SLOW RELAXATION OF 2-D STOCHASTIC ISING MODELS

WITH RANDOM AND NON-RANDOM BOUNDARY CONDITIONS

YASUNARI HIGUCHI

Department of Mathematics, Faculty of Science, Kobe University, Rokko, Kobe 657, Japan, e-mail:higuchi@math.s.kobe-u.ac.jp

NOBUO YOSHIDA

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

We consider finite volume stochastic Ising models corresponding to the usual Ising ferromagnet at sufficiently low temperature, with zero external magnetic field. We show that for a certain class of boundary conditions in which neither (+) nor (–) predominates the other, the spectral gap on a square shrinks exponentially fast in the side-length. We also discuss some consequences of this estimate, including an application to random (1/2-Bernoulli, for example) boundary conditions.

1 Introduction

We consider a finite volume stochastic Ising model with zero-external magnetic field on a cube Λ(l) = (−\frac{l}{2}, −\frac{l}{2}]^{d} ∩ Z^{d}. At sufficiently high temperature, where the Dobrushin-Shlosman mixing condition holds, the speed at which the stochastic Ising model relaxes to the equilibrium is uniformly positive in l and the boundary conditions ([3], [5], [9], [12], for example). This statement can be formulated more specifically as uniform bounds on spectral gap or on the logarithmic Sobolev constant( log-Sobolev constant, for short), because it is reasonable to interpret

spectral gap ≍ speed of relaxation,

and as is well known,

\frac{1}{log-Sobolev constant} ≤ spectral gap.

At low temperature, however, uniform positivity of the spectral gap is broken as a manifestation of phase-coexistence. In fact, the vanishment of the spectral gap as l goes to infinity is caused by the growth of the height of energy the barrier between local equilibriums. Rigorous estimate of vanishing spectral gap in the context of stochastic Ising model was initiated, to our knowlegde, by a work of L. E. Thomas ([10]), in which he showed the spectral gap on the cube Λ(l) shrinks exponentailly fast in l^{d−1} if free boundary condition is considered. His estimate is best possible up to constant factors, since it is known (cf. (2.3) below) that the spectral gap of any stochasic Ising model on Λ(l) shrinks at most exponentially fast in l^{d−1}. The purpose of this article is to extend the result of Thomas to wider class of boundary conditions and to discuss some applications, which are made possible by our generalization. In doing so, we had to restrict our attention to the two-dimentional case, where the geometry is more tractable than in higher dimensions. However, we strongly believe that similar analysis is possible in dimension three of greater.

In the remainder of this section, we introduce a sequence of definitions and notations.

The lattice. For x = (x_{j})_{j=1}^{d} ∈ Z^{d}, we will use the l_{1}-norm: ∥x∥_{1} = |x_{1}| + ⋯ + |x_{d}|, together with the l_{∞}-norm: ∥x∥_{∞} = max{|x_{1}|, ⋯, |x_{d}|}. A set Λ ⊂ Z^{d} is said to be l_{p}-connected ( p = 1 or ∞ ) if for each distinct x, y ∈ Λ, we can find some {x_{0}, ⋯, x_{n}} ⊂ Λ with x_{0} = x, x_{n} = y and ∥x_{j} − x_{j−1}∥_{p} = 1 (j = 1, ⋯, n). The interior and exterior boundaries of a set Λ ⊂ Z^{d} will be denoted respectively by

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

and

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

F will stand for the totality of all non-empty, finite subset of Z^{d}. The number of points contained in Λ ∈ F will be denoted by |Λ|. The set B of bonds in Z^{d} is defined by

B = {{x, y} ⊂ Z^{d} ; ∥x − y∥_{1} = 1}.

For a set Λ, we also define

B_{Λ} = {{x, y} ∈ B ; (x, y) ∈ Λ^{2}},

and

∂Λ = {{x, y} ∈ B ; (x, y) ∈ Λ × Λ^{c}}.

The number of bonds contained in a set γ ⊂ B will be denoted by |γ|. For a bond b = {x, y}, consider a unit (d − 1)-cell b^{∗} := Q(x) ∩ Q(y), where Q(x) = ∏_{j=1}^{d}[x_{j} − \frac{1}{2}, x_{j} + \frac{1}{2}] ⊂ R^{d}. Two bonds b_{1} and b_{2} are said to be adjacent if b_{1}^{∗} and b_{2}^{∗} have a (d − 2)-cell in common. It follows that any bond b has 6(d−1) bonds as its adjacent neibours. A set γ ⊂ B is said to be connected if for each distinct b, b′ ∈ γ, we can find some {b_{0}, ⋯, b_{n}} ⊂ γ with b_{0} = b, b_{n} = b′ and b_{j−1} and b_{j} is adjacent(j = 1, ⋯, n).

