Finite volume Glauber dynamics in a small magnetic field

Nobuo Yoshida

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

Abstract

We consider Glauber dynamics on a finite cube in the d-dimensional lattice (d ≥ 2), which is associated with basic Ising model at temperature T = 1/β ≪ 1 under a magnetic field h > 0. We prove that if the “effective magnetic field” is positive, then the relaxation of the Glauber dynamics in the uniform norm is exponentially fast, uniformly over the size of underlying cube. The result covers the case of the free-boundary condition with arbitrarily small positive magnetic field. This paper is a continuation of an attempt initiated in [SY97] to shed more light on the relaxation of the finite volume Glauber dynamics, when the thermodynamic parameter (β, h) is so near the phase transition line: {(β, h); β_{c} < β & h = 0} that the Dobrushin-Shlosman mixing condition is no longer available.

Key words and phrases: Ising model, mixing conditions, Basuev region, boundary conditions, Glauber dynamics, exponential relaxation, spectral gap

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_{Λc} denotes the following configuration:

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

The set of all real function 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} βσ_{x} \{h + ∑_{{y∉Λ \atop ∥x−y∥_{1}=1}} ω_{y}\} ,  (1.1)

where β > 0 and h ≥ 0 denote respectively the inverse temparature and the magnetic field. The real numbers;

(h + ∑_{{y∉Λ \atop ∥x−y∥_{1}=1}} ω_{y})_{x∈Λ} ,  (1.2)

which appear in the second summation on the right-hand-side of (1.1) is often called the effective magnetic field. 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),  (1.3)

where Z^{Λ,ω} is the normalizing constant. We will use the following common and convenient abbrebiation; μ^{Λ,ω}f = ∑_{σ∈S^{Λ}} μ^{Λ,ω}{σ}f(σ) for f ∈ C_{Λ}. The following facts are well known.(i); The limit m_{β,h}^{±} = lim_{Λ↑Z^{d}} μ_{β,h}^{Λ,±}(σ_{0}) exists. (ii); m_{β,h}^{+} = m_{β,h}^{−} > 0 if h > 0. (iii);There exists a critical inverse temperature β_{c} ∈ (0, ∞) such that m_{β,0}^{+} = m_{β,0}^{−} = 0 if β < β_{c} and m_{β,0}^{+} > 0 > m_{β,0}^{−} if β_{c} < β. Therefore, the (β, h)-plane is devided into two parts according as m_{β,h}^{+} = m_{β,h}^{−} 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}.  (1.4)

The former is called one phase region and the latter is called phase transition line.

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},  (1.5)

where

c_{x}^{Λ,ω}(σ) = exp −∇_{x}H^{Λ,ω}(σ).  (1.6)

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),

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.  (1.7)

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}) ,  (1.8)

for all f ∈ C_{Λ}, where

T_{t}^{Λ,ω} = exp tA^{Λ,ω}, t > 0.  (1.9)

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

there exist constants C_{i} = C_{i}(d, β, h, ω) ∈ (0, ∞) (i = 1, 2) such that for all cube
Λ, f ∈ C_{Λ} and t > 0
∥T_{t}^{Λ,ω}f − μ^{Λ,ω}f∥ ≤ C_{1}|||f||| exp −\frac{t}{C_{2}} .  (1.10)

Note first of all that the above exponential relaxation property can be true only in the one phase region. In fact, it is not difficult to see that, if (1.10) holds for some β, h and ω ∈ Ω, then m_{β,h}^{+} = m_{β,h}^{−}. Here we summerize some of known results and a conjecture;

(i) The exponential relaxation property (1.10) is true for all ω ∈ {−1, +1}^{Z^{d}}, whenever a mixing condition in the sense of the Dobrushin-Shlosman in a certain restricted sense is valid ( [MO94a, MO94b], See also [SZ92]). This is the case if d = 2 and (β, h) is in the one phase region( [SS95]) or d ≥ 3 and (β, h) is sufficiently deep inside the one phase region( [MO94a]).

(ii) It is conjectured that if d ≥ 3, β is sufficiently large and h is a certain small positive number depending on β, then there is ω ∈ Ω which violates the exponential relaxation property. The part of the (β, h)-plane reffered to above is often called “Basuev region”.

Though the conjecture (ii) above is not vindicated yet, there are some results on related models which suggest the existence of such “dangerous” boundary condition ω ∈ Ω ([CM96a, CM96b, DM94]). On the other hand, the conjecture motivated an attempt to find out a class of “safe” boundary conditions when (β, h) is sitting in the one phase region but near the phase transition line. In fact, it is proved in [SY97, Theorem 4.1] that the exponential relaxation property (1.10) is true (for any Λ ⊂⊂ Z^{d}, not only for cubes) at least in the following three cases: (a); d ≥ 2, β < β_{c}, h ≥ 0 and ω ∈ Ω is such that all components of the effective magnetic field (cf. (1.2)) are non-negative for any Λ, (b); d = 2, β ≥ β_{c}, h > 0 and ω ≡ +1, (c); d ≥ 3, β is sufficiently large, h > 0 and ω ≡ +1. In the present paper, we continue the program in [SY97] to understand more about the exponential relaxation property in the Basuev region. 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 relaxation property( [MO94b, SZ92]). We instead use the more specific structures of the ferromagnetic Ising model, especially attractivity and Ising percolation.

2 Results

The main result in this paper is the following

Theorem 2.1 There exists β_{0} = β_{0}(d) ∈ (0, ∞) for which the following hold.

