The spectral gap of the 2-D stochastic Ising model with mixed boundary conditions

Preliminary Draft

Kenneth S. Alexander^{1} and Nobuo Yoshida^{2}

Abstract

We establish upper bounds for the spectral gap of the stochastic Ising model at low temperatures in an l × l box with boundary conditions which are not purely plus or minus; specifically, we assume the magnitude of the sum of the boundary spins over each interval of length l in the boundary is bounded by δl, where δ < 1. We show that for any such boundary condition, when the temperature is sufficiently low (depending on δ), the spectral gap decreases exponentially in l.

1 Introduction

1.1 General background and heuristics

We begin with an informal description; full definitions will be given below. Consider the stochastic Ising model (Glauber dynamics) in an l ×l box Λ(l), below the critical temperature. At equilibrium, the typical configuration resembles one of the two infinite-volume pure phases (plus phase or minus phase) except very near the boundary. That is, the equilibrium distribution μ = μ_{Λ(l),ω}^{β} (at inverse temperature β, under boundary condition ω) is roughly either the plus phase, the minus phase or a distributional mixture of the two. This equilibrium may take a long time to be reached, if the box is large. The rate of convergence is described by the spectral gap, denoted gap(Λ(l), ω, β), which is the smallest positive eigenvalue of the negative of the generator of the dynamics. More precisely, for S(·) the associated semigroup and ∥ · ∥_{μ} the L^{2}(μ) norm, gap(Λ(l), ω, β) is the largest constant Δ such that

∥S(t)f − ∫ f dμ∥_{μ} ≤ ∥f − ∫ f dμ∥_{μ}e^{−Δt} for all f ∈ L^{2}(μ) and t ≥ 0.

For pure boundary conditions, say all plus, at subcritical temperatures the spectral gap is believed to be of order l^{−2} [FH87]. The spectral gap can be very sensitive to the boundary condition, however. For example, removing as few as O(log l) plus spins near each corner of Λ(l) (leaving the boudary there free, or minus) yields a gap much smaller than l^{−2}, and removing ϵl plus spins from each corner, for some positive ϵ, yields a gap which decreases exponentially in l [Al00]. These phenomena are outgrowths of the fact that the boundary conditions are not well mixed, the free boundary or minus spins being concentrated in short intervals at the corners. More mixed boundary conditions are considered in [HY97], where it is shown that if the boundary condition ω satisfies

1Department of Mathematics, University of Southern California, Los Angeles, CA 90089-1113, USA. email:

alexandr@math.usc.edu. Research supported by NSF grant DMS-9802368.

2Division of Mathematics, Graduate School of Science, Kyoto University, Kyoto 606-8502, Japan.

email:nobuo@kusm.kyoto-u.ac.jp

| ∑_{y∈I} ω_{y}| ≤ δl/2 for every interval I in ∂_{ex}Λ(l)  (1.1)

with δ < 1, then

gap(Λ(l), ω, β) ≤ B_{1.2} exp (−βl/C_{1.2}) , l = 1, 2, ⋯,  (1.2)

where B_{1.2} = B_{1.2}(β) > 0 and C_{1.2} > 0. Here ∂_{ex}Λ(l) denotes the exterior boundary; see (1.8). One can allow the boundary spins ω_{y} to take values in the continuum [−1, 1], with ω_{y} = 0 representing the free boundary condition at site y. The condition (1.1) is somewhat restrictive, however; for example, it does not allow the long intervals of boundary plus spins which appear in the above-mentioned results from [Al00]. In this paper we establish (1.2) under a “mixed boundary” hypothesis much weaker than (1.1).

The importance of the geometry of boundary spin locations can be seen in comparing the result in [Al00], giving exponential decay of the gap when ϵl plus spins are removed at each corner, to a result of Martinelli [Mar94] which states that when one side of the square has all-plus boundary condition, and the other 3 sides have free boundary, at sufficiently low temperatures one has

exp (−C(β, ϵ)l^{\frac{1}{2} +ϵ}) ≤ gap(Λ(l), ω, β) for ϵ > 0, l = 1, 2, · · · .  (1.3)

In the latter case there are many fewer plus spins but the gap is much larger, meaning the convergence to the equilibrium plus phase is much faster.

The heuristics of the gap are rooted in the ideas of energy barriers and traps. From certain starting configurations, to reach a typical equilibrium configuration, one must pass through a set of configurations for each of which the total energy is greater than either the typical starting or equilibrium total energies. An energy barrier is such a set of high-energy configurations; the height of the barrier is the typical additional energy of the barrier configurations relative to the starting configurations. A trap is a set of starting configurations from which one cannot reach equilibrium without crossing an energy barrier. (We do not make formal definitions here, as we will not use these concepts other than descriptively.) Typically one expects the gap to be exponentially small in the height of the energy barrier that must be crossed, for a trap of which the probability is “not too small.” Often traps are related to the existence of macroscopic regions of the “wrong phase,” that is, say, regions of minus phase when the equilibrium is purely the plus phase. For example, in the “corners-removed” context of [Al00], a trap is formed by the configurations in which there is an “X” of minus phase connecting the four free-boundary corner regions, and the height of the associated energy barrier is proportional to the length of the corner regions. In the above “three-sides-free” example of Martinelli, however, say with the plus spins on the right side of the square, there is no real energy barrier because, starting from the minus phase, a region of plus phase can sweep leftward, maintaining an approximately vertical interface, until it covers the full square.

Consider now a boundary condition ω which is “well-mixed” in the sense that

| ∑_{y∈I} ω_{y}| ≤ δ|I|  (1.4)

for every “sufficiently long” interval I in the boundary of Λ(l), with δ < 1, and suppose ω favors the plus phase (more precisely, the magnetization at the center of the square is nonnegative.) If the system is started entirely in the minus phase, we expect the region of minus phase (the “droplet”) to pull away from the boundary and then shrink to nothing, at which time equilibrium is essentially reached. When the droplet initially fills Λ(l), the energy associated to its surface (this surface being essentially ∂_{ex}Λ(l)) is at most 8δl, by (1.4). When the droplet has pulled only slightly away from the boundary, however, the surface energy becomes essentially twice the droplet boundary length (provided the temperature is very low), hence is at least about 8l. Thus there is an energy barrier; the droplet will tend to stick to the boundary, meaning the minus phase is a trap. Though we do not make these particular heuristics rigorous in our proofs, they are what underlie our main result.

For fixed ω satisfying (1.4), at higher but still subcritical temperatures, one does not expect this phenomenon of sticking to the boundary to occur. This is because the surface energy (appropriately defined using surface tension and coarse-graining) of the droplet is no longer essentially twice its length; a diagonal interface has significantly less surface energy than combined horizontal and vertical interfaces having the same endpoints. This means the droplet should be able to pull away from the boundary, first from the corners, without the crossing of an energy barrier. We will not investigate this type of behavior here.