The configurations and the Gibbs states. We define two kind of spin configurations;

Ω_{Λ} = {σ = (σ(x))_{x∈Λ} ; σ(x) = +1 or −1}, Λ ∈ F,

and

Ω_{b} = {ω = (ω(x))_{x∈Z^{d}} ; ω(x) ∈ [−1, 1]}.

We will refer an element ω of Ω_{b} as a boundary condition or a boundary configuration. The set of all real function on Ω_{Λ} is denoted by C_{Λ}. For Λ ∈ F and ω ∈ Ω_{b}, Hamiltonian H_{Λ}^{ω} ∈ C_{Λ} is defined by

H_{Λ}^{ω}(σ) = − ∑_{\{x,y\}∈B_{Λ}} σ(x)σ(y) − ∑_{\begin{matrix}(x, y) ∈ Λ × Λ^{c} \\ ∥x − y∥_{1} = 1 \end{matrix}} σ(x)ω(y).

We regard ω(y)’s as spins put on ∂_{ex}Λ(l), but one can also regard them as magnetic field, applied to ∂_{in}Λ(l), since we have allowed ω(y) to take value from continuous interval [−1, 1]. Gibbs state on Λ with the boundary condition ω ∈ Ω_{b} is defined as a probability distribution μ_{Λ}^{ω} on Ω_{Λ}, in which the probability of each configuration σ ∈ Ω_{Λ} is given by

μ_{Λ}^{ω}(\{σ\}) = \frac{1}{Z_{Λ}^{ω}} exp −βH_{Λ}^{ω}(σ),

where β > 0 is a parameter and Z_{Λ}^{ω} is a constant to make μ_{Λ}^{ω}(Ω_{Λ}) = 1.

Stochastic Ising Models.

For Λ ∈ F, we define a function q_{Λ} : Λ × Ω_{Λ} × Ω_{b} → ]0, ∞[ by

q_{Λ}(x, σ, ω) = exp −\frac{β}{2} \{H_{Λ}^{ω}(σ^{x}) − H_{Λ}^{ω}(σ)\}
= exp −βσ(x){  ∑_{y∈Λ:∥x−y∥_{1}=1} σ(y) + ∑_{y∉Λ:∥x−y∥_{1}=1} ω(y)},

where σ^{x} is a configuration obtained from σ by replacing σ(x) by −σ(x). Then, the following properties are easily verified;

(i) Boundedness : For each β > 0, there exist positive constants \underline{q}(d, β) and \overline{q}(d, β) such that

\underline{q}(d, β) ≤ q_{Λ}(x, σ, ω) ≤ \overline{q}(d, β),

for all Λ ∈ F, and (x, σ, ω) ∈ Λ × Ω_{Λ} × Ω_{b}.

(ii) Detailed Balance Condition:

q_{Λ}(x, σ, ω) exp −βH_{Λ}^{ω}(σ) = q(x, σ^{x}, ω) exp −βH_{Λ}^{ω}(σ^{x}),  (1.1)

for all Λ ∈ F and (x, σ, ω) ∈ Λ × Ω_{Λ} × Ω_{b}.

In fact, only these two properties are relevant in our exposition, rather than particular choice of q_{Λ}. Now, fix Λ ∈ F and ω ∈ Ω_{b}. We define a linear operator A_{Λ}^{ω} : C_{Λ} → C_{Λ} by

A_{Λ}^{ω}f(σ) = ∑_{x∈Λ} q_{Λ}(x, σ, ω){f(σ^{x}) − f(σ)}, f ∈ C_{Λ}.

It can easily be seen from (1.1) that

−μ_{Λ}^{ω}(fA_{Λ}^{ω}g) = −μ_{Λ}^{ω}(gA_{Λ}^{ω}f)
= \frac{1}{2} ∑_{x∈Λ} ∑_{σ∈Ω} μ_{Λ}^{ω}(σ)q_{Λ}(x, σ, ω)\{f(σ^{x}) − f(σ)\}\{g(σ^{x}) − g(σ)\}.

Finally, we define

gap(Λ, ω) = inf \{\frac{−μ_{Λ}^{ω}(fA_{Λ}^{ω}f)}{μ_{Λ}^{ω}(|f − μ_{Λ}^{ω}f|^{2})} ; f ∈ C_{Λ}\} ,  (1.2)

which is the smallest positive eigenvalue of −A_{Λ}^{ω}, and hence is called the spectral gap.