(a) For any β > β_{0} and h_{0} > 0, there exist constants C_{i} = C_{i}(d, β, h_{0}) ∈ (0, ∞) (i = 1, 2) such that if h > 0, ω ∈ Ω and a cube Λ satisfy

min_{x∈Λ} \{h + ∑_{{y∉Λ \atop ∥x−y∥_{1}=1}} ω_{y}\} ≥ h_{0},  (2.1)

then for all f ∈ C_{Λ} and t > 0

∥T_{t}^{Λ,ω}f − μ^{Λ,ω}f∥ ≤ C_{1}|||f||| exp −\frac{t}{C_{2}} .  (2.2)

(b) For β > β_{0},

0 < inf{gap_{β,h}^{Λ,ω} ; h > 0, ω ∈ Ω and a cube Λ satisfy (2.1)}.  (2.3)

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}} .  (2.4)

Remark 2.1 The condition (2.1) is satisfied for any h > 0, if all ω_{y}’s are non-negative. It is the case of very small h when the concluison of the theorem becomes really interesting because, for d ≥ 3, it covers the Basuev region reffered to in the previous section. Even for d = 2, our result seems to be new, since it also applies to the free boundary condition. Part(a) for pure (+)-boundary condition has already been obtained in [SY97, Theorem 4.1, part(b)] (in fact for any Λ ⊂⊂ Z^{d}, not only for cubes). The extension to the present case, which includes the case of free boundary condition with any positive magnetic field was a question left unsolved in that paper. The uniform positivity of the spectral gap in (2.4) is in sharp contrast with what happens at the phase transition line of h = 0. In fact, [T89, Proposition 2.4] says that there are C_{i} = C_{i}(d) ∈ (0, ∞) (i = 1, 2) such that for sufficiently large β and h = 0,

gap_{β,0}^{Q(l),0} ≤ exp (βC_{1} − βC_{2}l^{d−1})  (2.5)

for any l ≥ 1. On the other hand, the right-hand-side estimate in (2.4) is a manifestation of the phase-coexistence at h = 0. This contrasts with the situation at high temperature; [SY97, Theorem 4.1, part(a)] implies in particular that 0 < inf_{{Λ⊂⊂Z^{d} \atop h≥0}} gap_{β,h}^{Λ,0} if β < β_{c}.

Part (a) of Theorem 2.1 is obtained by combination of Proposition 2.2 and Proposition 2.3, which we present below.

Proposition 2.2 ([SY97]) 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_{Λc}}(σ_{x}) − μ^{Λ∩Q_{x}(l),(−)_{Λ}·ω_{Λc}}(σ_{x})\} ≤ B_{1} exp −\frac{l}{B_{2}} , l = 1, 2, ….  (2.6)

Then there exists {C_{1}, C_{2}} ⊂ (0, ∞) which depend only on d, β, h, B_{1} and B_{2} above such that (2.2) holds for all f ∈ C_{Λ} and all t ≥ 0.

Proposition 2.3 There exists β_{0} = β_{0}(d) ∈ (0, ∞) for which the following holds. For any β ≥ β_{0} and h_{0} ∈ (0, ∞), there exist constants B_{i} = B_{i}(d, β, h_{0}) ∈ (0, ∞) (i = 1, 2) such that (2.6) holds for all β ≥ β_{0}, h > 0, ω ∈ Ω and a cube Λ which satisfy (2.1).

In Section3, we will prove Proposition 2.3 by using Proposition 2.4 below, which is an extension of [Sch94a, Theorem3]. To state Proposition 2.4, we need to introduce some more notations. We take a subset E ⊂ {±1, …, ±d} to specify some of 2d basic directions in the lattice. We suppose that E is non-degenerate;

E ∩ {−j, j} ≠ ϕ, ∀j = 1, …, d.  (2.7)

We first describe a boundary condition around a box Λ, which equals (±) on faces in Edirections and equals (0) on the other faces. 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}.

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.

We next define a box Q(l; E) (l > 0) by

Q(l; E) = ∏_{j=1}^{d} Q^{(j)}(l),  (2.8)

where

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

As l increases, Q(l; E) grows larger only in the directions specified by E.

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 2.4 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,  (2.9)

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 2.2 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 2.4 is what can be read off directly from the proof of [Sch94a, 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 [Sch94a, Theorem 3] to cover the case of E ≠ {±1, …, ±d}, as far as (2.7) is satisfied. The proof of Proposition 2.4 will be sketched in section 4.

For the part (b) of Theorem 2.1, the uniform positivity of the spectral gap in both (2.3) and (2.4) is the consequence of the part (a). The right-hand-side estimate of (2.4) comes from the following

Proposition 2.5 There exist 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 exists 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}} ,  (2.10)

Remark 2.3 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, ∞) ([Sch94a, Theorem 5], for example). Thus, the inequality (2.10) is sharp up to constants. Similar upper and lower bound of the spectral gap for (−)-boundary condition can be found in [Sch94b, Theorem 4]

Remark 2.4 Proposition 2.5 says a little more than is needed to conclude the right-hand-side estimate of (2.4). In fact, the right-hand-side estimate of (2.4) can be obtaied just by proving (2.10) for small enough B. The proof of (2.10) in this case is easy as we present now. It is not difficult to see that

exp (−2h(B/h)^{d}) ≤ \frac{μ_{β,h}^{Q(B/h),0}\{σ\}}{μ_{β,0}^{Q(B/h),0}\{σ\}} ≤ exp (2βh(B/h)^{d}) .