Additional existing results at subcritical temperatures include the following. Thomas [Tho89] proved that in general dimension d, for free boundary conditions (ω ≡ 0), for sufficiently large β,

gap(Λ(l), ω, β) ≤ B_{1.5} exp (−βl^{d−1}/C_{1.5}) l = 1, 2, ⋯,  (1.5)

where B_{1.5} = B_{1.5}(β, d) > 0 and C_{1.5} = C_{1.5}(d) > 0. For d = 2, Cesi et al [CGMS96] prove (1.5) with ω ≡ 0 for all β > β_{c}, where β_{c} is the inverse critical temparature. For d = 2, in contrast with (1.5), it is known that for β > β_{c} and ω ≡ +1,

exp (−φ(l)) ≤ gap(Λ(l), ω, β), l = 1, 2, ⋯,  (1.6)

with a function φ(l) = o(l^{\frac{1}{2} +ϵ}) as l ↗ ∞, for all ϵ > 0. This result was first obtained by F. Martinelli [Mar94, Mar97]. More recently, Y. Higuchi and J. Wang [HW99] showed (1.6) with

φ(l) = C(β)(l ln l)^{\frac{1}{2}} . Schonmann [Sch94, Theorem 5] showed that gap(Λ(l), ω) can shrink no faster than exponential of O(l^{d−1}); specifically, the spectral gap has the following general lower bound for all d ≥ 2 and β > 0:

\underline{q}(β)l^{−d} exp (−4β ∑_{j=0}^{d−1} l^{j}) ≤ inf_{ω∈Ω_{b}} gap(Λ(l), ω, β), l = 1, 2, · · · .  (1.7)

Here \underline{q}(β) is a uniform lower bound for all flip rates.

1.2 Basic definitions

The lattice. For x = (x_{1}, x_{2}) ∈ Z^{2}, we will use both the l_{1}-norm ∥x∥_{1} = |x_{1}| + |x_{2}| and the l_{∞}-norm ∥x∥_{∞} = max{|x_{1}|, |x_{2}|}. A set Λ ⊂ Z^{2} 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^{2} will be denoted respectively by

∂_{in}Λ = {x ∈ Λ; ∥x − y∥_{1} = 1 for some y ∉ Λ },
∂_{ex}Λ = {y ∉ Λ; ∥x − y∥_{1} = 1 for some x ∈ Λ }.  (1.8)

The number of points contained in a set Λ ⊂ Z^{2} will be denoted by |Λ|. We will use the notation Λ ⊂⊂ Z^{2} to indicate that Λ ⊂ Z^{2} with |Λ| < ∞. A cube with the side-length l will be denoted by

Λ(l) = (−l/2, l/2]^{d} ∩ Z^{d}.  (1.9)

An l_{∞}-connected subset of ∂_{ex}Λ(l) will be called an interval of ∂_{ex}Λ(l).

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 ⊂⊂ Z^{2},
Ω_{b} = {ω = (ω_{y})_{y∈Z^{2}} ; ω_{y} ∈ [−1, 1]}.

We are mainly interested in ω_{y} ∈ {−1, 0, 1}, but there is no extra work in allowing ω_{y} ∈ [−1, 1]. We will refer an element ω of Ω_{b} as a boundary condition. The set of all real functions on Ω_{Λ} is denoted by C_{Λ}. For Λ ⊂⊂ Z^{2} and ω ∈ Ω_{b}, the Hamiltonian H_{Λ}^{ω} : Ω_{Λ} → R is defined by

H_{Λ}^{ω}(σ) = −\frac{1}{2} ∑_{\begin{matrix}x, y ∈ Λ \\ ∥x − y∥_{1} = 1 \end{matrix}} σ_{x}σ_{y} − ∑_{\begin{matrix}x ∈ Λ, y ∉ Λ \\ ∥x − y∥_{1} = 1 \end{matrix}} σ_{x}ω_{y}.

A finite-volume Gibbs state on Λ with the boundary condition ω ∈ Ω_{b} is defined to be 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 the inverse temparature and Z_{Λ}^{ω} is the normalizing constant.

Stochastic Ising Models. For Λ ⊂⊂ Z^{2}, we consider a function q_{Λ} : Λ × Ω_{Λ} × Ω_{b} → ]0, ∞[ which satisfies the following conditions;

(i) Boundedness : There exist positive constants \underline{q}(β) and \overline{q}(β) such that

\underline{q}(β) ≤ q_{Λ}(x, σ, ω) ≤ \overline{q}(β),  (1.10)

for all Λ ⊂⊂ Z^{2}, and (x, σ, ω) ∈ Λ × Ω_{Λ} × Ω_{b}.

(ii) the Detailed Balance Condition:

q_{Λ}(x, σ, ω) exp (−βH_{Λ}^{ω}(σ)) = q_{Λ}(x, σ^{x}, ω) exp (−βH_{Λ}^{ω}(σ^{x})) ,  (1.11)

for all Λ ⊂⊂ Z^{2} and (x, σ, ω) ∈ Λ × Ω_{Λ} × Ω_{b}, where σ^{x} is the configuration obtained from σ by replacing σ_{x} by −σ_{x}.

An example of such q_{Λ}(x, σ, ω) is given by;

q_{Λ}(x, σ, ω) = exp (−(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}}) .

Now, fix Λ ⊂⊂ Z^{2} 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}.

Thus q_{Λ}(x, σ, ω) represents the rate at which the spin at x flips to the opposite spin, when the configuration is σ. It can easily be seen from (1.11) that

−μ_{Λ,ω}^{β}(fA_{Λ}^{ω}g) = \frac{1}{2} ∑_{x∈Λ} ∑_{σ∈Ω} μ_{Λ,ω}^{β}(σ)q_{Λ}(x, σ, ω)\{f(σ^{x}) − f(σ)\}\{g(σ^{x}) − g(σ)\}.  (1.12)

Next, we define

gap(Λ, ω, β) = inf \{\frac{−μ_{Λ,ω}^{β}(fA_{Λ}^{ω}f)}{μ_{Λ,ω}^{β}(|f − μ_{Λ,ω}^{β}f|^{2})} ; f ∈ C_{Λ}\} ,  (1.13)

which is the smallest positive eigenvalue of −A_{Λ}^{ω}. Considering only indicator functions in (1.13) we obtain

gap(Λ, ω, β) ≤ \frac{\overline{q}(β)}{μ_{Λ(l),ω}^{β}(Γ)μ_{Λ(l),ω}^{β}(Γ^{c})} ∑_{x∈Λ(l)} ∑_{σ∈Γ:σ^{x}∉Γ} μ_{Λ(l),ω}^{β}(σ).  (1.14)

Thus any fixed event Γ gives an upper bound for the gap. Roughly, to obtain a good bound one wants to choose Γ to be a trap.

1.3 Statement of main results

The following is our main result, improving on the condition (1.1).

Theorem 1.1 Consider a stochastic Ising model on a square Λ(l) satisfying (1.10) and (1.11). Suppose that 0 < δ < 1 and the boundary condition ω_{y} ∈ [−1, +1], y ∈ ∂_{ex}Λ(l) satisfies

| ∑_{y∈I} ω_{y}| ≤ δ|I| for every interval I ⊂ ∂_{ex}Λ(l) with |I| = l.  (1.15)

Then, there exists β_{0} = β_{0}(δ) > 0 such that

gap(Λ(l), ω, β) ≤ B_{1.16} exp(−βl/C_{1.16}),  (1.16)

for β ≥ β_{0} and l = 1, 2, …, where B_{1.16} = B_{1.16}(β, δ) > 0 and C_{1.16} = C_{1.16}(δ) > 0.