2 Results

In what follows, we will be exclusively concerned with the stochastic Ising model on a cube

Λ(l) = (−\frac{l}{2}, −\frac{l}{2}]^{d} ∩ Z^{d}, l = 1, 2, · · · .

For each l = 1, 2, ⋯, we define a function V_{l} : Ω_{b} → [0, 2dl^{d−1}] by

V_{l}(ω) = max {| ∑_{x∈A} ω(x)|; A is l_{∞}-connected subset of ∂_{ex}Λ(l) } .

With this notation in place, we now state our main result as;

Theorem 2.1 Let d = 2 and define a class Ω_{b}(θ) (0 ≤ θ ≤ 1) of boundary conditions by

Ω_{b}(θ) = \{ω ∈ Ω_{b}; \overline{lim_{l→∞}} \frac{V_{l}(ω)}{4l} ≤ θ\} .  (2.1)

Then, for every θ < θ̂ < \frac{1}{8} and α ∈]0, 1 − 8θ̂[, there exist β_{0} > 0, c : [β_{0}, ∞[ → ]0, ∞[ and l_{0} : Ω_{b}(θ) → N such that

gap(Λ(l), ω) ≤ c(β) exp(−αβl),  (2.2)

if β ≥ β_{0}, ω ∈ Ω_{b}(θ) and l ≥ l_{0}(ω).

Remark 2.1 It is known([6], Theorem 5) that the following is true for d ≥ 2 and β > 0;

\underline{q}(d, β)l^{−d} exp (−4β ∑_{j=0}^{d−1} l^{j}) ≤ inf_{ω∈Ω_{b}} gap(Λ(l), ω), l = 1, 2, · · · .  (2.3)

Thus, in dimension two, gap(Λ(l), ω) shrinks at most exponentially fast in l. Theorem 2.1 says that the exponential decay does occur whenever the boundary condition is chosen from Ω_{b}(θ) with θ < 1/8.

Remark 2.2 Let us consider the following boundary condition ω ∈ Ω_{b} for δ > 0 ;

ω(x) = \{ \begin{matrix}+1 & if x_{1} = [\frac{l}{2}] and \frac{−δl}{2} < x_{2} ≤ \frac{−δl}{2}, \\ 0 & otherwise. \end{matrix}

This boundary condition seems to provide an interesting application of Theorem 2.1. From static point of view, a particular choice of δ > 0 makes no difference as l goes to infinity. In fact, it is not difficult to see that the μ^{Λ(l),ω} converges to the pure (+)-phase whenever δ > 0. On the other hand, the behaviour of gap(Λ(l), ω) as l → ∞ is quite different depending on whether δ < 1/2 or δ ≥ 1. Since ω ∈ Ω_{b}(θ) with θ = δ/4, we have (2.2) if δ < 1/2. However, (2.2) is no longer true for δ = 1. In fact, it follows from Corollary 4.1 of [4] that for δ = 1,

exp (−βC_{ε}l^{\frac{1}{2} +ε}) ≤ gap(Λ(l), ω), ε > 0, l = 1, 2, · · · .

This also shows that our condition θ < 1/8 has only a limited possibility for improvement.

Remark 2.3 Theorem 2.1 has an application to spectral gap of threedimensional stochastic Ising models ([7]). Let d = 3 and we set Λ_{3}(l) = (−\frac{l}{2}, −\frac{l}{2}]^{3}∩Z^{3}, Λ_{2}(l) = (−\frac{l}{2}, −\frac{l}{2}]^{2}∩Z^{2}. On a finite subset Λ in Z^{3}, consider the following Hamiltonian with the boundary condition ω ∈ Ω_{b} and the magnetic field h ∈ R;

H_{Λ}^{ω,h}(σ) = H_{Λ}^{ω}(σ) − h ∑_{x∈Λ} σ(x).

The corresponding finite volume Gibbs state and the spectral gap, denoted respectively by μ^{Λ,ω,h} and gap(Λ, ω, h), are defined in the same way as in the case h = 0. For any fixed h > 0, a result of Martinelli and Olivieri ([5], part(c) of Theorem 5.1) implies that

inf{gap(Λ_{3}(l), ω, h) ; ω ∈ {−1, +1}^{Z^{3}}, l ≥ 1} > 0,  (2.4)

if β is large enough (depending on h).

On the other hand, consider a boundary condition ω ∈ {−1, +1}^{Z^{3}} such that ω ≡ −1 on Z^{2} × {±1} and such that the restriction of ω to Z^{2} × {0}, as considered as an element of {−1, +1}^{Z^{2}}, belongs to Ω_{b}(0). We then have for large enough β that

gap(Λ_{2}(l) × \{0\}, ω, h)|_{h=2} ≤ c(β) exp −\frac{βl}{2} ,  (2.5)

if l is large. In fact, the magnetic field h = 2 is cancelled out by the (−)-spins sitting in Z^{2} × {±1}, which enables us to apply Theorem 2.1 to 2-dimensional square Λ_{2}(l).