From this and a standard comparison argument, we get

gap_{β,h}^{Q(B/h),0} ≤ gap_{β,0}^{Q(B/h),0} exp (βC_{0} + 4βh(B/h)^{d})

for some C_{0} = C_{0}(d) ∈ (0, ∞). We then apply (2.5) to see that the right hand side of the above inequality is bounded from above by

exp {β(C_{0} + C_{1}) − β(B/h)^{d−1}(C_{2} − 4B)} ,

which proves (2.10) when B < C_{2}/4.

3 Proof of Proposition 2.3

In this section, we give a proof of Proposition 2.3. We begin by reducing the proof to the case of ω ≡ 0. This is possible by the following inequality;

μ_{β,h}^{Λ∩Q_{x}(l),(+)_{Λ}·ω_{Λc}}(σ_{x}) − μ_{β,h}^{Λ∩Q_{x}(l),(−)_{Λ}·ω_{Λc}}(σ_{x}) ≤ μ_{β,h_{0}}^{Λ∩Q_{x}(l),(+)_{Λ}·0_{Λc}}(σ_{x}) − μ_{β,h_{0}}^{Λ∩Q_{x}(l),(−)_{Λ}·0_{Λc}}(σ_{x}) (3.1)

The above inequality is proved as follows. This is where the condition (2.1) is used. 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 this fact can be given in the same way as that of [Hig93, (2.15)]. 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_{Λc})_{y},

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

In this case (3.2) is obvious and (3.3) follows from (2.1). Since ν^{{h_{z}^{±}}} = μ_{β,h}^{Λ∩Q_{x}(l),(±)_{Λ}·ω_{Λc}} and

ν^{{k_{z}^{±}}} = μ_{β,h_{0}}^{Λ∩Q_{x}(l),(±)_{Λ}·0_{Λc}}, (3.4) implies (3.1).

The basic strategy to prove that the right-hand-side of (3.1) decays exponentially in l is similar to Appendix 1 of [MO94a], where the “effectiveness” of a mixing condition for Gibbs states is discussed. We apply coupling technique to derive a recursive inequality, which leads to desired exponential decay. An alternative proof, which does not rely on such coupling technique is presented in [Y97].

Let us abbrebiate Λ ∩ Q_{x}(l) and (±)_{Λ} · 0_{Λc} respectively by Γ and ζ^{±} for simplicity. It is convenient to introduce the following restricted configuration space; S^{Δ,η} = {σ_{Δ}·η_{Δc} ; σ ∈ S^{Δ}} (Δ ⊂⊂ Z^{d}, η ∈ Ω). Define;

F_{z}(l) = μ_{β,h_{0}}^{Γ,ζ^{+}}{σ_{z} = +1} − μ_{β,h_{0}}^{Γ,ζ^{−}}{σ_{z} = +1}, z ∈ Γ.

We are going to prove that F_{x}(l) decays exponentially in l. To this end, we may and will assume that

(d + 1)m + 1 < \frac{l}{2},  (3.5)

where m = m(h, d, β) ≥ 1/h is a large enough integer which will be specified at the end of the proof. It is enough to establish the following recursive inequality;

F_{z}(l) ≤ e^{−1} max\{F_{z′}(l) ; z′ ∈ Γ ∩ Q_{z}((d + 1)m + 1)\}, if z ∈ Q_{x}(\frac{l}{2}).  (3.6)

In fact, starting from z = x, we can iterate (3.6) at least n = ⌊\frac{l/2}{m(d+1)+1}⌋ times to obtain

F_{x}(l) ≤ e^{−n} max{F_{z′}(l) ; z′ ∈ Γ ∩ Q_{x}(n(d + 1)m + n)}
≤ e^{−n},

which is what we want to prove.

To prove (3.6), we begin with some geometry. For z ∈ Γ, define E(z) = {−2d ≤ j ≤ 2d; z + δ_{j} ∈ Γ}. If z ∉ ∂_{int}Γ, then E(z) = {±1, …, ±d}. If z ∈ ∂_{int}Γ, then E(z) is a collection of “ inward” directions to Γ. Note that E(z) ∩ {−j, +j} ≠ ϕ for any z ∈ Γ. Furthermore, for each z ∈ Γ, we can find \overline{z} ∈ Γ such that

z ∈ \overline{z} + Q(dm, E(\overline{z})) ⊂ \overline{z} + Q((d + 1)m, E(\overline{z})) ⊂ Γ.  (3.7)

If z is near ∂_{int}Γ, we have to choose \overline{z} from ∂_{int}Γ. We set Γ(z) = \overline{z} + Q((d + 1)m, E(\overline{z})). At this point, let us make the following observation for later use;

\{y ∈ ∂_{ext}Γ(z) ; η_{y}^{1} ≠ η_{y}^{2}\} ⊂ Γ ∩ ∂_{ext}Γ(z) = ∪_{j∈E(\overline{z})}∂_{ext}^{(j)}Γ(z),

if z ∈ Q_{x}(\frac{l}{2}), η^{1} ∈ S^{Γ,ζ^{−}} and η^{2} ∈ S^{Γ,ζ^{+}}. (3.8)

This can be seen as follows. Since z ∈ Q_{x}(\frac{l}{2}), we see from (3.5) that ∂_{ext}Γ(z) ⊂ Q_{x}(l). This implies that ∂_{ext}Γ(z)\Γ is contained in Q_{x}(l)\Γ, where both η^{1} and η^{2} are identical to 0.

