EXPONENTIAL CONVERGENCE TO EQUILIBRIUM OF FINITE VOLUME GLAUBER DYNAMICS NEAR THE BORDER OF THE ONE PHASE REGION

Nobuo YOSHIDA (Kyoto)

ABSTRACT

We consider Glauber dynamics on a finite set in the d-dimensional lattice (d ≥ 2), which is associated with basic Ising model. We are particularly interested in the case that the thermodynamic parameter (β, h) is sitting in a place in the one phase region, which is so near to the multi phase region that the Dobrushin-Shlosman mixing condition is no longer available. We prove that for some “nice” boundary conditions, the convergence of the Glauber dynamics in the uniform norm is exponentially fast, uniformly over the size of underlying subset in the lattice.

1 Introduction

We begin by reviewing the standard setup of the model. We then introduce some more specific backgrouds of our results.

The lattice. We will work on the d-dimensional integer lattice Z^{d} = {x = (x^{i})_{i=1}^{d} : x^{i} ∈ Z} with d ≥ 2, 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 interior and exterior boundaries of a set Λ ⊂ Z^{d} will be denoted respectively by

∂_{int}Λ = {x ∈ Λ; ∥x − y∥_{1} = 1 for some y ∉ Λ },
∂_{ext}Λ = {y ∉ Λ; ∥x − y∥_{1} = 1 for some x ∈ Λ }.

A set Λ ⊂⊂ Z^{d} is called a box when Λ = Z^{d} ∩ ∏_{j=1}^{d}[a^{j}, b^{j}) for some integers a^{j} < b^{j} (j = 1, ⋯, n). A box Λ, expressed as above is called a cube when all side-lengths b^{j}−a^{j} (j = 1, ⋯, n) are identical. For x ∈ Z^{d} and l = 1, 2, ⋯, a cube with the center x and the side-lengh 2l is denoted by Q_{x}(l); Q_{x}(l) = Z^{d} ∩ ∏_{i=1}^{d}[x^{i} − l, x^{i} + l). Q_{0}(l) will be simply denoted by Q(l).

The configurations and the Gibbs states. Configuration spaces are defined as follows;

S^{Λ} = {σ = (σ_{x})_{x∈Λ} ; σ_{x} ∈ {−1, +1}}, Λ ⊂⊂ Z^{d},
Ω = {ω = (ω_{x})_{x∈Z^{d}} ; ω_{x} ∈ {−1, 0, +1}}.

We will refer an element ω of Ω as a boundary condition. Three boundary conditions are specially relevant:(±) and (0), which are defined respectively by (±)_{x} ≡ ±1 and (0)_{x} ≡ 0. (±) is called pure (±)-boundary condition and (0) is called the free-boundary condition. They will often be abbreviated respectively by ± and 0 especially when they appear as superscript. For Λ ⊂⊂ Z^{d} and (σ, ω) ∈ S^{Λ} × Ω, 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}

The set of all real functions on S^{Λ} is denoted by C_{Λ}. For Λ ⊂⊂ Z^{d} and ω ∈ Ω, Hamiltonian H^{Λ,ω} ∈ C_{Λ} is defined by

H^{Λ,ω}(σ) = − ∑_{{\{x,y\}⊂Λ \atop ∥x−y∥_{1}=1}} βσ_{x}σ_{y} − ∑_{{x∈Λ,y∉Λ \atop ∥x−y∥_{1}=1}} βσ_{x}ω_{y} − ∑_{x∈Λ} βhσ_{x},  (1.1)

where β > 0 and h ≥ 0. Gibbs state on Λ with the boundary condition ω ∈ Ω is defined as a probability distribution μ^{Λ,ω} = μ_{β,h}^{Λ,ω} on S^{Λ}, in which the probability of each configuration σ ∈ S^{Λ} is given by

μ^{Λ,ω}(\{σ\}) = \frac{1}{Z^{Λ,ω}} exp −H^{Λ,ω}(σ),

where Z^{Λ,ω} is the normalizing constant. It is well known that the limit m^{±} = lim_{Λ↑Z^{d}} μ^{Λ,±}(σ_{0}) 1 exists and that

(i) m^{+} = m^{−} > 0 if h > 0,

(ii) For h = 0, there exists a critical inverse temperature β_{c} ∈ (0, ∞) such that m^{+} = m^{−} = 0 if β < β_{c} and m^{+} > 0 > m^{−} if β_{c} < β.

