition:

∇_{x}{c_{x}(σ) exp −H_{{x}}(σ)} ≡ 0, for all x ∈ Z^{d}.  (2.8)

(R-4) Attractiveness: For each x ∈ Z^{d},

σ ↦ c_{x}(σ)|_{σ_{x}=−1} is non-decreasing
and σ ↦ c_{x}(σ)|_{σ_{x}=+1} is non-increasing.

We now define A : C → C by Af = ∑_{x∈Z^{d}} A_{x}f. Then, by general arguments in [Lig, Chapter I], we know that A can be closed on (C(S^{Z^{d}}), ∥ · ∥) and that the closure generates a contraction, strongly continuous semi-group (exp tA)_{t≥0}, which we will refer to as Glauber dynamics . Further, we see from (2.8) that

−μ(fAg) = \frac{1}{2} ∑_{x∈Z^{d}} μ(c_{x}∇_{x}f∇_{x}g), \{f, g\} ⊂ C.  (2.9)

We also need to introduce the Glauber dynamics in a finite set Λ ⊂⊂ G with the boundary condition ω ∈ S^{Z^{d}}. The semi-group will be generated by an operator

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

We then have

−μ^{Λ,ω}(fA^{Λ,ω}g) = \frac{1}{2} ∑_{x∈Λ} μ^{Λ,ω}(c_{x}∇_{x}f∇_{x}g), \{f, g\} ⊂ C.  (2.10)

We set

T_{t}^{Λ,ω} = exp tA^{Λ,ω}, t > 0.

We have defined A^{Λ,ω} and T_{t}^{Λ,ω} as operators acting on C. But, we can also regard them as operators from C_{Λ} into itself.

3 An extension of a result of Martinelli and Olivieri

First we recall the following result:

Theorem 3.1 ([MO94a]) Suppose that there exists {C_{0}, C_{1}} ⊂ (0, ∞) such that

sup_{x∈Z^{d}}\{μ^{Q_{l}(x),+}(σ_{x}) − μ^{Q_{l}(x),−}(σ_{x})\} ≤ C_{0} exp −\frac{l}{C_{1}}  (3.1)

for all l = 1, 2, ⋯ (cf. (2.1)). Then, the set Gibbs(Φ) consists of a unique element μ and there exists {C_{2}, C_{3}} ⊂ (0, ∞) which depends only on d, M_{0}, r_{0}, r_{1}, C_{0}, and C_{1} (cf. (2.2),(2.3) and (2.7)) such that

∥(exp tA)f − μf∥ ≤ C_{2}|||f||| exp −\frac{t}{C_{3}}  (3.2)

for all f ∈ C and all t ≥ 0.

Remark: In [MO94a], the interaction Φ is supposed to be shift-invariant, but their proof works also without shift-invariance.

Next we state the main result in this paper. This result says that a stronger conclusion follows under a stronger type of mixing condition than (3.1) above. In the statement of this stronger mixing condition, a certain boundary condition ω is singled out and in the conclusion of the theorem the same special boundary condition ω is the only one that can be considered. In typical applications of this theorem the constants C_{0} and C_{1}, which appear in its statement, will not depend on Λ.

Theorem 3.2 Suppose that Λ ⊂⊂ Z^{d} and that {C_{0}, C_{1}} ⊂ (0, ∞) are such that for a certain configuration ω

sup_{x∈Λ}\{μ^{Λ∩Q_{l}(x),(+)_{Λ}·ω_{Λ^{c}}}(σ_{x}) − μ^{Λ∩Q_{l}(x),(−)_{Λ}·ω_{Λ^{c}}}(σ_{x})\} ≤ C_{0} exp −\frac{l}{C_{1}}  (3.3)

for all l = 1, 2, . . .. Then there exists {C_{2}, C_{3}} ⊂ (0, ∞) which depend only on d, M_{0}, r_{0}, r_{1}, C_{0}, and C_{1} above such that

∥T_{t}^{Λ,ω}f − μ^{Λ,ω}f∥ ≤ C_{2}|||f||| exp −\frac{t}{C_{3}}  (3.4)

for all f ∈ C_{Λ} and all t ≥ 0.

Proof: We define a new bounded, finite range interaction Φ̂ by setting

Φ̂_{F} (σ) = ∑_{X;X∩Λ=F} Φ_{X}(σ_{Λ} · ω_{Λ^{c}}).