We will use the following coupling (cf. [BM94, Theorem 2] ); for each Δ ⊂ Γ and (η^{1}, η^{2}) ∈ S^{Γ,ζ^{−}} × S^{Γ,ζ^{+}}, there is a a probability measure μ^{Δ,η^{1},η^{2}} on the coupled configuration space S^{Δ,η^{1}} × S^{Δ,η^{2}} = {(σ^{1}, σ^{2}) ; σ^{i} ∈ S^{Δ,η^{i}}} such that (i); μ^{Δ,η^{1},η^{2}} is a coupling of μ_{β,h_{0}}^{Δ,η^{1}} and μ_{β,h_{0}}^{Δ,η^{2}}, i.e., its first and second marginals are μ_{β,h_{0}}^{Δ,η^{1}} and μ_{β,h_{0}}^{Δ,η^{2}}, respectively. (ii); If η^{1} ≤ η^{2}, then μ^{Δ,η^{1},η^{2}} is above diagonal in that μ^{Δ,η^{1},η^{2}} {σ^{1} ≤ σ^{2}} = 1. (iii); For z ∈ Δ,

μ^{Δ,η^{1},η^{2}} \{σ_{z}^{1} ≠ σ_{z}^{2}\} = μ^{Δ,η^{1},η^{2}} \{ \begin{matrix}There is a path γ from z to a site adjacent \\ to the set \{y ∈ ∂_{ext}Δ ; η_{y}^{1} ≠ η_{y}^{2}\} such that \\ σ^{1} ≡ −1 and σ^{2} ≡ +1 on γ. \end{matrix} \} .  (3.9)

We now claim that for z ∈ Q_{x}(\frac{l}{2}) and (η^{1}, η^{2}) ∈ S^{Γ,ζ^{−}} × S^{Γ,ζ^{+}},

μ^{Γ(z),η^{1},η^{2}}\{σ_{z}^{1} ≠ σ_{z}^{2}\} \{ \begin{matrix}= 0 & if η^{1} ≡ η^{2} on Γ ∩ ∂_{ext}Γ(z), \\ ≤ exp (C_{1} − \frac{m}{C_{1}} ) , & otherwise, \end{matrix}  (3.10)

where C_{1} = C_{1}(β, d) ∈ (0, ∞) is a constant. This can be seen as follows. If η^{1} ≡ η^{2} on Γ ∩ ∂_{ext}Γ(z), then it follows from (3.8) that η^{1} ≡ η^{2} on ∂_{ext}Γ(z), which implies μ^{Γ(z),η^{1},η^{2}}{σ_{z}^{1} ≠ σ_{z}^{2}} = 0 by (3.9). If η^{1} ≢ η^{2} on Γ ∩ ∂_{ext}Γ(z), we can use (3.8) and (3.9) to see that

μ^{Γ(z),η^{1},η^{2}}\{σ_{z}^{1} ≠ σ_{z}^{2}\} ≤ μ^{Γ(z),η^{1},η^{2}} \{ \begin{matrix}There is a path γ from z to a site adjacent \\ to the set Γ ∩ ∂_{ext}Γ(z) such that \\ σ^{1} ≡ −1 and σ^{2} ≡ +1 on γ. \end{matrix} \}

≤ μ_{β,h}^{Γ(z),η^{1}} \{ \begin{matrix}There is a (−)-path γ from z to a site \\ adjacent to the set Γ ∩ ∂_{ext}Γ(z) \end{matrix} \}
≤ μ_{β,1/m}^{Γ(z),(−;E(\overline{z}))} \{ \begin{matrix}There is a (−)-path γ from z to a site \\ adjacent to the set Γ ∩ ∂_{ext}Γ(z) \end{matrix} \} , (3.11)

where we have used the FKG inequality in passage to the last line. By (3.7) and Proposition 2.4,the probability appearing in (3.11) is bounded from above by exp (C_{1} − \frac{m}{C_{1}} ) with some C_{1} = C_{1}(d, β) ∈ (0, ∞). This proves (3.10). We finally define a measure \overline{μ}^{Γ} on S^{Γ,ζ^{−}} × S^{Γ,ζ^{+}} by

\overline{μ}^{Γ}\{(σ^{1}, σ^{2})\} = ∑_{{(η^{1},η^{2}) \atop η^{i} ≡ σ^{i} on Γ\Γ(z)}} μ^{Γ,ζ^{+},ζ^{−}}\{(η^{1}, η^{2})\}μ^{Γ(z),η^{1},η^{2}}\{(σ^{1}, σ^{2})\}.

Then \overline{μ}^{Γ} is a coupling of μ_{β,h_{0}}^{Γ,ζ^{∓}} and is above diagonal. We thus have that

F_{z}(l) = \overline{μ}^{Γ}\{σ_{z}^{1} ≠ σ_{z}^{2}\}
= ∑_{(η^{1},η^{2})} μ^{Γ,ζ^{+},ζ^{−}}{(η^{1}, η^{2})}μ^{Γ(z),η^{1},η^{2}}{σ_{z}^{1} ≠ σ_{z}^{2}}.  (3.12)

When z ∈ Q_{x}(\frac{l}{2}), we can plug (3.10) in (3.12) to obtain

F_{z}(l) ≤ 2d(2(d + 1)m)^{d−1} exp (C_{1} − \frac{m}{C_{1}} ) max\{F_{z′}(l) ; z′ ∈ Γ ∩ Q_{z}((d + 1)m + 1)\},

which proves (3.6) by taking m large enough.

Q. E. D.

4 Proof of Proposition 2.4

In this section, we sketch the proof of Proposition 2.4, which is an extension of [Sch94a, Theorem 3]. Although the proof requires fairly long sequence of elaborate arguments and estimates as can be seen from that of [Sch94a], the same argument as in that paper works in the generality we need, with some modifications. So, we do not go further than to present some key lemmata, in which we indicate some essential changes we have to make to generalize the techniques in [Sch94a]. We begin by introducing some notations.