Condition (1.15) is much milder than (1.1). For example, (1.15) allows a boundary condition which is +1 for 99 % of the boundary with 1 % zero on each side. Moreover, condition (1.15) turns out to be optimal in the following example. For δ > 0, consider a boundary condition ω ∈ Ω_{b} defined by

ω_{x} = \{ \begin{matrix}+1 & if x_{1} = [l/2] + 1 and \frac{−δl}{2} < x_{2} ≤ \frac{δl}{2}, \\ 0 & otherwise. \end{matrix}  (1.17)

In this example, we see the transition from (1.2) to (1.3) depending on the value of δ. By Theorem 1.1, one sees that (1.2) is true for all δ < 1. On the other hand, it follows from [Mar94, Corollary 4.1] that (1.3) holds true for δ = 1.

Theorem 1.1 has the following application to random boundary conditions.

Corollary 1.2 Suppose that d = 2 and that ω_{y} ∈ [−1, 1], y ∈ Z^{2} are i.i.d. random variables with the mean m ∈ (−1, 1). Then, there exists β_{0} = β_{0}(m) > 0 as follows. For β ≥ β_{0}, there are constants B_{1.18} = B_{1.18}(β, m) > 0 and C_{1.18} = C_{1.18}(m) > 0 such that almost surely;

gap(Λ(l), ω, β) ≤ B_{1.18} exp(−βl/C_{1.18}) for l = 1, 2, ...  (1.18)

Proof of Corollary 1.2 is similar to that of [HY97, Corollary 2.2.] and hence is omitted.

2 Preliminaries for the proof of Theorem 1.1

2.1 Contours

The set B of all bonds in Z^{2} is defined by

B = {{x, y} ⊂ Z^{2} ; ∥x − y∥_{1} = 1}.

For a set Λ, we define

B_{Λ} = {{x, y} ∈ B ; (x, y) ∈ Λ^{2}},
∂B_{Λ} = {{x, y} ∈ B ; (x, y) ∈ Λ × Λ^{c}},
\overline{B}_{Λ} = B_{Λ} ∪ ∂B_{Λ}.

The dual lattice (Z^{2})^{∗} is Z^{2} shifted by (\frac{1}{2}, \frac{1}{2}); sites and bonds of this lattice are called dual sites and dual bonds. x^{∗} denotes x + (\frac{1}{2}, \frac{1}{2}). When necessary for clarity, bonds of Z^{2} are called regular bonds. To each regular bond b there is associated a unique dual bond b^{∗} which is its perpendicular bisector. For A ⊂ B we write A^{∗} for \{e^{∗} : e ∈ A\}. For γ ⊂ \overline{B}_{Λ}^{∗} we set

V (γ) = ∪_{e={x,y}:e^{∗}∈γ} {x, y}, V_{ex}(γ) = V (γ) ∩ ∂_{ex}Λ.

When convenient we view bonds and dual bonds as closed intervals in R^{2}, as when referrring to a connected set of (dual) bonds. The number of dual bonds contained in a set γ ⊂ B^{∗} will be denoted by |γ|.

For x ∈ R^{2} let Q(x) = ∏_{j=1}^{2}[x_{j} − \frac{1}{2}, x_{j} + \frac{1}{2}], and for Θ ⊂ Z^{2} let Q(Θ) = ∪_{x∈Θ} Q(x). A contour is a finite subset γ ⊂ B^{∗} which is of the form ∂Q(Θ) for some finite Θ ⊂ Z^{2} for which both Θ and Θ^{c} are l_{1}-connected. The set Θ is uniquely determined by γ and hence is denoted by Θ(γ). As is well known, for each b ∈ B and m = 1, 2, ⋯,

♯{γ : γ is a contour with |γ| = m and γ ∋ b } ≤ 3^{m−1}.  (2.1)

If a contour γ is a subset of \overline{B}_{Λ}^{∗} for some Λ ⊂ Z^{2}, we say γ is a contour in Λ. For σ ∈ Ω_{Λ}, ε = + or − and Λ ⊂ Z^{2}, an (ϵ)-cluster in Λ at σ is an l_{1}-connected component of {x ∈ Z^{2} : σ_{x} = ϵ1}. The outer boundary of a bounded subset A of R^{2} is the unique connected component of ∂A which is contained in the closure of the unique unbounded component of A^{c}. A contour γ is said to be an (ε)-contour in Λ at σ if γ is the outer boundary of Q(Θ) for some (ε)-cluster Θ. 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. Note that the boundary condition does not affect whether a given γ is an (ε)-contour in Λ, under our definition.

2.2 Reductions

We may assume slightly more restrictive condition than (1.15) to prove Theorem 1.1. First, we may replace (1.15) with

| ∑_{y∈I} ω_{y}| ≤ δ_{2.2}|I| for every interval I ⊂ ∂_{ex}Λ(l) with |I| ≥ l.  (2.2)

This is because (1.15) for a given δ implies (2.2) with δ_{2.2} = (1+δ)/2. We may next strengthen (2.2) as follows: there exists 0 < δ_{2.3} < 1 such that

| ∑_{y∈I} ω_{y}| ≤ δ_{2.3}|I| for every interval I ⊂ ∂_{ex}Λ(l) with |I| ≥ δ_{2.3}l.  (2.3)

In fact, suppose that the boundary condition ω satisfies (2.2) for some δ < 1. Let δ̃ < 1 satisfy

δ + (1 + δ)(1 − δ̃)δ̃^{−1} < δ̃ < 1.  (2.4)

Then (2.3) with δ_{2.3} = δ̃ holds. To see this, take an arbirary interval I ⊂ ∂_{ex}Λ(l) with |I| ≥ δ̃l. By expanding I, if necessary, we get an interval Ĩ ⊃ I such that |Ĩ| ≥ l and |Ĩ\I| ≤ (1 − δ̃)l ≤ (1 − δ̃)δ̃^{−1}|I|. We then have |Ĩ| ≤ (1 + (1 − δ̃)δ̃^{−1})|I|, so

|∑_{y∈I} ω_{y}| ≤ |∑_{y∈_{I}}ω_{y}| + (1 − δ̃)δ̃^{−1}|I|,
≤ δ|Ĩ| + (1 − δ̃)δ̃^{−1}|I|
≤ (δ + (1 + δ)(1 − δ̃)δ̃^{−1}) |I|
≤ δ̃|I|.

2.3 Outline of the proof of Theorem 1.1

We may and will assume (2.3). The basic strategy to prove Theorem 1.1 is rather standard [Tho89, HY97]. We define an event Γ_{l} ⊂ Ω_{Λ(l)} in which a “large” contour is present; by (1.14),

gap(Λ(l), ω, β) ≤ \frac{\overline{q}(β)}{μ_{Λ(l),ω}^{β}(Γ_{l})μ_{Λ(l),ω}^{β}(Γ_{l}^{c})} ∑_{x∈Λ(l)} ∑_{σ∈Γ_{l}:σ^{x}∉Γ_{l}} μ_{Λ(l),ω}^{β}(σ).  (2.5)

To prove Theorem 1.1, we will show that for large β, μ_{Λ(l),ω}^{β}(Γ_{l}) is uniformly positive in l (Lemma 2.1 below) and

μ_{Λ(l),ω}^{β}(Γ_{l}^{c})^{−1} ∑_{x∈Λ(l)} ∑_{σ∈Γ_{l},σ^{x}∉Γ_{l}} μ_{Λ(l),ω}^{β}(σ)

is exponentially small in l (Lemma 2.2 below).