We see from (2.4) and (2.5) that the behaviour of gap(Λ, ω, h) when the diameter of Λ goes to infinity can be quite different depending on whether Λ is fat of thin, which disproves the equivalence among different formulations of the Dobrushin-Shlosman mixing condition (cf. [3], [5], [9], [12]).

The next corollary is an application to Theorem 2.1 to boundary conditions, which are randomly distributed. This is possible because the class Ω_{b}(θ) with θ < 1/8 is ample enough to accommodate i.i.d. random boundary conditions with the mean of ω(x) within ] − \frac{1}{8}, \frac{1}{8}[.

Corollary 2.2 Suppose that d = 2 and that ω = (ω(x))_{x∈Z^{2}} ∈ Ω_{b} is an i.i.d. random variable with mean m ∈] − \frac{1}{8}, \frac{1}{8}[. Then, for every m̂ ∈]|m|, \frac{1}{8}[ and α ∈]0, 1 − 8m̂[, there exist β_{0} > 0, c : [β_{0}, ∞[ →]0, ∞[ such that if β ≥ β_{0}, the following holds almost surely;

gap(Λ(l), ω) ≤ c(β) exp(−αβl) for sufficiently large l’s.  (2.6)

Proof: It can be seen from the following simple argument that ω ∈ Ω_{b}(|m|) almost surely. We set

S_{i,k}^{(l)}(ω) = ∑_{j=i+1}^{k}{ω(x_{j}^{(l)}) − m},

where {x_{j}^{(l)}}_{j=1}^{4l} is the clockwise enumeration of ∂_{ex}Λ(l) starting from x_{1}^{(l)} = (0, [l/2]). Now take η ∈]\frac{1}{2}, 1[, ε ∈]0, 1[ and p ∈ N such that p(2η − 1) > 1. We have

max_{0≤i<k≤4l} |S_{i,k}^{(l)}(ω)| ≤ |S_{0,4l}^{(l)}(ω)| + max_{0≤i<4l} |S_{0,i}^{(l)}(ω)| + max_{0<k≤4l} |S_{k,4l}^{(l)}(ω)|,

and thus,

P ( max_{0≤i<k≤4l} |S_{i,k}^{(l)}(ω)| ≥ 3εl^{η})

≤ P (|S_{0,4l}^{(l)}(ω)| ≥ εl^{η}) + P ( max_{0≤i<4l} |S_{0,i}^{(l)}(ω)| ≥ εl^{η})
+ P ( max_{0<k≤4l} |S_{k,4l}^{(l)}(ω)| ≥ εl^{η})
=: (I) + (II) + (III).

Each term in the right hand side can be estimated as follows;

(I) ≤ \frac{E|S_{0,4l}^{(l)}|^{2p}}{(εl)^{2ηp}} ≤ C_{p}ε^{−2ηp}l^{−p(2η−1)},
(III) = (II)
≤ 4P (|S_{0,4l}^{(l)}(ω)| ≥ εl^{η} − \sqrt{8l})
≤ 4C_{p}l^{p}(εl^{η} − \sqrt{8l})^{−2p},

where in the third line we have used the maximal inequality ([2], p.45) for a summation of i.i.d. random variables. Therefore,

∑_{l=1}^{∞} P ( max_{0≤i<k≤4l} |S_{i,k}^{(l)}(ω)| ≥ 3εl^{η}) < ∞

and consequently,

max_{0≤i<k≤4l} |S_{i,k}^{(l)}(ω)| = o(l^{η}) a.s..

by the Borel-Cantelli lemma. This implies

V_{l}(ω) ≤ 4l|m| + 2 max_{0≤i<k≤4l} |S_{i,k}^{(l)}(ω)|
= 4l|m| + o(l^{η}), a.s.,

which is what we wanted to prove.

Q.E.D.

3 Proof of Theorem 2.1

Although we are mainly interested in two-dimensional case, we will state some lemmata in d(≥ 2)-dimensional setting. Our approach here is close to that in [10]. An alternative proof, which is based on some techniques used in [6] is also available ([11]). The use of contour is essential in our proof as it is in the above quoted references. So we begin by explaining the notion of contour. A contour is a finite subset γ ⊂ B with the following properties; there exists a finite subset Θ ⊂ Z^{d} such that

(i) Both Θ and Θ^{c} is l_{1}-connected.