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

The number of bonds contained in a set γ ⊂ B will be denoted by |γ|.

A contour is a finite subset γ ⊂ B with the following properties; there exists a finite subset Θ ⊂ Z^{d} such that both Θ and Θ^{c} is connected and that (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 in the sense explained in the bottom paragraph of [Sch94a, page 7], it follows that for each b ∈ B and m = 1, 2, …,

♯{γ : contour with |γ| = m and γ ∋ b } ≤ exp ((m − 1)κ_{1}) ,  (4.1)

for some κ_{1} = κ_{1}(d) ∈ (0, ∞) (See (4.24) in [Gri89], for example). For 1 ≤ l and a finite set S ⊂ B, let F_{l}(S) be the all possible choice of a family {γ_{i}}_{i=1}^{k} (k = 1, …, l) of contours such that ∑_{i=1}^{k} |γ_{i}| = l and γ_{i} ∩ S ≠ ϕ for i = 1, …, k. Starting from (4.1), it is not difficult to prove that

♯F_{l}(S) ≤ exp ((|S| + l)κ_{2})  (4.2)

for some κ_{2} = κ_{2}(d) ∈ (0, ∞) ( [Sch94a, page 7–8]). 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 ∈ S^{Λ}, ε = + or − and Λ ⊂ Z^{d}, a contour γ is said to be an (ε)-contour in Λ at σ if it satisfies

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

A contour γ is said to be a contour in Λ at σ ∈ S^{Λ} if it is either (+)-contour in Λ at σ or (−)-contour in Λ at σ. We take E ⊂ {±1, …, ±d} such that (2.7) holds. For a contour γ, we define

|γ|_{E} = ∑_{{1≤j≤d \atop \{−j,j\}⊂E}} |γ^{j}| + ∑_{{1≤j≤d \atop \{−j,j\}⊄E}} \frac{|γ^{j}|}{2},  (4.3)

where γ^{j} = {b ∈ γ; b is parallel to x^{j}-axis.}. We will need the following isoperimetric inequality;

|γ|_{E} ≥ d2^{\frac{|E|}{d} −1}|Θ(γ)|^{\frac{d−1}{d}} .  (4.4)

The constant d2^{\frac{|E|}{d} −1} is sharp. In fact, (4.4) becomes equality for a rectangular contour γ of the form; Θ(γ) = Q(l; E), (l = 1, 2, …,) We will need exactly the best constant in (4.4), rather than just the existence of some multiplicative constant for which an inequality of the form (4.4) is true.

Lemma 4.1 For a box Λ and contours {γ_{i}}_{i=1}^{k} in Λ, such that Θ(γ_{i}) ∩ Θ(γ_{j}) = ϕ if i ≠ j, define

R_{Λ,c,E} = \{σ ∈ S^{Λ} ; |Θ(γ)| ≤ (c2^{\frac{|E|}{d} −1})^{d} for all contour in Λ at σ \},  (4.5)
S^{Λ}(γ_{1}, …, γ_{k}) = {σ ∈ S^{Λ} ; {γ_{i}}_{i=1}^{k} appear as (+)-contours in Λ at σ }. (4.6)

Then

μ_{β,h}^{Λ,(−;E)} (S^{Λ}(γ_{1}, …, γ_{k})|R_{Λ,c,E}) ≤ exp (−2β (1 − \frac{ch}{d}) ∑_{i=1}^{k} |γ_{i}|_{E}) .  (4.7)

Proof: We first prove the following energy gap estimate; For σ ∈ R_{Λ,c,E} ∩ S^{Λ}(γ_{1}, …, γ_{k})

H^{Λ,(−;E)}(σ) − H^{Λ,(−;E)}(T σ) ≥ 2β (1 − \frac{ch}{d}) ∑_{i=1}^{k} |γ_{i}|_{E},  (4.8)

where T σ ∈ S^{Λ} is defined by

T σ_{x} = \{ \begin{matrix}−σ_{x}, & if x ∈ ∪_{i=1}^{k}Θ(γ_{i}) \\ σ_{x}, & if x ∉ ∪_{i=1}^{k}Θ(γ_{i}). \end{matrix}  (4.9)

Clearly,

H^{Λ,(−;E)}(σ) − H^{Λ,(−;E)}(T σ) ≥ 2β ∑_{i=1}^{k}(|γ_{i}|_{E} − h|Θ(γ_{i})|).  (4.10)

On the other hand, by the definition of R_{Λ,c,E} and the isoperimetric inequality (4.4),

|Θ(γ_{i})| = |Θ(γ_{i})|^{\frac{1}{d}} · |Θ(γ_{i})|^{\frac{d−1}{d}}
≤ c2^{\frac{|E|}{d} −1} · \frac{2^{1− \frac{|E|}{d}}}{d} |γ_{i}|_{E}
= \frac{c}{d}|γ_{i}|_{E}.  (4.11)

Plugging this into (4.10), we get (4.8). Once (4.8) is proved, the rest of the proof is the same as that of [Sch94a, Lemma3], hence is omitted.