To define Γ_{l}, we choose ϵ_{l,ω} = ± as follows;

ϵ_{l,ω} = \{ \begin{matrix}+ & if μ_{Λ(l),ω}^{β}(σ_{0}) ≥ 0, \\ − & if μ_{Λ(l),ω}^{β}(σ_{0}) < 0. \end{matrix}  (2.6)

We fix δ_{1} such that δ_{2.3} < δ_{1} < 1. The event Γ_{l} is defined by

Γ_{l} = {σ ∈ Ω_{Λ(l)} ; C_{l}(σ) ≠ ∅},  (2.7)

where

C_{l}(σ) = \{γ ; \begin{matrix}γ is an (ϵ_{l,ω})-contour in Λ(l) at σ such that \\ γ ∩ ∂Q(Λ(l)) ≠ ∅ and |γ| ≥ 2δ_{1}l \end{matrix} \} .  (2.8)

To bound (2.5) from above, we use the following two lemmas.

Lemma 2.1 Assume (2.2). There exists β_{1} = β_{1}(δ_{2.3}) > 0 such that

inf_{l≥1} μ_{Λ(l),ω}^{β}(Γ_{l}) ≥ \frac{1}{3} for β ≥ β_{1}.  (2.9)

Lemma 2.2 Assume (2.2). There exists β_{2} = β_{2}(δ_{2.3}) > 0 such that

∑_{x∈Λ(l)} ∑_{σ∈Γ_{l}:σ^{x}∉Γ_{l}} μ_{Λ(l),ω}^{β}(σ) ≤ μ_{Λ(l),ω}^{β}(Γ_{l}^{c})B_{2.10} exp (−βl/C_{2.10})  (2.10)

for β ≥ β_{2} and l = 1, 2, …, where B_{2.10} = B_{2.10}(β, δ_{2.3}) > 0 and C_{2.10} = C_{2.10}(δ_{2.3}) > 0.

Theorem 1.1 follows immediately by plugging (2.9) and (2.10) into (2.5). In fact, we have for β ≥ max{β_{1}, β_{2}} that

gap(Λ(l), ω, β) ≤ 3\overline{q}(β)B_{2.10} exp (−βl/C_{2.10}) .  (2.11)

□ 3 Proof of Lemmas 2.1 and 2.2

3.1 Energy estimates for contours

The proofs of Lemma 2.1 and Lemma 2.2 are based on energy estimates for contours, which we present in this subsection. We have to introduce additional definitions. The right, left, top and bottom sides of the square Q(Λ(l)) are denoted by F_{l}^{+1}, F_{l}^{−1}, F_{l}^{+2} and F_{l}^{−2}, respectively. A set of dual bonds γ ⊂ \overline{B}_{Λ}^{∗} is said to be horizontally crossing if

A ∩ F_{l}^{1} ≠ ∅ and A ∩ F_{l}^{−1} ≠ ∅.  (3.1)

Similarly, γ is said to be vertically crossing if

A ∩ F_{l}^{2} ≠ ∅ and A ∩ F_{l}^{−2} ≠ ∅.  (3.2)

The set A is said to be crossing if it is either horizontally crossing or vertically crossing. Suppose that γ_{1}, …, γ_{p} are contours in Λ(l). We set

Δ_{γ_{1},...,γ_{p}}H_{Λ(l)}^{ω}(σ) = H_{Λ(l)}^{ω}(σ) − H_{Λ(l)}^{ω}(T_{γ_{1}} ∘ ⋯ ∘ T_{γ_{p}}σ), σ ∈ Ω_{Λ(l)}  (3.3)

where we have defined a map T_{γ} : Ω_{Λ(l)} → Ω_{Λ(l)} for a contour γ by

(T_{γ}σ)_{x} = \{ \begin{matrix}−σ_{x}, & if x ∈ Θ(γ) \\ σ_{x}, & if x ∉ Θ(γ). \end{matrix}  (3.4)

Suppose that a contour γ is non-crossing. Then γ ∩ (F_{l}^{i} ∪ F_{l}^{j}) = ∅ for some i, j with |i| = 1 and |j| = 2. Then there exists a unique connected component \underline{γ} of γ\∂Q(Λ(l)) which divides Λ(l) into two l_{1}-connected components Θ̃ and Λ(l)\Θ̃ such that Θ(γ) ⊂ Θ̃ and F_{l}^{i} ∪ F_{l}^{j} ⊂ ∂Q (Λ(l)\Θ̃). We define \overline{γ} ⊂ ∂Q(Λ(l)) and the interval I(γ) ⊂ ∂_{ex}Λ(l) respectively by

\overline{γ} = ∂Q(Λ(l)) ∩ ∂Q(Θ̃),  (3.5)
I(γ) = V_{ex}(\overline{γ}).  (3.6)

Note that

\overline{γ} ⊃ γ ∩ ∂Q(Λ(l)).  (3.7)

Note also that bonds in \overline{γ} are in one-to-one correspondence with sites in I(γ) in an obvious way.

Lemma 3.1 a) Let γ be a a non-crossing (ϵ)-contour at a configuration σ ∈ Ω_{Λ(l)}. Then

|γ\∂Q(Λ(l))| ≥ \frac{1}{2}|γ| + \frac{1}{2}|γ\(∂Q(Λ(l)) ∪ \underline{γ})|,  (3.8)
\frac{1}{2}Δ_{γ}H_{Λ(l)}^{ω}(σ) ≥ \frac{1}{2}|γ| + \frac{1}{2}|γ\(∂Q(Λ(l)) ∪ \underline{γ})| − ϵ ∑_{y∈V_{ex}(γ)} ω_{y},  (3.9)
\frac{1}{2}Δ_{γ}H_{Λ(l)}^{ω}(σ) ≥ |\underline{γ}| − ϵ ∑_{y∈V_{ex}(\overline{γ})} ω_{y} ≥ 0.  (3.10)

b) Suppose {γ_{j}}_{j=1}^{p} are (ϵ)-contours in Λ(l) at σ such that

\frac{1}{2}Δ_{γ_{1},...,γ_{p}}H_{Λ(l)}^{ω}(σ) ≥ c_{1}l − c_{2} for some c_{i} ≥ 0 (i = 1, 2).  (3.11)

Then,

\frac{1}{2}Δ_{γ_{1},...,γ_{p}}H_{Λ(l)}^{ω}(σ) ≥ \frac{c_{1}}{c_{1}+8} ∑_{j=1}^{p} |γ_{j}| − c_{2}.  (3.12)

c) Let γ, γ_{1}, …, γ_{p} be non-crossing (ϵ)-contours at a configuration σ ∈ Ω_{Λ(l)}. Suppose that condition (2.3) is satisfied and that I is an interval in ∂_{ex}Λ(l) such that

∪_{j=1}^{p}I(γ_{j}) ⊂ I,  (3.13)
δ_{2.3}l ≤ |I| ≤ ∑_{j=1}^{p} |\underline{γ_{j}}| + c,  (3.14)

where c ≥ 0 is a constant. Then

\frac{1}{2}Δ_{γ_{1},...,γ_{p}}H_{Λ(l)}^{ω}(σ) ≥ ε_{3.15} max\{l, ∑_{j=1}^{p} |γ_{j}|\} − c,  (3.15)

where the constant ε_{3.15} > 0 depends only on δ_{2.3}.