(ii) γ = ∂Θ.

The set Θ is uniquely determined by γ and hence is denoted by Θ(γ). Since a contour γ is a set of connected bonds, it follows that for each b ∈ B and m = 1, 2, ⋯,

♯{γ : contour with |γ| = m and γ ∋ b } ≤ κ(d)^{m−1},  (3.1)

where κ(d) is a constant which depends only on d ( (4.24) in [1], for example). Obviously, we can take κ(d) = 3 if d = 2. If a contour γ is a subset of B_{Λ} ∪ ∂Λ for some Λ ⊂ Z^{d}, we say γ is a contour in Λ. For σ ∈ Ω_{Λ}, ε = + or − and Λ ⊂ Z^{d}, a contour γ is said to be an (ε)-contour in Λ at σ if it satisfies

∂_{in}Θ(γ) ⊂ {x ∈ Λ; σ(x) = ε1} and ∂_{ex}Θ(γ) ⊂ {x ∈ Λ; σ(x) = −ε1} ∪ Λ(l)^{c}.

A contour γ is said to be a contour in Λ at σ ∈ Ω_{Λ} if it is either (+)-contour in Λ at σ or (−)-contour in Λ at σ.

Lemma 3.1 Suppose that d = 2 and that γ is a contour in Λ(l) at σ ∈ Ω_{Λ(l)}, which intersects with no pair of opposite sides of ∂_{in}Λ(l). Then,

H_{Λ(l)}^{ω}(σ) − H_{Λ(l)}^{ω}(T_{γ}σ) ≥ |γ| − 2V_{l}(ω),  (3.2)

where T_{γ}σ ∈ Ω_{Λ(l)} is defined by

T_{γ}σ(x) = \{ \begin{matrix}−σ(x), & if x ∈ Θ(γ) \\ σ(x), & if x ∉ Θ(γ). \end{matrix}  (3.3)

Proof: Let S_{1}, S_{2}, ⋯ be those sides of ∂_{in}Λ(l) which do not intersect with γ. Then, there must be a connected component λ_{0} of B_{Λ(l)} ∩ γ which divide Λ(l) into two connected components, one contains Θ(γ) and the other contains S_{1}, S_{2}, · · ·. We denote the former component by Θ̃. Next, we decompose the set \overline{γ} := ∂Λ(l) ∩ ∂Θ̃ as follows;

\overline{γ} = (⋃_{i≥1} ρ_{i}) ∪ (⋃_{i≥1} δ_{i}) ,

where {ρ_{i}} are connected components of γ ∩ ∂Λ(l) and {δ_{i}} are connected components of \overline{γ}\γ. Finally, we let \{λ_{0}, λ_{1}, ⋯\} denote the set of connected components of B_{Λ(l)} ∩ γ. We then have following relations;

H_{Λ(l)}^{ω}(σ) − H_{Λ(l)}^{ω}(T_{γ}σ) = 2 ∑_{i≥0} |λ_{i}| − 2 ∑_{i≥1} ∑_{\begin{matrix}y ∉ Λ(l) \\ \{x, y\} ∈ ρ_{i} \end{matrix}} ω(y) (3.4)

|\overline{γ}| = ∑_{i≥1} |ρ_{i}| + ∑_{i≥1} |δ_{i}|,  (3.5)
|γ| = ∑_{i≥1} |ρ_{i}| + ∑_{i≥0} |λ_{i}|,  (3.6)

and

|\overline{γ}| ≤ \frac{|γ|}{2}.  (3.7)

(3.4),(3.5) and (3.6) are obvious. (3.7) can also be seen from geometric considerations. ¿From these observations, we obtain

∑_{i≥1} ∑_{\begin{matrix}y ∉ Λ(l) \\ \{x, y\} ∈ ρ_{i} \end{matrix}} ω(y) = ∑_{\begin{matrix}y ∉ Λ(l) \\ \{x, y\} ∈ \overline{γ} \end{matrix}} ω(y) − ∑_{i≥1} ∑_{\begin{matrix}y ∉ Λ(l) \\ \{x, y\} ∈ δ_{i} \end{matrix}} ω(y)

≤ V_{l}(ω) + ∑_{i≥1} |δ_{i}|
= V_{l}(ω) + |\overline{γ}| − ∑_{i≥1} |ρ_{i}| by (3.5)

= V_{l}(ω) + |\overline{γ}| − |γ| + ∑_{i≥0} |λ_{i}| by (3.6)

≤ V_{l}(ω) − \frac{|γ|}{2} + ∑_{i≥0} |λ_{i}| by (3.7).  (3.8)

Here, in the second line, we have used the fact that \{y ∉ Λ(l); \{x, y\} ∈ \overline{γ}\} is l_{∞}-connected. Plugging (3.8) into (3.4), we obtain (3.2).