Q. E. D.

Lemma 4.2 Suppose that β > \frac{κ_{2}}{2} , where κ_{2} is a constant in (4.2) Then for any 0 < A < d(1 − \frac{κ_{2}}{2β}),

\overline{lim_{h↘0}}h^{d−1} log μ^{Q(\frac{B}{h};E),(−;E)} (R_{Q(\frac{B}{h};E), \frac{A}{h} ,E}) ≤ −2^{|E|−d} (m^{(+;E)}(β)B^{d} − dB^{d−1}) ,  (4.12)

where

m^{(+;E)}(β) = inf_{l≥1} inf_{z∈Q(l;E)} μ_{β,0}^{Q(l;E),(+;E)}(σ_{z}).  (4.13)

Proof: We have from the same argument as [Sch94a, Lemma 8] that

\overline{lim_{h↘0}}h^{d−1} log μ_{β,h}^{Q(\frac{B}{h};E),(−;E)} (R_{Q(\frac{B}{h};E), \frac{A}{h} ,E})

≤ \overline{lim_{h↘0}}h^{d−1} \{m^{(+;E)}(β)h |Q (\frac{B}{h}; E)| − |∂Q (\frac{B}{h}; E)|_{E}\} .  (4.14)

Since

|Q (\frac{B}{h}; E)| ∼ 2^{|E|−d} (\frac{B}{h} )^{d} ,

|∂Q (\frac{B}{h}; E)|_{E} ∼ d2^{|E|−d} (\frac{B}{h} )^{d−1}

as h ↘ 0, the RHS of (4.14) is identified with that of (4.12).

Q. E. D.

Lemma 4.3 Suppose that two strictly monotone sequences B_{n} ↘ d and β_{n} ↗ ∞ satisfy B_{n}m^{(+;E)}(β_{n}) > d for each n = 1, 2, … (cf. (4.13)). Then for each ε > 0, there is n_{0} ≥ 1 depending on the choice of ε, {B_{n}}_{n=1}^{∞} and {β_{n}}_{n=1}^{∞} such that for each n ≥ n_{0} and β ≥ β_{n}

\overline{lim_{h↘0}}h^{d−1} log μ_{β,h}^{Q(\frac{Bn}{h};E),(−;E)} (|C (\frac{B_{n}}{h}; E)| ≥ (\frac{ε}{h})^{d}) < 0,  (4.15)

where C (\frac{B_{n}}{h}; E) is the set of sites in Q (\frac{B_{n}}{h}; E) which are connected to a site adjacent to

∪_{j∈E}∂_{ext}^{(j)}Q(\frac{B_{n}}{h}; E) by a (−)-path.

Proof: Since d (1 − \frac{κ_{2}}{2β_{n}} ) ↗ d ↙ B_{n} as n ↗ ∞, we can pick n_{0} ≥ 1 and 0 < A_{n} < d (1 − \frac{κ_{2}}{2β_{n}} ) such that

2^{|E|−d}B_{n}^{d} ≤ 2^{|E|−d}A_{n}^{d} + ε^{d}

for n ≥ n_{0}. Suppose that |C (\frac{B_{n}}{h}; E) (σ)| ≥ (\frac{ε}{h})^{d} and that γ = γ(σ) is the (+)-contour in Q(\frac{B_{n}}{h}; E) which maximaizes |Θ(γ)|. Then, σ ∈ R_{Q( \frac{Bn}{h};E); \frac{An}{h} ,E}, since

|Θ(γ)| ≤ |Q (\frac{B_{n}}{h} , E)| − |C (ε, E) (σ)|
≤ 2^{|E|−d} (\frac{B_{n}}{h} )^{d} − (\frac{ε}{h})^{d}
≤ 2^{|E|−d} (\frac{A_{n}}{h} )^{d} .

Therefore,

μ_{β,h}^{Q(\frac{Bn}{h};E),(−;E)} (|C (\frac{B_{n}}{h}; E)| ≥ (\frac{ε}{h})^{d}) ≤ μ_{β,h}^{Q(\frac{Bn}{h};E),(−;E)} (R_{\frac{Bn}{h} , \frac{An}{h} ,E}) .

(4.15) now follows from Lemma 4.2.

Q. E. D.

The rest of the proof of Proposition 2.4 is sketched as follows. It is enough to consider the case A < d. In fact, the case d ≤ A can easily be reduced to the case of A < d as follows. Define A′, B′ and h′ by \frac{A′}{A} = \frac{B′}{B} = \frac{h′}{h} = \frac{2d}{A+B} ≤ 1. Then, A′ < d < B′ and

μ_{β,h}^{Q( \frac{B}{h};E),(E;−)} (B (\frac{A}{h}; E)) ≤ μ_{β,h′}^{Q( \frac{B′}{h′};E),(E;−)} (B (\frac{A′}{h′}; E))

by the FKG inequality. For A < d, we consider μ^{Q( \frac{B}{h};E),(E;−)}-probabilities of following two events separatedly;

\{|C (\frac{d}{h}; E)| ≥ (\frac{ε}{h})^{d}\} , B (\frac{A}{h}; E) ∩ \{|C (\frac{d}{h}; E)| ≤ (\frac{ε}{h})^{d}\} .

By Lemma 4.3, the probability of the first event has an estimate of the form C exp(−\frac{C}{h^{d−1}} ) for large enough β. For the second probability, we can proceed in line with [Sch94a, Lemma 10, 11] to obtain an upper bound of the form C exp(−\frac{C}{h}) if ε is small and β is large. Putting these two together, we get Proposition 2.4.