Clearly r(Φ̂) ≤ r(Φ) and ∥Φ̂∥ is dominated by a constant which depends on M_{0} and r_{0}, but not on the choice of Λ. We will denote the corrresponding Hamiltonian and the finite volume Gibbs states respectively by Ĥ_{Y} and μ̂^{Y,η} ( Y ⊂⊂ Z^{d}). Then, Ĥ_{Y} (σ_{Y} · η_{Y^{c}}) can be seen to be equal to H_{Λ∩Y} (σ_{Λ∩Y} · η_{Λ\Y} · ω_{Λ^{c}}). In fact,

−Ĥ_{Y} (σ_{Y} · η_{Y^{c}}) = ∑_{F;F ∩Y ≠Ø} ∑_{X;X∩Λ=F} Φ_{X}(σ_{Λ∩Y} · η_{Λ\Y} · ω_{Λ^{c}})
= ∑_{X;X∩Λ∩Y ≠Ø} Φ_{X}(σ_{Λ∩Y} · η_{Λ\Y} · ω_{Λ^{c}}).

From this, we see that

μ̂^{Y,η} = μ^{Λ∩Y,η_{Λ\Y} ·ω_{Λ^{c}}} ⊗ ρ^{Y \Λ},  (3.5)

which implies that

Gibbs(Φ̂) = {μ^{Λ,ω} ⊗ ρ^{Λ^{c}}},

where ρ^{X} denotes the 1/2-Bernoulli measure on S^{X} for X ⊂ Z^{d}. Also, using (3.3) and (3.5),we can verify the condition (3.1) for the finite volume Gibbs states {μ̂^{Y,η}} with the same constants C_{0} and C_{1} that appear in (3.3).

On the other hand, we define a new generator Â of an attractive Glauber dynamics corresponding to the interaction Φ̂ by

Â = A^{Λ,ω} + ∑_{x∉Λ} ∇_{x}.  (3.6)

Then, it is clear that

(exp tÂ)f = T_{t}^{Λ,ω}f, for f ∈ C_{Λ}.  (3.7)

We now apply Theorem 3.1 to Â, to conclude

∥ (exp tÂ) f − (μ^{Λ,ω} ⊗ ρ^{Λ^{c}})f∥ ≤ C_{2}|||f||| exp −\frac{t}{C_{3}}  (3.8)

for all f ∈ C and for all t ≥ 0. In view of (3.7), (3.4) follows from (3.8).

Q.E.D.

4 Application to the Ising ferromagnet

In this section, we turn our attention to the most classical example of the interaction Φ;

Φ_{X}(σ) = \{ \begin{matrix}βσ_{x}σ_{y} & if X = \{x, y\} and ∥x − y∥_{1} = 1, \\ βhσ_{x} & if X = \{x\}, \end{matrix}  (4.1)

where β > 0 and h ∈ R are fixed parameters. By β_{c} we denote, as usual, the critical inverse temperature. The finite volume Gibbs state μ^{Λ,ω} is defined by (2.5), with a slight extension. Namely, we will take the boundary condition ω from {−1, 0, +1}^{Z^{d}}, instead of S^{Z^{d}}. The major reason for this extension is to include an important special case ω_{x} ≡ 0, i.e., the free boundary condition. One can easily check that Theorem 3.2 applies also to these extended boundary conditions.

Theorem 4.1 Suppose that one of the following holds.

(a) β < β_{c}, ω ∈ {−1, 0, +1}^{Z^{d}} and

inf_{x∈Λ} {h + ∑_{y∉Λ:∥x−y∥_{1}=1} ω_{y}} ≥ 0.  (4.2)

(b) d = 2, β ≥ β_{c}, h > 0, and ω = +.

(c) β is sufficiently large, h > 0, and ω = +.

Then there exist {C_{0}, C_{1}} ⊂ (0, ∞) which is independent of the choice of Λ such that (3.3) holds for any l ≥ 1, and hence there exists {C_{2}, C_{3}} ⊂ (0, ∞) which depends only on d, M_{0}, r_{0}, r_{1}, C_{0}, and C_{1} above such that (3.4) holds for all f ∈ C_{Λ} and t ≥ 0. In case (a), it is possible to choose {C_{0}, C_{1}}, and hence {C_{2}, C_{3}} uniformly in h and ω within a class defined by (4.2). If β > β_{c} in case (b) and (c), it is possible to choose C_{1} uniformly in h > 0.