Q.E.D.

For l = 1, 2, ⋯, and ε = + or −, We set

Γ_{l,ε} = \{σ ∈ Ω_{Λ(l)}; \begin{matrix}There exists an (ε)-contour γ in Λ(l) at σ \\ with |γ| ≥ 2[l/2] \end{matrix} \} . (3.9)

Lemma 3.2

lim_{\overline{β→∞}} inf\{μ_{Λ(l)}^{ω}(Γ_{l,+}) + μ_{Λ(l)}^{ω}(Γ_{l,−}); l ≥ 1, ω ∈ Ω_{b}\} ≥ 1.  (3.10)

Proof: Note that

∑_{ε=±} μ_{Λ(l)}^{ω}(Γ_{l,ε}) ≥ ∑_{ε=±} μ_{Λ(l)}^{ω}({σ(0) = ε1} ∩ Γ_{l,ε})
= ∑_{ε=±} {μ_{Λ(l)}^{ω}({σ(0) = ε1}) − μ_{Λ(l)}^{ω}({σ(0) = ε1} ∩ Γ_{l,ε}^{c})}
≥ 1 − ∑_{ε=±} δ(β, ε),

where δ(β, ε) = sup_{l,ω} μ_{Λ(l)}^{ω}({σ(0) = ε1} ∩ Γ_{l,ε}^{c}). So let us prove that δ(β, ±) → 0, as β → ∞. For each σ ∈ {σ(0) = ε1} ∩ Γ_{l,ε}^{c}, there is an (ε)-contour γ at σ such that Θ(γ) ∋ 0 and |γ| < 2[l/2]. Since γ is not long enough to contain a point in ∂_{ex}Λ(l), we have

H_{Λ(l)}^{ω}(σ) − H_{Λ(l)}^{ω}(T_{γ}σ) ≥ 2|γ|,

and thus,

μ_{Λ(l)}^{ω}(γ appears ) ≤ e^{−2β|γ|}.

Combining all these, we obtain

μ_{Λ(l)}^{ω}({σ(0) = ε1} ∩ Γ_{l,ε}^{c}) ≤ ∑_{γ:contour in Λ(l) Θ(γ) ∋ 0, |γ| < 2[l/2]} μ_{Λ(l)}^{ω}(γ appears )

≤ ∑_{m≥2d} 2dmκ(d)^{m−1}e^{−2βm} ⟶^{β→∞} 0,

which implies that δ(β, ±) →^{β→∞} 0.

Q.E.D.

By (3.10), there exists a β_{1}(d) > 0, and ε = ε(l, ω) : such that

inf\{μ_{Λ(l)}^{ω}(Γ_{l,ε(l,ω)}) ; β ≥ β_{1}(d), l ≥ 1, ω ∈ Ω_{b}\} ≥ \frac{1}{3}  (3.11)

We now set

Γ_{l} = Γ_{l,ε(l,ω)},  (3.12)

and for σ ∈ Ω_{Λ(l)},

C_{l}^{ω}(σ) = \{γ ; \begin{matrix}(ε(l, ω))-contour in Λ(l) at σ, \\ |γ| ≥ 2[l/2] \end{matrix} \} .  (3.13)

Finally, for a contour γ, we define

Γ_{l}(γ) = \{σ ∈ Γ_{l} ; \begin{matrix}γ appears as the maximal element of C_{l}^{ω}(σ) \\ in the sense that if C_{l}^{ω}(σ) ∋ λ ≠ γ, then λ ⊂ Θ(γ). \end{matrix} \} .

(3.14)

Lemma 3.3

lim_{β→∞} sup \{μ_{Λ(l)}^{ω}(Γ_{l}|T_{γ}Γ_{l}(γ)) ; \begin{matrix}l ≥ 1, ω ∈ Ω_{b}, \\ γ is a contour in Λ(l), |γ| ≥ 2[l/2] \end{matrix} \} = 0,

(3.15)

where T_{γ}Γ_{l}(γ) = {T_{γ}σ; σ ∈ Γ_{l}(γ)} (cf. 3.3). Proof: Note that

Γ_{l} ∩ T_{γ}Γ_{l}(γ) ⊂ {T_{γ}σ ; σ ∈ Γ_{l}(γ), T_{γ}σ ∈ Γ_{l} }
⊂ \{T_{γ}σ ; \begin{matrix}There is an (ε(l, ω))-contour γ′ in Λ(l) at T_{γ}σ \\ such that γ′ ⊂ Θ(γ) and |γ′| ≥ 2[l/2] \end{matrix} \} .