Therefore, the (β, h)-plane is divided into two parts according as m^{+} = m^{−} or not, up to the critical point (β_{c}, 0);

{(β, h); β < β_{c} or h > 0}, {(β, h); β_{c} < β & h = 0}.  (1.2)

The former is called one phase region and the latter is called multi phase region.

The Glauber dynamics. For f ∈ C_{Λ}, we introduce a difference operator

∇_{x}f(σ) = f(σ^{x}) − f(σ), x ∈ Λ,

where σ^{x} is defined by;

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

We also introduce a uniform norm; ∥f∥ = sup_{σ} |f(σ)| and a semi-norm; |||f||| = ∑_{x∈Λ} ∥∇_{x}f∥. Now, fix Λ ⊂⊂ Z^{d} and ω ∈ Ω. We define a linear operator A^{Λ,ω} : C_{Λ} → C_{Λ} by

A^{Λ,ω}f(σ) = ∑_{x∈Λ} c_{x}^{Λ,ω}(σ)∇_{x}f(σ), f ∈ C_{Λ},  (1.3)

where

c_{x}^{Λ,ω}(σ) = exp −∇_{x}H^{Λ,ω}(σ).  (1.4)

It can easily be seen that

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

1For a measure μ and a function f, we let μ(f) stand for ∫ fdμ.

which shows that A^{Λ,ω} is μ^{Λ,ω}-symmetric and non-positive definite. Finally, we define

gap_{β,h}^{Λ,ω} = inf {λ > 0; Ker (λ − (−A^{Λ,ω})) ≠ {0}} > 0.  (1.5)

which is called the spectral gap. gap_{β,h}^{Λ,ω} can be characterized as the largest number that satisfies

∥T_{t}^{Λ,ω}f − μ^{Λ,ω}f∥_{L^{2}(μ^{Λ,ω})} ≤ ∥f − μ^{Λ,ω}f∥_{L^{2}(μ^{Λ,ω})} exp (−tgap_{β,h}^{Λ,ω}) ,  (1.6)

for all f ∈ C_{Λ}, where

T_{t}^{Λ,ω} = exp tA^{Λ,ω}, t > 0.  (1.7)

Some backgrounds of our result. For given β > 0, h ∈ R and ω ∈ Ω, we are interested in whether the following uniform exponential convergence properties are true;

UEC(ω): There exist constants C_{i} = C_{i}(d, β, h, ω) ∈ (0, ∞) (i = 1, 2) such that for all Λ ⊂⊂ Z^{d}, f ∈ C_{Λ} and t > 0

∥T_{t}^{Λ,ω}f − μ^{Λ,ω}f∥ ≤ C_{1}|||f||| exp −\frac{t}{C_{2}} .  (1.8)

UEC(ω;□): There exist constants C_{i} = C_{i}(d, β, h, ω) ∈ (0, ∞) (i = 1, 2) such that (1.8) holds for all cube Λ, f ∈ C_{Λ} and t > 0.

Here we summarize some observations, known results and conjectures;

(i) The property UEC(ω;□), and hence UEC(ω)cannot be true for any ω ∈ Ω if (β, h) is in the multi phase region. In fact, it is not difficult to see that UEC(ω)is strong enough to guarantee m^{+} = m^{−}.

(ii) UEC(ω)and UEC(ω;□)are true for all ω ∈ {−1, +1}^{Z^{d}}, whenever a mixing condition in the sense of the Dobrushin-Shlosman is valid ( [10, 11], See also [16]). For example, UEC(ω)holds if (β, h) is sufficiently deep inside the one phase region([6]) and UEC(ω;□)holds if d = 2 and (β, h) is in the one phase region( [14]).

(iii) It is conjectured that if d ≥ 3, β is sufficiently large and h is a certain small positive number depending on β, there is ω ∈ Ω which violate UEC(ω;□). The part of the (β, h)-plane reffered to above is often called “Basuev region”. Though the conjecture is not vindicated yet, there are some results on related models which suggest the existence of such “dangerous” boundary condition ω ∈ Ω ([3, 4, 5]).