We will use the following variational characterization of of the spectral gap;

gap_{β,h}^{Λ,ω} = inf \{\frac{−μ^{Λ,ω}(fA^{Λ,ω}f)}{μ^{Λ,ω}(|f − μ^{Λ,ω}f|^{2})} ; f ∈ C_{Λ}\} .  (5.1)

To describe an appropriate trial function f ∈ C_{Λ} to plug in, we make the following definitions. For c > 0 and a box Λ, define

R_{Λ,c} = {σ ∈ S^{Λ} ; |Θ(γ)| ≤ (2c)^{d} for all (+)-contour in Λ at σ },  (5.2)
∂R_{Λ,c} = {σ ∈ R_{Λ,c} ; σ^{x} ∉ R_{Λ,c} for some x ∈ Λ}.  (5.3)

Now, by plugging the indicator function χ of the event R_{Q( \frac{B}{h}), \frac{A}{h}} (0 < A < B) in (5.1), we obtain the following estimate of the spectral gap, which is reminiscient of the use of Cheeger constant in differential geometry.

gap_{β,h}^{Q( \frac{B}{h}),0} ≤ \frac{−μ^{Q( \frac{B}{h}),0}(χA^{Q( \frac{B}{h}),0}χ)}{μ^{Q( \frac{B}{h}),0}(|χ − μ^{Q( \frac{B}{h}),0}χ|^{2})}
≤ \frac{C(d, β)}{μ^{Q( \frac{B}{h}),0} (R_{Q( \frac{B}{h}), \frac{A}{h}} ) μ^{Q( \frac{B}{h}),0} (Rc_{Q( \frac{B}{h}), \frac{A}{h}} )}

· ∑_{x∈Q( \frac{B}{h})} ∑_{σ∈R_{Q( \frac{B}{h}), \frac{A}{h}}} μ^{Q( \frac{B}{h}),0}(σ)|χ(σ^{x}) − 1|

≤ \frac{C(d, β)|Q(\frac{B}{h})|μ^{Q( \frac{B}{h}),0} (∂R_{Q( \frac{B}{h}), \frac{A}{h}} |R_{Q( \frac{B}{h}), \frac{A}{h}} )}{μ^{Q( \frac{B}{h}),0} (Rc_{Q( \frac{B}{h}), \frac{A}{h}} )}.  (5.4)

We will estimate the probabilities in the numerator and the denominator of (5.4) in a series of lemmata.

Lemma 5.1 For a box Λ and contours {γ_{i}}_{i=1}^{k} in Λ, such that Θ(γ_{i}) ∩ Θ(γ_{j}) = ϕ if i ≠ j, recall that we have defined an event S^{Λ}(γ_{1}, …, γ_{k}) by (4.6). Suppose now that 0 < A < \frac{d}{2(d+1)}. Then for any B ≥ 1 and h ∈ (0, 1)

μ_{β,h}^{Q( \frac{B}{h}),0} (S^{Q( \frac{B}{h})}(γ_{1}, …, γ_{k})|R_{Q( \frac{B}{h}), \frac{A}{h}} ) ≤ exp \{−β (1 − \frac{2A(d + 1)}{d}) ∑_{i=1}^{k} |γ_{i}|\} .  (5.5)

Proof: We first prove the following; for σ ∈ R_{Q( \frac{B}{h}), \frac{A}{h}} ∩ S^{Q( \frac{B}{h})}(γ_{1}, …, γ_{k})

H^{Q( \frac{B}{h}),0}(σ) − H^{Q( \frac{B}{h}),0}(T σ) ≥ β (1 − \frac{2A(d + 1)}{d}) ∑_{i=1}^{k} |γ_{i}|,  (5.6)

where T σ ∈ S^{Λ} is defined by (4.9). Clearly,

H^{Λ,0}(σ) − H^{Λ,0}(T σ) ≥ 2β ∑_{i=1}^{k}(|γ_{i} ∩ B_{Λ}|_{E} − h|Θ(γ_{i})|).  (5.7)

Since |Θ(γ_{i})| ≤ A^{d}|Q(\frac{B}{h})|, we see from (a careful reading of the proof of) [T89, Lemma A.1] that

|γ_{i} ∩ ∂Q (\frac{B}{h} )| ≤ (\frac{1}{2} + A) |γ_{i}|.  (5.8)

On the other hand, we have from |Θ(γ_{i})| ≤ (\frac{2A}{h})^{d} that |Θ(γ_{i})| ≤ \frac{A}{dh}|γ_{i}|, (cf. the proof of (4.11)). Plugging this and (5.8)into (5.7), we get (5.6). Once (5.6) is proved, the rest of the proof is the same as that of [Sch94a, Lemma3], hence is omitted.

Q. E. D.

Lemma 5.2 Suppose that 0 < A < \frac{d}{4(d+1)} and that β (\frac{1}{2} − \frac{2A(d+1)}{d}) > κ_{2}, where κ_{2} is the constant in (4.2). Then, there is C(d, β) ∈ (0, ∞) such that for any B ≥ 1 and h ∈ (0, 1),

μ_{β,h}^{Q( \frac{B}{h}),0} (∂R_{Q( \frac{B}{h}), \frac{A}{h}} |R_{Q( \frac{B}{h}), \frac{A}{h}} ) ≤ C(d, β) (\frac{2B}{h} )^{d−1} exp \{−βd (\frac{2A}{h})^{d−1}\} .  (5.9)

Proof: We begin by observing how a configuration in ∂R_{Λ,c} looks like. For σ ∈ R_{Λ,c}, σ^{x} ∉ R_{Λ,c} is possible only when the following occurs; there is a colloection of (+)-contours {γ_{i}}_{i=1}^{k} in Λ at σ for some k = 1, 2, … such that γ_{i} ∋ x (1 ≤ i ≤ k) and such that the flipping of the spin at x makes them coalesce into a new (+)-contour γ in Λ at σ^{x} with |Θ(γ)| > (2c)^{d}. We then have Θ(γ) = {x} ∪ (∪_{i=1}^{k}Θ(γ_{i})), and thus