Proof of part (a): We have the following relations:

\frac{1}{2}Δ_{γ}H_{Λ(l)}^{ω}(σ) = |γ\∂Q(Λ(l))| − ϵ ∑_{y∈V_{ex}(γ)} ω_{y},  (3.16)
|I(γ)| = |\overline{γ}| ≤ |\underline{γ}|,  (3.17)
|\overline{γ}\γ| ≤ |γ\(∂Q(Λ(l)) ∪ \underline{γ})|.  (3.18)

The equality (3.16) is obvious. The inequalities (3.17) and (3.18) can be seen from geometric considerations as follows. We decompose γ\∂Q(Λ(l)) into connected components {λ_{i}}_{i≥0}, where λ_{0} = \underline{γ}. Let τ_{0} = \overline{γ} and let \{τ_{i}\}_{i≥1} be connected components of \overline{γ}\γ. We can arrange the enumeration so that λ_{i} and τ_{i} have common endpoints for each i ≥ 0. From this observation and the fact that γ is non-crossing, it follows that |λ_{i}| ≥ |τ_{i}| for each i ≥ 0. In particular, |λ_{0}| ≥ |τ_{0}| and ∑_{i≥1} |τ_{i}| ≤ ∑_{i≥1} |λ_{i}| which prove, respectively, (3.17) and (3.18). By (3.7) and (3.17), we have that |\underline{γ}| ≥ |γ ∩ ∂Q(Λ(l))| and hence that

2|γ\∂Q(Λ(l))| = 2|\underline{γ}| + 2|γ\(∂Q(Λ(l)) ∪ \underline{γ})|
≥ |\underline{γ}| + |γ ∩ ∂Q(Λ(l))| + 2|γ\(∂Q(Λ(l)) ∪ \underline{γ})|
= |γ| + |γ\(∂Q(Λ(l)) ∪ \underline{γ})|.

This proves (3.8), which together with (3.16) implies (3.9).

On the other hand, we have by (3.18) that |\overline{γ}\γ| ≤ |γ\∂Q(Λ(l))| − |\underline{γ}| and hence that

| ∑_{y∈V_{ex}(γ)} ω_{y} − ∑_{y∈I(γ)} ω_{y}| = | ∑_{y∈V_{ex}(\overline{γ}\γ)} ω_{y}|
≤ |γ\∂Q(Λ(l))| − |\underline{γ}|.  (3.19)

The inequality (3.10) follows from (3.16) and (3.19).

Proof of part (b): Let α = 1/(c_{1} + 8). Case 1: α ∑_{j=1}^{p} |γ_{j}| ≤ l. In this case, we obviously have that

\frac{1}{2}Δ_{γ_{1},...,γ_{p}}H_{Λ(l)}^{ω}(σ) ≥ c_{1}α ∑_{j=1}^{p} |γ_{j}| − c_{2}.

Case 2: α ∑_{j=1}^{p} |γ_{j}| ≥ l. In this case,

\frac{1}{2}Δ_{γ_{1},...,γ_{p}}H_{Λ(l)}^{ω}(σ) ≥ ∑_{j=1}^{p} (|γ_{j}\∂Q(Λ(l))| − |γ_{j} ∩ ∂Q(Λ(l))|)
= ∑_{j=1}^{p} (|γ_{j}| − 2|γ_{j} ∩ ∂Q(Λ(l))|)
≥ ∑_{j=1}^{p}|γ_{j}| − 8l
≥ (1 − 8α)∑_{j=1}^{p}|γ_{j}|.

Therefore (3.12) follows.

Proof of part (c): It is enough to prove (3.11) with some c_{1} > 0 and c_{2} = c. Recall that δ_{2.3} < δ_{1} < 1. Case 1: ∑_{j=1}^{p} |I(γ_{j})| ≤ δ_{1}|I|. In this case,

\frac{1}{2}Δ_{γ_{1},...,γ_{p}}H_{Λ(l)}^{ω}(σ) ≥ ∑_{j=1}^{p} (|\underline{γ_{j}}| − |I(γ_{j})|)

≥ |I| − ∑_{j=1}^{p} |I(γ_{j})| − c
≥ (1 − δ_{1})|I| − c
≥ (1 − δ_{1})δ_{2.3}l − c,

which implies (3.11) with c_{1} = (1 − δ_{1})δ_{2.3}. Case 2: ∑_{j=1}^{p} |I(γ_{j})| ≥ δ_{1}|I|. We set A = I\ ∪_{j=1}^{p} I(γ_{j}) so that |A| ≤ (1 − δ_{1})|I|. We then have by (3.10), (3.14), (2.3) that

\frac{1}{2}Δ_{γ_{1},...,γ_{p}}H_{Λ(l)}^{ω}(σ) ≥ ∑_{j=1}^{p} (|\underline{γ_{j}}| − ϵ ∑_{y∈I(γ_{j})} ω_{y})
≥ |I| − c − ϵ ∑_{y∈I} ω_{y} − |A|
≥ |I| − c − δ_{2.3}|I| − (1 − δ_{1})|I|
≥ (δ_{1} − δ_{2.3})δ_{2.3}l − c,

which implies (3.11) with c_{1} = (δ_{1} − δ_{2.3})δ_{2.3}. □

Lemma 3.2 Let γ be an (ϵ)- contour in Λ(l) at a configuration σ.

a) If γ intersects with exactly one of the sides F_{l}^{j} (j = ±1, ±2), then

Δ_{γ}H_{Λ(l)}^{ω}(σ) ≥ \{ \begin{matrix}|\{horizontal bonds in γ\}| & if j = ±1, \\ |\{vertical bonds in γ\}| & if j = ±2. \end{matrix}  (3.20)

b) If Θ(γ) ∋ 0 and |γ| < 2l, then

Δ_{γ}H_{Λ(l)}^{ω}(σ) ≥ 2|γ|/9.  (3.21)

Proof of part (a): Suppose for example that γ intersects only with F_{l}^{1}. Then,

{horizontal bonds in γ} ⊂ γ\∂Q(Λ(l)),
|γ ∩ F_{l}^{1}| ≤ |{vertical bonds in γ\∂Λ(l)}|.

Therefore,

\frac{1}{2}Δ_{γ}H_{Λ(l)}^{ω}(σ) ≥ |\{horizontal bonds in γ\}|
+|{vertical bonds in γ\∂Q(Λ(l))}| − |γ ∩ F_{l}^{1}|
≥ |{horizontal bonds in γ}|.