Motivated by the conjecture mentioned in (iii) above, an attempt has been initiated in [15, 18] to find out a class of “safe” boundary conditions when (β, h) is sitting in the Basuev region. In this report, we review results in these two papers. A technical constraint in working in the region where the Dobrushin-Shlosman mixing condition is no longer available is that we have to abandon the use of logarithmic Sobolev inequality, which is in a very powerful analytic tool to establish the exponential convergence property( [11, 16]). We instead use the more specific structures of the ferromagnetic Ising model, especially attractivity and Ising percolation.

2 Results

The main results in this report can be summarized as follows

Theorem 2.1 ([15, Theorem 4.1],[18, Theorem 2.1])

(a) If β < β_{c}, h ≥ 0 and ω ≥ 0 (i.e., ω_{x} ≥ 0, ∀x ∈ Z^{d}), then UEC(ω)holds with constants C_{i} = C_{i}(β, d) ∈ (0, ∞) (i = 1, 2).

(b) If β is sufficiently large, h > 0, and ω ≥ 0, then UEC(ω;□)holds with constants C_{i} = C_{i}(β, d, h) ∈ (0, ∞) (i = 1, 2).

(c) If β is sufficiently large, then

0 < inf{gap_{β,h}^{Λ,ω} ; Λ is a cube, ω ≥ 0}, for any h > 0.  (2.1)

Furthermore, in the case of the free-boundary condition, there exist C_{3} = C_{3}(d, β) ∈ (0, ∞) and C_{4} = C_{4}(d) ∈ (0, ∞) such that for any h ∈ (0, 1)

0 < inf\{gap_{β,h}^{Λ,0} ; Λ is a cube \} ≤ C_{3} exp −\frac{β}{C_{4}h^{d−1}} .  (2.2)

Remark 2.1 Part(a) and part(b) for the pure (+)-boundary condition have been obtained in [15, Theorem 4.1]. For the pure (+)-boundary condition, stronger result is proved there than it stands here in part(b); Not only UEC(ω;□)but UEC(ω)holds. Also, if d = 2, UEC(ω)holds for all β ≥ β_{c}. The extension of part(b) to the general case and the upper bound of the spectral gap in part(c) are progresses made in [18]. The uniform positivity of the spectral gap in (2.2) contrasts with [17, Proposition 2.4], which says that for sufficiently large β and h = 0, gap_{β,0}^{Q(l),0} decays exponentially in l^{d−1}. On the other hand, the right-hand-side estimate in (2.2) is a manifestation of the phase-coexistence at h = 0. This contrasts with part(a) of the theorem; it implies that for β < β_{c}, inf_{Λ⊂⊂Z^{d}} gap_{β,h}^{Λ,0} stays positive even for h = 0.

UEC(ω)and UEC(ω;□)in Theorem 2.1 are obtained by putting Proposition 2.2 and Proposition 2.3 together. The first one is a reiteration of [15, Theorem 3.1];

Proposition 2.2 Suppose that Λ ⊂⊂ Z^{d} and that {B_{1}, B_{2}} ⊂ (0, ∞) are such that for a certain boundary condition ω ∈ Ω

sup_{x∈Λ} \{μ^{Λ∩Q_{x}(l),(+)_{Λ}·ω_{Λ^{c}}}(σ_{x}) − μ^{Λ∩Q_{x}(l),(−)_{Λ}·ω_{Λ^{c}}}(σ_{x})\} ≤ B_{1} exp −\frac{l}{B_{2}} , l = 1, 2, ….  (2.3)

Then there exists {C_{1}, C_{2}} ⊂ (0, ∞) which depend only on d, β, h, B_{1} and B_{2} above such that (1.8) holds for all f ∈ C_{Λ} and all t ≥ 0.

By Proposition 2.2, the proof of UEC(ω)and UEC(ω;□)in Theorem 2.1 comes down to checking the mixing condition (2.3), which we present as a proposition.

Proposition 2.3 (a) For any β < β_{c}, there exist constants B_{i} = B_{i}(d, β) ∈ (0, ∞) (i = 1, 2) such that (2.3) holds for all h ≥ 0 and ω ≥ 0 and Λ ⊂⊂ Z^{d}.