∑_{i=1}^{k} |Θ(γ_{i})| ≥ (2c)^{d} − 1.  (5.10)

Since |Θ(γ_{i})| ≤ (2c)^{d} for 1 ≤ i ≤ k, we have that |Θ(γ_{i})| ≤ \frac{c|γ_{i}|}{d} (cf. the proof of (4.11)). Combining this and (5.10), we obtain

∑_{i=1}^{k} |γ_{i}| ≥ \frac{d}{c}((2c)^{d} − 1).  (5.11)

From what we have observed above, we see

μ_{β,h}^{Q( \frac{B}{h}),0} (∂R_{Q( \frac{B}{h}), \frac{A}{h}} |R_{Q( \frac{B}{h}), \frac{A}{h}} )
≤ ∑_{x∈Q( \frac{B}{h})} ∑_{l≥l_{0}( \frac{A}{h})} ∑_{\{γ_{i}\}∈F_{l}(x)} μ_{β,h}^{Q( \frac{B}{h}),0} (S^{Q( \frac{B}{h})}(\{γ_{i}\})|R_{Q( \frac{B}{h}), \frac{A}{h}} )  (5.12)

where l_{0}(\frac{A}{h}) = 2d\{(\frac{2A}{h})^{d−1} − \frac{1}{2A}\} and F_{l}(x) stands for all possible choice of a family \{γ_{i}\} of contours in Q(\frac{B}{h}) such that ∑_{i=1}^{k} |γ_{i}| = l and γ_{i} ∋ x. By (4.2), Lemma 5.1 and our assumption on A and β,

∑_{\{γ_{i}\}∈F_{l}(x)} μ_{β,h}^{Q( \frac{B}{h}),0} (S^{Q( \frac{B}{h})}(\{γ_{i}\})|R_{Q( \frac{B}{h}), \frac{A}{h}} )

≤ exp \{κ_{2}(l + 1) − β (1 − \frac{2A(d + 1)}{d}) l\}

≤ exp (κ_{2} − \frac{βl}{2} ) .

From this and (5.12), it is straightforward to conclude (5.9).

Q. E. D.

Lemma 5.3 For any ε ∈ (0, 1) and 0 < A < \frac{d}{2(d+1)}, there is β_{0}(d, ε, A) ∈ (0, ∞) such that for β ≥ β_{0}

sup_{{B≥1 \atop h∈(0,1)}} μ_{β,h}^{Q( \frac{B}{h}),0} (R_{Q( \frac{B}{h}), \frac{A}{h}} ) ≤ \frac{1 + ε}{2}.  (5.13)

Proof: We start with the following obvious inequality;

μ_{β,h}^{Q( \frac{B}{h}),0} (R_{Q( \frac{B}{h}), \frac{A}{h}} ) ≤ μ_{β,h}^{Q( \frac{B}{h}),0} \{σ_{0} = −1\} + μ_{β,h}^{Q( \frac{B}{h}),0} (\{σ_{0} = 1\} ∩ R_{Q( \frac{B}{h}), \frac{A}{h}} ) .  (5.14)

The first probability in the RHS of (5.14) is less than 1/2 since h > 0. In a configuration of R_{Q( \frac{B}{h}), \frac{A}{h}} ∩ \{σ_{0} = 1\}, the origin must be surrounded by a (+)-contour γ in Q(\frac{B}{h}). Since γ neccessarily crosses the positive half of the x^{1}-axis, we have from Lemma 5.1 and (4.1) that

μ_{β,h}^{Q( \frac{B}{h}),0} (\{σ_{0} = 1\} ∩ R_{Q( \frac{B}{h}), \frac{A}{h}} )

≤ ∑_{m=1}^{∞} ∑_{s≥m} ∑_{{γ \atop γ∋(m,0,...,0), |γ|=s}} μ_{β,h}^{Q( \frac{B}{h}),0} (S^{Q( \frac{B}{h})}(γ) ∩ R_{Q( \frac{B}{h}), \frac{A}{h}} )

≤ ∑_{m=1}^{∞} ∑_{s≥m} ∑_{{γ \atop γ∋(m,0,...,0), |γ|=s}} exp \{−β (1 − \frac{2A(d + 1)}{d}) s\}

≤ ∑_{m=1}^{∞} ∑_{s≥m} exp \{sκ_{1} − β (1 − \frac{2A(d + 1)}{d}) s\} ,

which converges to 0 as β ↗ ∞, uniformly in B and h. This completes the proof of the lemma.

Q. E. D.

We now conclude the proof of Proposition 2.5. We take A = \frac{d}{8(d+1)}. Then there is β_{0}(d) ∈ (0, ∞) such that (5.9) holds for β ≥ β_{0}. On the other hand, we see from Lemma 5.3 that, there is β_{1}(d) ∈ (0, ∞) such that for β ≥ β_{1}

inf_{{B≥1 \atop h∈(0,1)}} μ_{β,h}^{Q( \frac{B}{h}),0} (Rc_{Q( \frac{B}{h}), \frac{A}{h}} ) ≥ \frac{1}{3}.  (5.15)

Therefore, for β ≥ max{β_{0}, β_{1}}, we may plug (5.9) and (5.15) into (5.4) to get (2.10).

Q. E. D.

Acknowledgement: The author thanks two anonymous referees for valuable comments.

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