Proof of part (b): If γ ∩ ∂_{ex}Λ(l) = ∅, then (3.21) is obvious. We therefore assume that γ ∩ ∂Q(Λ(l)) ≠ ∅. Since 0 ∈ Θ(γ) and |γ| < 2l, γ must intersect with exactly one of F_{l}^{j} (j = ±1, ±2). Therefore, we see from (3.20) that \frac{1}{2}Δ_{γ}H_{Λ(l)}^{ω}(σ) ≥ l, which implies (3.21) by Lemma 3.1. □

3.2 Proof of Lemma 2.1

We have that

μ_{Λ(l),ω}^{β}(Γ_{l}) ≥ μ_{Λ(l),ω}^{β}({σ_{0} = ϵ_{l,ω}1} ∩ Γ_{l})
= μ_{Λ(l),ω}^{β}({σ_{0} = ϵ_{l,ω}1}) − μ_{Λ(l),ω}^{β}({σ_{0} = ϵ_{l,ω}1} ∩ Γ_{l}^{c})
≥ \frac{1}{2} − μ_{Λ(l),ω}^{β}(\{σ_{0} = ϵ_{l,ω}1\} ∩ Γ_{l}^{c}).  (3.22)

At a configuration σ ∈ {σ_{0} = ϵ_{l,ω}1} ∩ Γ_{l}^{c}, the point 0 is enclosed by a (ϵ_{l,ω})-contour γ such that either γ ∩ ∂Q(Λ(l)) = ∅ or |γ| < 2δ_{1}l. If γ ∩ ∂Q(Λ(l)) = ∅, then

Δ_{γ}H_{Λ(l)}^{ω}(σ) = 2|γ|.

If otherwise, γ satisfies the condition for (3.21). We therefore have in both cases

Δ_{γ}H_{Λ(l)}^{ω}(σ) ≥ 2|γ|/9.

By the standard Peierl’s argument,

μ_{Λ(l),ω}^{β}({σ_{0} = ϵ_{l,ω}1} ∩ Γ_{l}^{c}) ≤ ∑_{γ} μ_{Λ(l),ω}^{β} {Δ_{γ}H_{ω} ; Λ(l)(σ) ≥ 2|γ|/9}
≤ ∑_{γ} exp (−2β|γ|/9) ,  (3.23)

where ∑_{γ} stands for summation over all contours γ which satisfies Θ(γ) ∋ 0. By using the counting inequality (2.1), we see that

lim_{β↗∞} ∑_{γ} exp (−2β|γ|/9) = 0,

which, together with (3.22) and (3.23), implies Lemma 2.1. □

3.3 Proof of Lemma 2.2

We may assume that ϵ_{l,ω} = +.

Step 1: Suppose that σ ∈ Γ_{l} and σ^{x} ∉ Γ_{l} for some x ∈ Λ(l). We consider two cases separately at first: σ_{x} = 1 and σ_{x} = −1.

Consider first σ_{x} = 1. Let γ be the outer boundary of the (+)-cluster at σ which contains x. The way the transition from σ ∈ Γ_{l} to σ^{x} ∉ Γ_{l} occurs is that the set C_{l}(σ) contains only the one element γ, and the flipping of σ_{x} shortens γ or separates γ from ∂Q(Λ(l)) or makes γ break into new shorter contours. Some of these shorter contours may include dual bonds which were not part of γ at σ, but rather were part of (−)-contours inside γ at σ. We have then

C_{l}(σ) = {γ},  (3.24)
x is in or adjacent to V (γ);  (3.25)

in fact if either (3.24) or (3.25) fails, then C_{l}(σ) = C_{l}(σ^{x}), contradicting our assumption that σ ∈ Γ_{l} and σ^{x} ∉ Γ_{l}. Further, there are (+)-contours γ_{1}, …, γ_{m} and γ′_{1}, …, γ′_{n} (m ≥ 0, n ≥ 0, 1 ≤ m+n ≤ 4) at the flipped configuration σ^{x}, and (−)-contours α_{1}, …, α_{k} (0 ≤ k ≤ 2) inside γ at σ, such that

γ_{j} ∩ ∂Q(Λ(l)) ≠ ∅, |γ_{j}| < 2δ_{1}l, for j = 1, …, m,  (3.26)
γ′_{j} ∩ ∂Q(Λ(l)) = ∅, for j = 1, …, n,  (3.27)
(γ ∪ (∪_{j=1}^{k}α_{j})) △ ((∪_{j=1}^{m}γ_{j}) ∪ (∪_{j=1}^{n}γ′_{j})) ⊂ ∂Q(x),  (3.28)
Θ(γ)△ ((∪_{j=1}^{m}Θ(γ_{j})) ∪ (∪_{j=1}^{n}Θ(γ′_{j})) ∪ (∪_{j=1}^{k}Θ(α_{j}))) = {x},  (3.29)

where △ stands for the symmetric difference of two sets. Each α_{j}, γ_{j} and γ′_{j} must surround at least one neighbor of x. γ_{1}, …, γ_{m} and γ′_{1}, …, γ′_{n} are precisely the (+)-contours at σ_{x} which include bonds of γ. Let

S_{−} = ∑_{j=1}^{m}|γ_{j}| + ∑_{j=1}^{n}|γ′_{j}|, S_{+} = |γ| + ∑_{j=1}^{k}|α_{j}|.

Using (3.28) it is easy to see that

S_{+} ≤ S_{−} ≤ S_{+} + 4.  (3.30)

We will show that

Δ_{γ,α_{1},...,α_{k}}H_{Λ(l)}^{ω}(σ) ≥ ε_{3.31}S_{+} − C_{3.31},  (3.31)

where ε_{3.31} = ε_{3.31}(δ) > 0 and C_{3.31} = C_{3.31}(δ) > 0, by using (3.28) and studying the contours γ_{1}, …, γ_{m} and γ′_{1}, …, γ′_{n}.

Now consider σ_{x} = −1. In this case, one possibility is that the flipping of σ_{x} connects together two or three (+)-clusters to create a (+)-cluster which has a shorter outer boundary than the longest of the original (+)-clusters had. If we let γ denote the outer boundary of the (+)-cluster of x at σ^{x}, this means there are again (+)-contours γ_{1}, …, γ_{m} and γ′_{1}, …, γ′_{n} (m ≥ 0, n ≥ 0, 1 ≤ m + n ≤ 2), and (−)-contours α_{1}, …, α_{k} (0 ≤ k ≤ 2) inside γ, such that (3.27)–(3.29) hold, but now γ and the α_{j} exist at σ^{x} and the γ_{j} and γ′_{j} exist at σ. Further, |γ| < 2δ_{1}l, and in place of (3.26),

γ_{j} ∩ ∂Q(Λ(l)) ≠ ∅, for j = 1, …, m, |γ_{j}| ≥ 2δ_{1}l for some j.  (3.32)

(The other possiblity when σ_{x} = −1 is that only one (+)-cluster (call it C) at σ is contained in the (+)-cluster of x at σ^{x}, and the flipping of σ_{x} to 1 shortens the boundary of C; this may be taken as another case of the above with m + n = 1 and 0 ≤ k ≤ 3.) Here in place of (3.24) we have

C_{l}(σ) = {γ_{j} : 1 ≤ j ≤ m, |γ_{j}| ≥ 2δ_{1}l}.  (3.33)

Statement (3.25) still holds, and in place of (3.31) we will prove

Δ_{γ_{1},...,γ_{m},γ′_{1},...,γ′_{n}}H_{Λ(l)}^{ω}(σ) ≥ ε_{3.34}S_{−} − C_{3.34},  (3.34)

It is easy to see that for fixed x, both for σ_{x} = 1 and for σ_{x} = −1, the sets {γ_{1}, … γ_{m}, γ′_{1}, … γ′_{n}} and {γ, α_{1}, …, α_{k}} uniquely determine each other.