(b) There exists β_{0} = β_{0}(d) ∈ (0, ∞) for which the following holds. For any β ≥ β_{0} and h ∈ (0, ∞), there exist constants B_{1} = B_{1}(d, β, h) ∈ (0, ∞) and B_{2} = B_{2}(d, β) ∈ (0, ∞) such that (2.3) holds for all β ≥ β_{0}, ω ≥ 0 and a cube Λ.

The proof of Proposition 2.3 will be given in section 3.

For the part (c) of Theorem 2.1, the uniform positivity of the spectral gap in both (2.1) and (2.2) is the consequence of the part (b). The right-hand-side estimate of (2.2) comes from the following

Proposition 2.4 There exists β_{0} = β_{0}(d) ∈ (0, ∞) and C_{1} = C_{1}(d) ∈ (0, ∞) for which the following holds. For any β ≥ β_{0} and B ≥ 1 there exist C_{2} = C_{2}(d, β, B) ∈ (0, ∞) such that for any h ∈ (0, 1),

gap_{β,h}^{Q(B/h),0} ≤ C_{2} exp −\frac{β}{C_{1}h^{d−1}} ,  (2.4)

Remark 2.2 It is not difficult to see that

gap_{β,h}^{Q(B/h),0} ≥ \frac{1}{C(d, β, B)} exp −\frac{C(d, β, B)}{h^{d−1}}

for some C(d, β, B) ∈ (0, ∞) ([12, Theorem 5], for example). Thus, the bound (2.4) is sharp up to constants. Similar estimate for (−)-boundary condition can be found in [13, Theorem 4]

The proof of Proposition 2.4 can be found in [18, section 5].

3 Proof of Proposition 2.3

In this section, we give a proof of Proposition 2.3. To handle case (a), the following estimate is available ([1, 2]); 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},  (3.1)

where μ_{β,0} denotes the infinite volume Gibbs state with h = 0. The proof of (2.3) will be built on an argument used in the proof of [8, Theorem 2], which makes use of several inequalities and of (3.1). (Part (ii) of Theorem 2 in [8] is actually the particular case of (2.3), in which ω = +.) For an inhomogeneous magnetic field {h_{z} ∈ R; z ∈ Λ∩Q_{x}(l)}, we define a probability measure ν^{{h_{z}}} on S^{Λ∩Q_{x}(l)} 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_{x}(l)} h_{z}σ_{z},

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

0 ≤ h_{z}^{+} − h_{z}^{−} ≤ k_{z}^{+} − k_{z}^{−}  (3.2)

and

0 ≤ k_{z}^{+} + k_{z}^{−} ≤ h_{z}^{+} + h_{z}^{−},  (3.3)

then

ν^{{h_{z}^{+}}}(σ_{x}) − ν^{{h_{z}^{−}}}(σ_{x}) ≤ ν^{{k_{z}^{+}}}(σ_{x}) − ν^{{k_{z}^{−}}}(σ_{x}).  (3.4)

The proof of the above fact, which is based on the Lebowitz’ inequality for duplicated spins ([9]), can be found in [8]. At this point, we make the following particular choice of {h_{z}^{±}} and {k_{z}^{±}};

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

and

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

In this case (3.2) is obvious and (3.3) follows from h ≥ 0 and ω ≥ 0. Since ν^{{h_{z}^{±}}} = μ^{Λ∩Q_{x}(l),(±)_{Λ}·ω_{Λ^{c}}} and ν^{{k_{z}^{±}}} = μ^{Λ∩Q_{x}(l),(±)_{Λ}·0_{Λ^{c}}}|_{h=0}, (3.4) implies in this case that

μ^{Λ∩Q_{x}(l),(+)_{Λ}·ω_{Λ^{c}}}(σ_{x}) − μ^{Λ∩Q_{x}(l),(−)_{Λ}·ω_{Λ^{c}}}(σ_{x})
≤ μ^{Λ∩Q_{x}(l),(+)_{Λ}·0_{Λ^{c}}}(σ_{x})|_{h=0} − μ^{Λ∩Q_{x}(l),(−)_{Λ}·0_{Λ^{c}}}(σ_{x})|_{h=0}
= ∫_{−1}^{+1} \frac{d}{dθ}ν^{\{θk_{z}^{+}\}}(σ_{x})dθ.  (3.5)