Proof: We will prove that (3.3) holds in each one of the cases. To handle case (a), the following estimate is available ([Aiz81]); if β < β_{c}, there exists C = C(β, d) > 0 such that

μ_{β,0}(σ_{x}σ_{y}) ≤ C exp −\frac{∥x − y∥_{1}}{C} , for all \{x, y\} ⊂ Z^{d},  (4.3)

where μ_{β,0} denote the infinite volume Gibbs state with h = 0. The proof of (3.3) will be built on an argument used in the proof of [Hig93, Theorem 2], which makes use of several inequalities and of (4.3). (Part (ii) of Theorem 2 in [Hig93] is actually the particular case of (3.3), in which ω = +.) For an inhomogeneous magnetic field {h_{z} ∈ R; z ∈ Λ ∩ Q_{l}(x)}, we define a probability measure ν^{{h_{z}}} on S^{Λ∩Q_{l}(x)} by

ν^{\{h_{z}\}}(\{σ\}) = \frac{exp −H_{\{h_{z}\}}(σ)}{normalization},

where

−H_{{h_{z}}}(σ) = β ∑_{⟨ v,w ⟩} σ_{v}σ_{w} + β ∑_{z∈Λ∩Q_{l}(x)} h_{z}σ_{z},

with ∑_{⟨ v,w ⟩} denoting the summation over all nearest neighbour pairs in the set Λ ∩ Q_{l}(x). We will need the following fact; if inhomogeneous magnetic fields {h_{z}^{±}} and {k_{z}^{±}} are given such that for each z ∈ Λ ∩ Q_{l}(x),

0 ≤ h_{z}^{+} − h_{z}^{−} ≤ k_{z}^{+} − k_{z}^{−}  (4.4)

and

0 ≤ k_{z}^{+} + k_{z}^{−} ≤ h_{z}^{+} + h_{z}^{−},  (4.5)

then

ν^{{h_{z}^{+}}}(σ_{x}) − ν^{{h_{z}^{−}}}(σ_{x}) ≤ ν^{{k_{z}^{+}}}(σ_{x}) − ν^{{k_{z}^{−}}}(σ_{x}).  (4.6)

The proof of the above fact, which is based on the Lebowitz’ inequality for duplicated spins ([Leb74]), can be found in [Hig93]. At this point, we make the following particular choice of {h_{z}^{±}} and {k_{z}^{±}};

h_{z}^{±} = h + ∑_{\begin{matrix}y ∉ Λ ∩ Q_{l}(x) \\ ∥y − z∥_{1} = 1 \end{matrix}} ((±)_{Λ} · ω_{Λ^{c}})_{y}

and

k_{z}^{±} = ∑_{\begin{matrix}y ∉ Λ ∩ Q_{l}(x) \\ ∥y − z∥_{1} = 1 \end{matrix}} ((±)_{Λ} · 0_{Λ^{c}})_{y}.

In this case (4.4) is obvious and (4.5) follows from (4.2). Since ν^{{h_{z}^{±}}} = μ^{Λ∩Q_{l}(x),(±)_{Λ}·ω_{Λ^{c}}} and ν^{{k_{z}^{±}}} = μ^{Λ∩Q_{l}(x),(±)_{Λ}·0_{Λ^{c}}}|_{h=0}, (4.6) implies in this case that

μ^{Λ∩Q_{l}(x),(+)_{Λ}·ω_{Λ^{c}}}(σ_{x}) − μ^{Λ∩Q_{l}(x),(−)_{Λ}·ω_{Λ^{c}}}(σ_{x})
≤ μ^{Λ∩Q_{l}(x),(+)_{Λ}·0_{Λ^{c}}}(σ_{x})|_{h=0} − μ^{Λ∩Q_{l}(x),(−)_{Λ}·0_{Λ^{c}}}(σ_{x})|_{h=0}
= ∫_{−1}^{+1} \frac{d}{dθ}ν^{\{θk_{z}^{+}\}}(σ_{x})dθ.  (4.7)