We now turn to the proof of (3.31) for σ_{x} = 1 and (3.34) for σ_{x} = −1. For σ_{x} = 1 we have using (3.29) that

\frac{1}{2}Δ_{γ,α_{1},...,α_{k}}H_{Λ(l)}^{ω}(σ) ≥ \frac{1}{2}Δ_{γ,α_{1},...,α_{k}}H_{Λ(l)}^{ω}(σ) − 4
≥ \frac{1}{2}Δ_{γ_{1},...,γ_{m},γ′_{1},...,γ′_{n}}H_{Λ(l)}^{ω}(σ^{x}) − 8
= \frac{1}{2}Δ_{γ_{1},...,γ_{m}}H_{Λ(l)}^{ω}(σ^{x}) + \frac{1}{2}Δ_{γ′_{1},...,γ′_{n}}H_{Λ(l)}^{ω}(T σ^{x}) − 8  (3.35)

where T = T_{γ_{1}} ∘⋯∘T_{γ_{m}} (Recall (3.4)). Each contour in {γ_{j}} is non-crossing, since |γ_{j}| < 2δ_{1}l. Therefore, we see from (3.9) that for any 0 ≤ p ≤ m,

\frac{1}{2}Δ_{γ_{1},...,γ_{m}}H_{Λ(l)}^{ω}(σ^{x}) ≥ \frac{1}{2}Δ_{γ_{1},...,γ_{p}}H_{Λ(l)}^{ω}(σ^{x}) + ∑_{j=p+1}^{m} (\frac{1}{2}|γ_{j}| − |γ_{j} ∩ ∂Λ(l)|)
≥ \frac{1}{2}Δ_{γ_{1},...,γ_{p}}H_{Λ(l)}^{ω}(σ^{x})
≥ 0.  (3.36)

On the other hand, we have

Δ_{γ′_{1},...,γ′_{n}}H_{Λ(l)}^{ω}(T σ^{x}) = 2∑_{j=1}^{n}|γ′_{j}|,  (3.37)

since γ′_{j} ∩ ∂Λ(l) = ∅. We have as a consequence that

\frac{1}{2}Δ_{γ,α_{1},...,α_{k}}H_{Λ(l)}^{ω}(σ) ≥ \frac{1}{2}Δ_{γ_{1},...,γ_{p}}H_{Λ(l)}^{ω}(σ^{x}) + ∑_{j=1}^{n}|γ′_{j}| − 8.  (3.38)

Note also that the first term on the right-hand-side of (3.38) is non-negative by (3.36). For σ_{x} = −1, γ is non-crossing since |γ| < 2δ_{1}l, and each α_{j} is inside γ, so it follows from (3.28) that

∂Q(x) ∪ (∪_{j=1}^{m}γ_{j}) is non-crossing in Q(Λ(l));  (3.39)

in particular each γ_{j} is non-crossing. Therefore (3.35)–(3.38) remain valid but with σ and σ^{x} interchanged; in fact we may replace (3.38) with

\frac{1}{2}Δ_{γ_{1},...,γ_{m},γ′_{1},...,γ′_{n}}H_{Λ(l)}^{ω}(σ) ≥ \frac{1}{2}Δ_{γ_{1},...,γ_{p}}H_{Λ(l)}^{ω}(σ) + ∑_{j=1}^{n}|γ′_{j}|.  (3.40)

To bound (3.38) or (3.40) from below, we pick a number δ_{2} such that δ_{2.3} < δ_{2} < δ_{1} and consider the following three cases separately.

Case 1: S_{+} ≥ 9l. Here the possible energy gain along ∂Λ(l) when T is applied is not enough to cancel the energy reduction in the interior. Specifically, for σ_{x} = 1,

\frac{1}{2}Δ_{γ_{1},...,γ_{p}}H_{Λ(l)}^{ω}(σ^{x}) ≥ ∑_{j+1}^{m}|γ_{j}\∂Q(Λ(l))| − ∑_{y∈∂Λ(l)} |ω_{y}|
≥ ∑_{j+1}^{m}|γ_{j}| − 8l.  (3.41)

With (3.30) and (3.38) this shows

\frac{1}{2}Δ_{γ,α_{1},...,α_{k}}H_{Λ(l)}^{ω}(σ) ≥ S_{+} − 8l − 8 ≥ \frac{1}{9}S_{+} − 8,  (3.42)

which proves (3.31). The same argument with (3.40) replacing (3.38) proves (3.34) when σ_{x} = −1.

Case 2: ∑_{j=1}^{m}|γ_{j}| ≤ (δ_{2}/δ_{1})S_{+}. Consider first σ_{x} = 1. Here by (3.30)

∑_{j=1}^{n}|γ′_{j}| ≥ S_{+} − ∑_{j=1}^{m}|γ_{j}| ≥ (1 − \frac{δ_{2}}{δ_{1}} ) S_{+},  (3.43)

which, together with (3.38), proves (3.31) in this case. Using again (3.30), the same argument with (3.40) replacing (3.38) proves (3.34) when σ_{x} = −1.

Case 3: ∑_{j=1}^{m}|γ_{j}| > (δ_{2}/δ_{1})S_{+} and S_{+} < 9l. By (3.30), (3.38) and (3.40) it is enough to prove that

Δ_{γ_{1},...,γ_{p}}H_{Λ(l)}^{ω}(σ^{x}) ≥ εl − c  (3.44)

for some ε > 0, c ≥ 0 and 1 ≤ p ≤ m. We consider several subcases as follows.

Case 3.1: ∑_{j=1}^{m}|γ_{j} ∩ ∂Q(Λ(l))| ≤ δ_{2.3}l. Consider first σ_{x} = 1. Since

δ_{2}l ≤ \frac{1}{2}(δ_{2}/δ_{1})|γ| ≤ \frac{1}{2}(δ_{2}/δ_{1})S_{+} ≤ \frac{1}{2}∑_{j=1}^{m}|γ_{j}|,  (3.45)

we have by (3.36) that

\frac{1}{2}Δ_{γ_{1},...,γ_{m}}H_{Λ(l)}^{ω}(σ^{x}) ≥ (δ_{2} − δ_{2.3})l

which proves (3.44). For σ_{x} = −1, in place of (3.45) we use (3.32) to obtain

δ_{2}l ≤ δ_{1}l ≤ \frac{1}{2}∑_{j=1}^{m}|γ_{j}|;

otherwise the argument for (3.44) is the same.

Case 3.2: The set ∂Q(x) ∪ (∪_{j=1}^{m}γ_{j}) is non-crossing in Λ(l) and

∑_{j=1}^{m}|γ_{j} ∩ ∂Q(Λ(l))| ≥ δ_{2.3}l.  (3.46)

In this case we have

(∪_{j=1}^{m}γ_{j}) ∩ (F_{l}^{i} ∪ F_{l}^{k}) = ∅

for some i, k with |i| = 1 and |k| = 2. Then there exists a connected subset, say λ, of

(∪_{j=1}^{m}\underline{γ_{j}}) ∪ ∂Q(x)

which divides Λ(l) into two connected components Θ̃ and Λ(l)\Θ̃ such that ∪_{j=1}^{m}Θ(γ_{j}) ⊂ Θ̃ and F_{l}^{i} ∪ F_{l}^{k} ⊂ ∂_{ex} (Λ(l)\Θ̃). Note that the set I defined by I = ∂_{ex}Λ(l) ∩ ∂Θ̃ is an interval. To prove (3.44) by applying (3.15), let us check (3.13) and (3.14) with p = m. We see from the construction of I that

∪_{j=1}^{m}I(γ_{j}) ⊂ I,
|I| ≤ |λ| ≤ ∑_{j=1}^{m} |\underline{γ_{j}}| + 4.