On the other hand, we have

\frac{d}{dθ}ν^{\{θk_{z}^{+}\}}(σ_{x}) = β ∑_{z∈Λ∩Q_{x}(l)} k_{z}^{+}ν^{\{θk_{z}^{+}\}}(σ_{x}; σ_{z})
≤ 2dβ ∑_{z∈∂_{int}Q_{x}(l)} ν^{{θk_{z}^{+}}}(σ_{x}; σ_{z})
≤ 2dβ ∑_{z∈∂_{int}Q_{x}(l)} μ_{β,0}(σ_{x}σ_{z})
≤ 4d^{2}βl^{d−1}C exp −\frac{l}{C} .  (3.6)

Here, the second inequality comes from an observation that k_{z}^{+} = 0 if z ∉ ∂_{int}Q_{x}(l), while we have used the GHS and GKS inequalities to pass from the second line to the third, to which we can apply (3.1). Plugging (3.6) in (3.5) we obtain (2.3). The constants {B_{1}, B_{2}} obtained from the above proof is obviously uniform in h and ω under consideration.

To prove case(b) of Proposition 2.3, we need Proposition 3.1 below, which is an extension of [12, Theorem3]. To state Proposition 3.1, we need to introduce some more notations. We take E ⊂ {±1, ⋯, ±d} and suppose that

E ∩ {−j, j} ≠ ϕ, ∀j = 1, ⋯, d.  (3.7)

We define a box Q(l; E) (l = 1, 2, ⋯) by

Q(l; E) = ∏_{j=1}^{d} Q^{(j)}(l),  (3.8)

where

Q^{(j)}(l) = \{ \begin{matrix}Z ∩ [−l, l) & if ± ∈ E, \\ Z ∩ [0, l) & if −j ∉ E and j ∈ E, \\ Z ∩ [−l, 0) & if −j ∈ E and j ∉ E. \end{matrix}

Let δ_{j} = (δ_{jk})_{k=1}^{d} ∈ Z^{d} and δ_{−j} = −δ_{j} (j = 1, 2, ⋯, d). For a box Λ ⊂ Z^{d}, ∂_{ext}Λ is the disjoint union of ∂_{ext}^{(j)}Λ for j = ±1, ⋯ ± d, where

∂_{ext}^{(j)}Λ = {y ∈ ∂_{ext}Λ ; y − δ_{j} ∈ Λ}.

In this case, we let (±; E) denote a configuration on ∂_{ext}Λ which is ≡ ±1 on ∪_{j∈E}∂_{ext}^{(j)}Λ and ≡ 0 on ∪_{j∉E}∂_{ext}^{(j)}Λ. Finally for A ⊂ Λ ⊂ Z^{d}, σ ∈ S^{Λ} and ε = + or −, (ε)-path in A at a configuration σ is a sequence γ = (x_{i} ∈ A)_{i=0}^{n} such that ∥x_{i} − x_{i−1}∥_{1} = 1 for 1 ≤ i ≤ n and σ_{x} ≡ ε1 for x ∈ γ.

Proposition 3.1 For each B ∈ (d, ∞) and A ∈ (0, B), there exists β_{0} = β_{0}(d, A, B) ∈ (0, ∞) such that for β ≥ β_{0},

\overline{lim_{h↘0}}h log μ_{β,h}^{Q(\frac{B}{h};E),(−;E)} (B (\frac{A}{h}; E)) < 0,  (3.9)

where B(\frac{A}{h}; E) stands for an event that there is a (−)-path in Q(\frac{B}{h}; E) which connects Q(\frac{A}{h}; E) to a site adjacent to ∪_{j∈E}∂_{ext}^{(j)}Q(\frac{B}{h}; E).