On the other hand, we have

\frac{d}{dθ}ν^{\{θk_{z}^{+}\}}(σ_{x}) = β ∑_{z∈Λ∩Q_{l}(x)} k_{z}^{+}ν^{\{θk_{z}^{+}\}}(σ_{x}; σ_{z})
≤ 2dβ ∑_{z∈∂_{int}Q_{l}(x)} ν^{{θk_{z}^{+}}}(σ_{x}; σ_{z})
≤ 2dβ ∑_{z∈∂_{int}Q_{l}(x)} μ_{β,0}(σ_{x}σ_{z})
≤ 4d^{2}βl^{d−1}C exp −\frac{l}{C} .  (4.8)

Here, the second inequality comes from an observation that k_{z}^{+} = 0 if z ∉ ∂_{int}Q_{l}(x), while we have used the GHS and GKS inequalities to pass from the second line to the third, to which we can apply (4.3). Plugging (4.8) in (4.7) we obtain (3.3). The constants {C_{0}, C_{1}} obtained from the above proof is obviously uniform in h and ω under consideration.

To prove that (3.3) also holds in cases (b) and (c) of the theorem, we will use some percolation notions. As usual we say that a chain is a sequence of distinct sites x_{1}, …, x_{n}, with the property that for i = 1, …, n−1, ∥x_{i} −x_{i+1}∥_{1} = 1. The sites x_{1} and x_{n} are called the end-points of the chain x_{1}, …, x_{n}. A chain is said to connect two sets if it has one end-point in each set.

Let B be the event that there is a chain of − spins connecting x to Λ ∩ (Q_{l}(x))^{c}. The statement that we want to prove is reduced to an upper estimate on μ^{Λ∩Q_{l}(x),(−)_{Λ}·ω_{Λ^{c}}}(B), via an argument that we present next. We partition B^{c} according to the set of sites in Q_{l}(x) which are connected to Λ ∩ (Q_{l}(x))^{c} by chains of − spins. We will use the notation {F_{α}} to denote this partition. Using a self-explanatory notation for conditional expectations, from the Markov property and the FKG inequalities, we obtain for each α

μ^{Λ∩Q_{l}(x),(−)_{Λ}·ω_{Λ^{c}}}(σ_{x}|F_{α}) ≥ μ^{Λ∩Q_{l}(x),(+)_{Λ}·ω_{Λ^{c}}}(σ_{x})

Therefore

μ^{Λ∩Q_{l}(x),(−)_{Λ}·ω_{Λ^{c}}}(σ_{x})

= ∑_{α} μ^{Λ∩Q_{l}(x),(−)_{Λ}·ω_{Λ^{c}}}(σ_{x}|F_{α}) μ^{Λ∩Q_{l}(x),(−)_{Λ}·ω_{Λ^{c}}}(F_{α})

+ μ^{Λ∩Q_{l}(x),(−)_{Λ}·ω_{Λ^{c}}}(σ_{x}|B) μ^{Λ∩Q_{l}(x),(−)_{Λ}·ω_{Λ^{c}}}(B) ≥ μ^{Λ∩Q_{l}(x),(+)_{Λ}·ω_{Λ^{c}}}(σ_{x}) μ^{Λ∩Q_{l}(x),(−)_{Λ}·ω_{Λ^{c}}}(B^{c}) − μ^{Λ∩Q_{l}(x),(−)_{Λ}·ω_{Λ^{c}}}(B).

Hence

μ^{Λ∩Q_{l}(x),(+)_{Λ}·ω_{Λ^{c}}}(σ_{x}) − μ^{Λ∩Q_{l}(x),(−)_{Λ}·ω_{Λ^{c}}}(σ_{x})
≤ μ^{Λ∩Q_{l}(x),(−)_{Λ}·ω_{Λ^{c}}}(B) + μ^{Λ∩Q_{l}(x),(+)_{Λ}·ω_{Λ^{c}}}(σ_{x}) μ^{Λ∩Q_{l}(x),(−)_{Λ}·ω_{Λ^{c}}}(B)
≤ 2μ^{Λ∩Q_{l}(x),(−)_{Λ}·ω_{Λ^{c}}}(B).  (4.9)

But since ω = +, we have from the FKG inequalities,

μ^{Λ∩Q_{l}(x),(−)_{Λ}·ω_{Λ^{c}}}(B) ≤ μ^{Q_{l}(x),−}(B).  (4.10)

The inequality (3.3) now follows from (4.9) and (4.10), since it is known that under the conditions in parts (b) or (c) of the theorem that we are proving, the probability in the right hand side of (4.10) decays exponentially with l. For the two-dimensional case this result can be found in [SS95] (combine inequalities (4.24) and (4.27) in that paper). For higher dimensions it is enough to observe that the renormalization argument used in [SS95] to prove this result in two dimensions works in the same way in higher dimensions, if one uses as input Theorem 1 in [Mar87], in the place of part (b) of Corollary 1 in [SS95]. (For an alternative proof of Martirosyan’s result the reader can consult [Sch94]; see Theorem 3 there.) If β > β_{c}, the the above argument yields a uniform choice of C_{1} in h > 0.