Thus,

μ_{Λ(l)}^{ω}(Γ_{l} ∩ T_{γ}Γ_{l}(γ)) ≤ ∑_{\begin{matrix}γ′ ; γ′ ⊂ Θ(γ) \\ |γ′| ≥ 2[l/2] \end{matrix}} ∑_{σ; σ ∈ Γ_{l}(γ) γ′ appears at T_{γ}σ} μ_{Λ(l)}^{ω}(T_{γ}σ). (3.16)

Since each γ′ in the summation is contained in B_{Λ(l)}, we have

μ_{Λ(l)}^{ω}(T_{γ}σ) ≤ e^{−2β|γ′|}μ_{Λ(l)}^{ω}(T_{γ′}T_{γ}σ)
= e^{−2β|γ′|}μ_{Λ(l)}^{ω}(T_{γ}T_{γ′}σ).

¿From this and the fact that T_{γ′} maps Γ_{l}(γ) injectively into itself, we have

∑_{σ ∈ Γ_{l}(γ) γ′ appears at T_{γ}σ} μ_{Λ(l)}^{ω}(T_{γ}σ) ≤ e^{−2β|γ′|}μ_{Λ(l)}^{ω}(T_{γ}Γ_{l}(γ)).

By the last inequality and (3.16), we now conclude

μ_{Λ(l)}^{ω}(Γ_{l}|T_{γ}Γ_{l}(γ)) ≤ ∑_{\begin{matrix}γ′ ; γ′ ⊂ Θ(γ) \\ |γ′| ≥ 2[l/2] \end{matrix}} e^{−2β|γ′|}

≤ ∑_{m≥2[l/2]} 2dl^{d}κ(d)^{m−1}e^{−2βm}

≤ c(d) exp {(1 + log κ(d) − 2β)l} →^{β→∞} 0 uniformly in l.

Q.E.D.

Proof of Theorem 2.1: In what follows, we assume d = 2. We will divide the proof into four steps.

Step 1: Suppose l ≥ 6, σ ∈ Γ_{l} and Λ(l, σ) := {x ∈ Λ(l) ; σ^{x} ∉ Γ_{l}} ≠ ϕ. We claim that

(i) there exists a unique (ε(l, ω))-contour γ = γ(σ) in Λ(l) at σ such that |γ| ≥ 2[l/2]

(ii) γ does not intersect with any pair of opposite sides in ∂_{in}Λ(l).

(iii) |Λ(l, σ)| ≤ 2|γ|.

The uniqueness of the contour mentioned in (i) is obvious; if there were two such contours, it would be impossible that σ^{x} ∉ Γ_{l} for some x. (iii) is also obvious. (ii) can be seen as follows: For x ∈ Λ(l), consider a set of contours in Λ(l) such that |λ| < 2[l/2] and x ∈ λ, say C_{x}. Then, for any pair of opposite sides in ∂_{in}Λ(l), the farther side from x cannot be touched by any element of C_{x}. Now, let

\{γ_{1}, ⋯, γ_{j}\} = \{γ ; \begin{matrix}γ is an (ε(l, ω))-contour in Λ(l) at σ^{x} \\ γ ∋ x \end{matrix} \} .

Then, it follows that

γ ⊂ ∪_{i=1}^{j}γ_{i} ∪ {b ∈ B ; b ∋ x}.

Thus, we see (ii) since {γ_{1}, ⋯, γ_{j}} ⊂ C_{x}.

Step 2: We now prove that

gap(Λ(l), ω) ≤ 6\overline{q}(d, β) ∑_{γ∈C_{l}} |γ|\frac{μ_{Λ(l)}^{ω}(Γ_{l}(γ))}{μ_{Λ(l)}^{ω}(Γ_{l}^{c})},  (3.17)

where Γ_{l}(γ) is defined by (3.14) and C_{l} is the set of all contours in Λ(l) which satisfy |γ| ≥ 2[l/2] and the property described in Step1-(ii).