Remark 3.1 In the the case of E = \{±1, ⋯, ±d\}, in which the smaller box Q(\frac{A}{h}; E) goes far away from ∂_{ext}Q(\frac{B}{h}; E) as h ↘ 0, Proposition 3.1 is what can be read off directly from the proof of [12, Theorem 3]. We also need to consider the case of E ≠ {±1, ⋯, ±d}, in which the situation is slightly different, since as h ↘ 0, the smaller box Q(\frac{A}{h}; E) goes far away from ∪_{j∈E}∂_{ext}^{(j)}Q(\frac{B}{h}; E) where (−)-boundary condition is put, but it stays adjacent to ∪_{j∉E}∂_{ext}^{(j)}Q(\frac{B}{h}; E) where free-boundary condition is put. The upshot is that it is possible to modify the argument in the proof of [12, Theorem 3] to cover the general case. The proof of Proposition 3.1 is sketched in [18, section 4]

Now we turn to the proof of case(b), which will be built on an argument used in [14] to pass from Corollary 1 to Corollary 2 in that paper.

Let B be the event that there is a path of − spins connecting x to Λ ∩ (Q_{x}(l))^{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 the following inequality;

μ^{Λ∩Q_{x}(l),(+)_{Λ}·ω_{Λ^{c}}}(σ_{x}) − μ^{Λ∩Q_{x}(l),(−)_{Λ}·ω_{Λ^{c}}}(σ_{x}) ≤ 2μ^{Λ∩Q_{x}(l),(−)_{Λ}·ω_{Λ^{c}}}(B).  (3.10)

The proof of this inequality can be read quickly in the proof of [15, Theorem 4.1, part(b)]. We will prove that the probability appearing in (3.10) decays exponentially in l. We take a large enough integer m = m(d, β, h_{0}) ≥ 1/h_{0} which will be specified later in the proof. We may and will assume that l ≥ (d + 4)m. By the FKG inequality, we have

μ^{Λ∩Q_{x}(l),(−)_{Λ}·ω_{Λ^{c}}}(B) = μ_{β,h}^{Λ∩Q_{x}(l),(−)_{Λ}·ω_{Λ^{c}}}(B)
≤ μ_{β,1/m}^{Λ∩Q_{x}(l),(−)_{Λ}·(0)_{Λ^{c}}}(B).  (3.11)

Let Λ ∩ Q_{x}(l) = Z^{d} ∩ ∏_{j=1}^{d}[a^{j}, b^{j}), where x ∈ Λ. Note that b^{j} − a^{j} ≥ l for all j. For each j = 1, ⋯, d, take a partition

a^{j} = c_{0}^{j} < c_{1}^{j} < ⋯ < c_{n(j)−1}^{j} < c_{n(j)}^{j} = b^{j}

for some n(j) such that

\begin{matrix}c_{k}^{j} − c_{k−1}^{j} = 2m, & if k = 1 or n(j), \\ m ≤ c_{k}^{j} − c_{k−1}^{j} < 2m & if 1 < k < n(j). \end{matrix}

This procedure induces a partition of Λ ∩ Q_{x}(l) into smaller boxes {Λ(k) ; k ∈ K}, where

K = {k = (k^{j})_{j=1}^{d} ∈ Z^{d}; 1 ≤ k_{j} ≤ n(j)},

Λ(k) = ∏_{j=1}^{d}[c_{k^{j}−1}^{j}, c_{k^{j}}^{j}).

For each z ∈ Λ∩Q_{x}(l), define E(z) = {−2d ≤ j ≤ 2d; z+δ_{j} ∈ Λ∩Q_{x}(l)}. For z ∉ ∂_{int}(Λ∩Q_{x}(l)) E(z) = {±1, ⋯, ±d}. For z ∈ ∂_{int}(Λ ∩ Q_{x}(l)), E(z) is a collection of “ inward” directions to Λ ∩ Q_{x}(l). Note that E(z) ∩ {−j, +j} ≠ ϕ for any z ∈ Λ ∩ Q_{x}(l). Furthermore, we can find z_{k} ∈ Λ ∩ Q_{x}(l) for each k ∈ K such that

Λ(k) ⊂ z_{k} + Q(dm, E(z_{k})) ⊂ z_{k} + Q((d + 1)m + 1, E(z_{k})) ⊂ Λ ∩ Q_{x}(l).

If k ∈ ∂_{int}K, we have to choose z_{k} in ∂_{int}(Λ ∩ Q_{x}(l)). We set

Λ_{1}(k) = z_{k} + Q(dm, E(z_{k})),
Λ_{2}(k) = z_{k} + Q((d + 1)m, E(z_{k})),
Λ_{3}(k) = z_{k} + Q((d + 1)m + 1, E(z_{k})).