On the other hand, we see from (3.46) that

|I| ≥ ∑_{j=1}^{m} |γ_{j} ∩ ∂Λ(l)| ≥ δ_{2.3}l.

We therefore have (3.13) and (3.14) with p = m.

Case 3.3: The set ∂Q(x) ∪ (∪_{j=1}^{m}γ_{j}) is crossing in Λ(l). By (3.39) this is possible only when σ_{x} = 1. There exist 1 ≤ i < j ≤ 4 and k ∈ {1, 2} such that

γ_{i} ∩ F_{l}^{k} ≠ ∅ and γ_{j} ∩ F_{l}^{−k} ≠ ∅.  (3.47)

Let us assume (3.47) with i = 1, j = 2, and k = 1. Then, the set γ_{1} ∪ γ_{2} cannot be vertically crossing, since |γ_{1}| + |γ_{2}| < 4δ_{1}l and γ_{1} ∪ γ_{2} is already horizontally crossing. Let us therefore assume that

(γ_{1} ∪ γ_{2}) ∩ F_{l}^{−2} = ∅.  (3.48)

We are now left with two possibilities.

Case 3.3.1: γ_{1} ∩ F_{l}^{2} ≠ ∅ and γ_{2} ∩ F_{l}^{2} ≠ ∅. In this case, I = I(γ_{1}) ∪ I(γ_{2}) ∪ F_{l}^{2} is an interval. To prove (3.44) by applying (3.15), we will check (3.13) and (3.14) with p = 2. We obviously have

∪_{j=1}^{2}I(γ_{j}) ⊂ I,
|I| ≥ l.

On the other hand, it is easy to see we have the following injections:

I(γ_{1}) ∩ F_{l}^{1} ⟶ \{vertical dual bonds in \underline{γ_{1}}\},
I(γ_{2}) ∩ F_{l}^{−1} ⟶ \{vertical bonds in \underline{γ_{2}}\},
F_{l}^{2} ⟶ \{horizontal bonds in (\underline{γ_{1}} ∪ \underline{γ_{2}}) ∪ ∂Q(x)\}.

We get |I| ≤ |\underline{γ_{1}}| + |\underline{γ_{2}}| + 4 as a consequence. We therefore have (3.13) and (3.14) with p = 2.

Case 3.3.2: γ_{1} ∩ F_{l}^{2} = ∅ or γ_{2} ∩ F_{l}^{2} = ∅. Let us assume γ_{1} ∩ F_{l}^{2} = ∅, so that γ_{1} does not intersect with F_{l}^{j}, j ≠ 1. The distance from x to F_{l}^{−1} is at most δ_{1}l, and hence the distance from x to F_{l}^{1} is at least (1 − δ_{1})l. This implies that γ_{1} deviates from F_{l}^{1} at least by distance (1 − δ_{1})l − 1. Therefore, by (3.20),

\frac{1}{2}Δ_{γ_{1}}H_{Λ(l)}^{ω}(σ) ≥ |\{horizontal bonds in γ\}|
≥ 2(1 − δ_{1})l − 2,

which establishes (3.44) with p = 1.

Step 2: Suppose again that σ ∈ Γ_{l} and σ^{x} ∉ Γ_{l}. If σ_{x} = 1, then every (+)-contour outside γ at σ has length at most 2δ_{1}l, and every (+)- or (−)-contour inside γ does not intersect ∂Q(Λ(l)). It follows that

T_{γ} ∘ T_{α_{1}} ∘ ⋯ ∘ T_{α_{k}}σ ∈ Γ_{l}^{c}.  (3.49)

Similarly if σ_{x} = −1 then

T_{γ_{1}} ∘ ⋯ ∘ T_{γ_{m}} ∘ T_{γ′_{1}} ∘ ⋯ ∘ T_{γ′_{n}}σ ∈ Γ_{l}^{c}.  (3.50)

Step 3: For x ∈ Λ(l) and σ ∈ Γ_{l} with σ^{x} ∉ Γ_{l}, let C_{+}(σ, x) denote the set of contours {γ, α_{1}, ⋯, α_{k}}, defined previously, if σ_{x} = 1, and let C_{−}(σ, x) = {γ_{1}, ⋯, γ_{m}, γ′_{1}, ⋯, γ′_{n}} if σ_{x} = −1. We have by obervations made in Step 1 that

∑_{x∈Λ(l)} ∑_{σ∈Γ_{l}:σ^{x}∉Γ_{l},σ_{x}=1} μ_{Λ(l),ω}^{β}(σ)  (3.51)
≤ ∑_{x∈Λ(l)} ∑_{γ,α_{1},···,α_{k}} μ_{Λ(l),ω}^{β} {σ : C_{+}(σ, x) = {γ, α_{1}, ⋯, α_{k}}, Δ_{γ,α_{1},...,α_{k}}H_{Λ(l)}^{ω}(σ) ≥ ε_{3.31}S_{+} − C_{3.31}}

where ∑_{γ,α_{1},···,α_{k}} stands for the summation over all possible values of C_{+}(σ, x). Now for fixed x and n there are at most c · 3^{n} possible values of C_{+}(σ, x) for which S_{+} = n. By the standard Peierl’s argument and the obervation made in Step 2, we can proceed as follows, provided β is sufficiently large:

∑_{x∈Λ(l)} ∑_{γ,α_{1},···,α_{k}} μ_{Λ(l),ω}^{β}{σ : C_{+}(σ, x) = {γ, α_{1}, ⋯, α_{k}}, Δ_{γ,α_{1},...,α_{k}}H_{Λ(l)}^{ω}(σ) ≥ ε_{3.31}S_{+} − C_{3.31}}

≤ ∑_{x∈Λ(l)} ∑_{γ,α_{1},···,α_{k}} exp(−β(ε_{3.31}S_{+} − C_{3.31}))

·μ_{Λ(l),ω}^{β}{T_{γ} ∘ T_{α_{1}} ∘ ⋯ ∘ T_{α_{k}}σ : C_{+}(σ, x) = {γ, α_{1}, ⋯, α_{k}}}
≤ ∑_{x∈Λ(l)} ∑_{n≥2δ_{1}l} c · 3^{n} exp(−β(ε_{3.31}n − C_{3.31}))μ_{Λ(l),ω}^{β}(Γ_{l}^{c})

≤ B_{3.52} exp (−βl/C_{3.52}) μ_{Λ(l),ω}^{β}(Γ_{l}^{c}).  (3.52)

Essentially the same argument, using C_{−}(σ, x) and S_{−} in place of C_{+}(σ, x) and S_{+}, shows that

∑_{x∈Λ(l)} ∑_{σ∈Γ_{l}:σ^{x}∉Γ_{l},σ_{x}=−1} μ_{Λ(l),ω}^{β}(σ) ≤ B_{3.53} exp (−βl/C_{3.53}) μ_{Λ(l),ω}^{β}(Γ_{l}^{c}).  (3.53)

We conclude (2.10) from (3.51), (3.52) and (3.53). □

Acknowledgement: The authors thank Y. Higuchi and N. Sugimine for helpful discussions.

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