Q.E.D.

Remarks: Part (a) of the Theorem 4.1 can be generalized. As the proof we presented suggests, the nearest neighbor interaction can be replaced by a more general translation-invariant finiterange ferromagnetic two-point interaction.

Part (a) of the Theorem 4.1 includes the case in which h = 0 and the boundary condition is free. The result that the system approaches equilibrium exponentially fast in this case for all β < β_{c}, uniformly in the size of the system can be contrasted with the result that for β > β_{c} the rate of exponential convergence to equilibrium for the system in a square box with free boundary conditions vanishes as an exponential of the side-length of the box, with the rate of this exponential decay being given by the surface tension (see [Mar94] and [CGMS]).

Regarding parts (b) and (c) of Theorem 4.1, we expect the same statements also to be true with free boundary conditions, but unfortunately the proof given above does not work in this case.

Acknowledgement: N.Y. thanks Y. Higuchi for discussions. The authors also thank two anonymous referees for their comments.

References

[Aiz81] Aizenman, M. : Rigorous studies of critical behavior II, In : Fritz, J., Jaffe, A., Szasz, D. (eds.) Statistical Physics and dynamical systems. Proceedings, Köseg 1984, pp. 453–481. Boston, Basel, Stuttgart: Birkhäuser (1985)

[CGMS] Cesi, F., Guadagni, G., Martinelli, F. and Schonmann, R. H.: On the 2D dynamical Ising model in the phase coexistence region near the critical point. J. Stat. Phys. (to appear).

[CM96a] Cesi, F. and Martinelli, F.: On the layering transition of an SOS surface interacting with a wall. I. Equilibrium results, J. Stat. Phys. 82, 823–913 (1996).

[CM96b] Cesi, F. and Martinelli, F.: On the layering transition of an SOS surface interacting with a wall. II. The Glauber dynamics, Commun. Math. Phys. (to appear).

[DM94] Dinaburg, E. I. and Mazel, A. E.: Layering transition in SOS model with external magnetic field. J. Stat. Phys. 74, 533–563 (1994).

[DS85] Dobrushin, R. and Shlosman, S.: Completely analytical Gibbs fields, in “Statistical Physics and Dynamical Systems”, Fritz, J., Jaffe, A., Szász, D., eds.. Birkhauser (1985), pp.371–403.

[Hig93] Higuchi,Y.: Coexistence of infinite (∗)-clusters II:– Ising percolation in two dimension, Prob. Th. Rel. Fields 97, 1–33 (1993).

[HY95] Higuchi,Y. and Yoshida, N.: Slow relaxation of stochastic Ising models with random and non-random boundary conditions, To appear in Proceeding of Taniguchi Symposium, (1995).

[Leb74] Lebowitz, J. L. : GHS and other inequalities, Commun. Math. Phys. 35, pp. 87–92 (1974).

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

[Mar94] Martinelli, F.: On the two dimensional dynamical Ising model in the phase coexistence region, J. Stat. Phys. 76, 1179–1246 (1994).

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

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

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

[Mar87] Martirosyan, D. G.: Theorems on strips in the classical Ising ferromagnetic model, Sov. J. Comtemp. Math. 22, 59–83 (1987).

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

[Sch94] Schonmann, R. H.: Slow droplet-driven relaxation of stochastic Ising models in the vicinity of the phase coexistence region, Commun. Math. Phys. 161, 1–49 (1994).

[Shl86] Shlosman, S. B.: Uniqueness and half space non-uniqueness of Gibbs states in Czech models, Theor. Math. Phys. 66, 284–293 (1986).

[SZ92] Stroock, D. W. and Zegarlinski, B.: The logarithmic Sobolev inequality for discrete spin systems on a lattice, Commun. Math. Phys. 149, 175–193 (1992).