In fact, letting χ_{l} : Ω_{Λ(l)} → {0, 1} be the indicator function of Γ_{l}, we have for β ≥ β_{1}(d) (cf. (3.11)) that

gap(Λ(l), ω) ≤ \frac{−μ_{Λ(l)}^{ω}(χ_{l}A_{Λ(l)}^{ω}χ_{l})}{μ_{Λ(l)}^{ω}(|χ_{l} − μ_{Λ(l)}^{ω}χ_{l}|^{2})}

≤ \frac{\overline{q}(d, β)}{μ_{Λ(l)}^{ω}(Γ_{l})μ_{Λ(l)}^{ω}(Γ_{l}^{c})} ∑_{x∈Λ(l)} ∑_{σ} μ_{Λ(l)}^{ω}(σ)χ_{l}(σ)|χ_{l}(σ^{x}) − χ_{l}(σ)|

≤ 3\overline{q}(d, β) ∑_{x∈Λ(l)} ∑_{σ∈Γ_{l},σ^{x}∉Γ_{l}} \frac{μ_{Λ(l)}^{ω}(σ)}{μ_{Λ(l)}^{ω}(Γ_{l}^{c})}

= 3\overline{q}(d, β) ∑_{σ∈Γ_{l}} |Λ(l, σ)|\frac{μ_{Λ(l)}^{ω}(σ)}{μ_{Λ(l)}^{ω}(Γ_{l}^{c})}

≤ 6\overline{q}(d, β) ∑_{γ∈C_{l}} |γ|\frac{μ_{Λ(l)}^{ω}(Γ_{l}(γ))}{μ_{Λ(l)}^{ω}(Γ_{l}^{c})},

where we have used Step1 to get the final inequality.

Step 3: Next, Let us prove that there exists l_{0} : Ω_{b}(θ) → N such that for all l ≥ l_{0}(ω) and γ ∈ C_{l},

μ_{Λ(l)}^{ω}(Γ_{l}(γ)) ≤ μ_{Λ(l)}^{ω}(T_{γ}Γ_{l}(γ)) exp −α̂β|γ|,  (3.18)

where α̂ = 1 − 4dθ̂. In fact, there exists l_{0} : Ω_{b}(θ) → N such that

\frac{V_{l}(ω)}{8[l/2]} < θ̂, for l ≥ l_{0}(ω),  (3.19)

as is immidiate from the definition of Ω_{b}(θ). We then have for γ ∈ C_{l},

V_{l}(ω) < 4θ̂|γ|,

and thus by Lemma 3.1,

H_{Λ(l)}^{ω}(σ) − H_{Λ(l)}^{ω}(T_{γ}σ) ≥ α̂|γ|,

for all σ ∈ Γ_{l}(γ). This implies (3.18).

Step 4: We are now, in position to conclude the proof. By Lemma 3.3, there exist a β_{2}(d) > 0 such that

inf \{μ_{Λ(l)}^{ω}(Γ_{l}^{c}|T_{γ}Γ_{l}(γ)) ; \begin{matrix}l ≥ 1, ω ∈ Ω_{b}, β ≥ β_{2}(d) \\ γ is a contour in Λ(l), |γ| ≥ 2[l/2] \end{matrix} \} ≥ \frac{1}{2}.

Thus, \frac{μ_{Λ(l)}^{ω}(Γ_{l}(γ))}{μ_{Λ(l)}^{ω}(Γ_{l}^{c})} in the summation in (3.17) can be dominated as follows;

\frac{μ_{Λ(l)}^{ω}(Γ_{l}(γ))}{μ_{Λ(l)}^{ω}(Γ_{l}^{c})} ≤ \frac{μ_{Λ(l)}^{ω}(T_{γ}Γ_{l}(γ))}{μ_{Λ(l)}^{ω}(Γ_{l}^{c})} exp −α̂β|γ|
≤ \frac{1}{μ_{Λ(l)}^{ω}(Γ_{l}^{c}|T_{γ}Γ_{l}(γ))} exp −α̂βl|γ|
≤ 2 exp −α̂β|γ|.

Therefore we have

gap(Λ(l), ω) ≤ 12\overline{q}(d, β) ∑_{γ∈C_{l}} |γ| exp −α̂β|γ|
≤ 12\overline{q}(d, β) ∑_{m≥2[l/2]} ml^{d}3^{m−1} exp −α̂βm
≤ c(d, β, θ̂) exp −l{βα̂ − log 3 − 1}
≤ c(d, α, β, θ̂) exp −αβl,

if β > max\{β_{1}(d), β_{2}(d), \frac{log 3+1}{α̂−α}\}.

Q.E.D.

Acknowledgments

We are indepted to Hal Tasaki for many variable comments, discussions and especially for the improvement of the proof of Lemma 3.1. The difference between statics and dynamics discussed in Remark 2.2 was pointed out by Aernout van Enter, who also kindly explained many related results to us. We also would like to thank Roberto Schonmann for communicating the content of Remark 2.3. Finally, N.Y. thanks Taniguchi Foundation for an opportunity to take part in stimulating meetings in Warick, UK.

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