For each k ∈ K, we donote by B(k) an event that there is a (−)-path in Λ_{2}(k) which connects Λ_{1}(k) to a site adjacent to ∪_{j∈E(z_{k})}∂_{ext}^{(j)}Λ_{2}(k). Let k(x) ∈ K be a unique element in K such that x ∈ Λ(k) and K(x) = \{k ∈ K ; Λ_{3}(k) ∩ Q_{x}(\frac{l}{2}) ≠ ϕ\} . We then observe that if B occurs, there is a path γ in K with the following properties;

(i) γ connects k(x) to a point in ∂_{int}K and

(ii) B(k) occurs for every k ∈ γ.

By (i), the path γ alluded above has the length at least \frac{l}{2(d+1)m}. Therefore, denoting by C_{s}(x) the set of all paths with k(x) as an end point and with the length s, we have

μ_{β,1/m}^{Λ∩Q_{x}(l),(−)_{Λ}·(0)_{Λ^{c}}}(B) ≤ ∑_{s≥ \frac{l}{2(d+1)m}} ∑_{γ∈C_{s}(x)} μ_{β,1/m}^{Λ∩Q_{x}(l),(−)_{Λ}·(0)_{Λ^{c}}} ( ⋂_{k∈γ} B(k)) .  (3.12)

Note that there is C_{1} = C_{1}(d) ∈ (0, ∞) such that

max_{k∈K} ♯{k′ ∈ K ; Λ_{3}(k) ∩ Λ_{3}(k′)} ≤ C_{1}.

Therefore, for each γ ∈ C_{s}(x), one can pick γ_{0} ⊂ γ in such a way that |γ_{0}| ≥ \frac{s}{C_{1}} and that {Λ_{3}(k) ; k ∈ γ_{0}} are disjoint. Thus, we have by the Markov property, FKG inequality and the shift-invariance that

μ_{β,1/m}^{Λ∩Q_{x}(l),(−)_{Λ}·(0)_{Λ^{c}}} ( ⋂_{k∈γ} B(k)) ≤ ∏_{k∈γ_{0}} μ_{β,1/m}^{Λ_{2}(k),(−;E(z_{k}))} (B(k))

= ∏_{k∈γ_{0}} μ_{β,1/m}^{Q((d+1)m;E(z_{k})),(−;E(z_{k}))} (B(dm; E(z_{k}))) .  (3.13)

Here, we have used the notations introduced in the paragraph which precedes Proposition 3.1. By Proposition 3.1, each factor in (3.13) is bounded from above by exp (C_{2} − \frac{m}{C_{2}} ) with some C_{2} = C_{2}(d, β) ∈ (0, ∞). We thus conclude that

μ_{β,1/m}^{Λ∩Q_{x}(l),(−)_{Λ}·(0)_{Λ^{c}}} ( ⋂_{k∈γ} B(k)) ≤ exp \{\frac{s}{C_{1}} (C_{2} − \frac{m}{C_{2}} )\} .  (3.14)

Combining (3.14) and that ♯C_{s}(x) ≤ (2d)^{s}, we now arrive at an estimate;

∑_{γ∈C_{s}(x)} μ_{β,1/m}^{Λ∩Q_{x}(l),(−)_{Λ}·(0)_{Λ^{c}}} ( ⋂_{k∈γ} B(k)) ≤ exp (\frac{s}{C_{3}} (C_{4} − m))  (3.15)

with some C_{i} = C_{i}(d, β) ∈ (0, ∞), (i = 3, 4). At this point, we assume that m > 2C_{4} so that we have from (3.12) and (3.15) that

μ_{β,1/m}^{Λ∩Q_{x}(l),(−)_{Λ}·(0)_{Λ^{c}}} (B) ≤ ∑_{s≥ \frac{l}{2(d+1)m}} exp −\frac{ms}{2C_{3}}

≤ C_{5} exp −\frac{l}{C_{6}} ,

with C_{5} = C_{5}(d, β, m) ∈ (0, ∞) and C_{6} = C_{6}(d, β) ∈ (0, ∞). This, in conjunction with (3.10) and (3.11), completes the proof of Proposition 2.3.

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