Ising Models on the Lattice Sierpinski Gasket

Yasunari Higuchi^{1} and Nobuo Yoshida^{2,3}

1 Department of Mathematics, Faculty of Science, Kobe University, Rokko, Kobe 657, Japan, e-mail:higuchi@math.s.kobe-u.ac.jp 2 Division of Mathematics, Graduate School of Science, Kyoto University, Kyoto 606-01, Japan, e-mail:nobuo@kusm.kyoto-u.ac.jp 3 Mathematics Department, 2-101, MIT, Cambridge, MA 02139, USA. e-mail:nobuo@math.mit.edu

Abstract. Ferromagnetic Ising models on the lattice Sierpinski gasket are considered. We prove the Dobrushin-Shlosman mixing condition and discuss corresponding properties of the stochastic Ising models.

Key Words: Ising model, Lattice Sierpinski gasket, Dobrushin-Shlosman mixing condition.

1 Introduction

This work is originated in an attempt to understand more about the enormous difference between one-dimensional and two-dimensional Ising models. Our interest in the lattice Sierpinski gasket comes from its geometrical character as a cross of the one-dimensional and the two-dimensional lattice. The lattice Sierpinski gasket is locally two dimensional, since each point in the lattice has four nearest neighbours. However, it is in a sense closer to one-dimensional lattice. For example, there exist a sequence {W_{l}}_{l=1}^{∞} of finite sets which increases to the whole lattice keeping |∂W_{l}| = 8. It is therefore reasonable to expect that fairly strong mixing property survives for an Ising model on the lattice Sierpnski gasket even at low temperature. The purpose of this article is to vindicate this expectation from both static and dynamical point of views. Before we can state the results precisely, we have to introduce a long sequence of definitions and notations.

The lattice Sierpinski gasket. We will denote the usual Euclidean distance on R^{2} by d(x, y) = |x − y|. We define x^{∗} = (−x_{1}, x_{2}) for x = (x_{1}, x_{2}) ∈ R^{2} and A^{∗} = \{x^{∗}; x ∈ A\} for A ⊂ R^{2}. Set x_{l} = 2^{l}(1, 0) and y_{l} = 2^{l}(\frac{1}{2}, \frac{\sqrt{3}}{2}), (l = 0, 1, ⋯). Starting with Λ_{0} = {0, x_{0}, y_{0}}, we construct a sequence (Λ_{l})_{l=0}^{∞} of finite subsets in R^{2} inductively by;

Λ_{l+1} = ⋃_{ν=0}^{2} Λ_{l,ν},

where

Λ_{l,ν} = \{ \begin{matrix}Λ_{l} & if ν = 0, \\ Λ_{l} + x_{l} & if ν = 1, \\ Λ_{l} + y_{l} & if ν = 2. \end{matrix}

Now, we set W_{l} = Λ_{l}^{∗} ∪ Λ_{l} and define the lattice Sierpinski gasket G by

G = ⋃_{l=0}^{∞} W_{l}.

The number of points contained in Λ ⊂ G will be denoted by |Λ| and we write Λ ⊂⊂ G when Λ ⊂ G and 1 ≤ |Λ| < ∞. A set W ⊂⊂ G is said to be a l-pair with the joint q ∈ G if there exists {x, x′} ⊂ G such that

W = (Λ_{l} + x) ∪ (Λ_{l} + x′),
and {q} = (Λ_{l} + x) ∩ (Λ_{l} + x′).

For example, W_{l} is a l-pair with the origin as its joint.

The set B of bonds in G is a subset of {{x, y} ⊂ G ; |x − y| = 1 } defined as follows. Let B_{0} = { b ⊂ Λ_{0} ; |b| = 2 } and

B_{l+1} = ⋃_{ν=0}^{2} B_{l,ν}, l = 0, 1, ⋯,

where

B_{l,ν} = \{ \begin{matrix}B_{l} & if ν = 0, \\ \{b + x_{l}; b ∈ B_{l}\} & if ν = 1, \\ \{b + y_{l}; b ∈ B_{l}\} & if ν = 2. \end{matrix}

Finally, define B by

B = ⋃_{n=0}^{∞} B_{n}^{∗} ∪ B_{n},

where B_{n}^{∗} = {b^{∗} : b ∈ B_{n}}. For a set Λ ⊂ G, we also define

B_{Λ} = {{x, y} ∈ B ; (x, y) ∈ Λ^{2}},
∂Λ = {{x, y} ∈ B ; x ∈ Λ, y ∉ Λ},
∂_{int}Λ = {x ∈ Λ; x ∈ b for some b ∈ ∂Λ },
and ∂_{ext}Λ = {x ∉ Λ; x ∈ b for some b ∈ ∂Λ }.

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

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

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

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

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

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

For S^{Λ} → R, we introduce the notations

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

∥f∥ = sup_{σ∈S^{Λ}} |f(σ)|,

and |||f||| = ∑_{x∈Λ} ∥∇_{x}f∥.

Function spaces C and C_{Λ} (Λ ⊂ G) are defined respectively by

C = {f : S^{G} → R ; |Δ_{f}| < ∞ },
and C_{Λ} = {f : S^{G} → R ; Δ_{f} ⊂ Λ }.

The local specification. For a measure m on some measurable space, we will use the following abbreviations

mf = ∫ fdm,
and m(f ; g) = m(fg) − mf · mg,

whenever the integrations make sense. Fix J = (J_{b})_{b} ∈ R^{B} and h = (h_{x})_{x} ∈ R^{G}. We will refer J, h, and the pair (J, h) respectively to the coupling constants , the magnetic field and the interaction. For Λ ⊂⊂ G and ω ∈ S^{G}, the Hamiltonian H_{Λ,ω} ∈ C_{Λ} is defined by;

−H_{Λ,ω}(σ) = ∑_{{x,y}∈B_{Λ}} J_{{x,y}}σ_{x}σ_{y} + ∑_{x∈Λ} σ_{x}(h_{x} + ∑_{y:{x,y}∈∂Λ} J_{{x,y}}ω_{y}).

The (J, h)-local specification is a family {μ^{Λ,ω}; Λ ⊂⊂ G, ω ∈ S^{G}} of Borel probability measures on Σ^{G} defined by

μ^{Λ,ω}f = δ_{ω} (\frac{ρ^{Λ}(f exp −H_{Λ,ω})}{ρ^{Λ} exp −H_{Λ,ω}}) ,

where ρ^{Λ} is the 1/2-Bernoulli measure on Σ^{Λ} and δ_{ω} is the Dirac delta measure concentrated on the configuration ω.

The infinite volume Gibbs state. A Borel probability measure μ is called a (J, h)- infinite volume Gibbs state or simply a (J, h)-Gibbs state if it solves the following Dobrushin-Lanford-Ruelle equation;

μ(μ^{Λ,·}f) = μf, ∀Λ ⊂⊂ G, ∀f ∈ C.  (1.1)

By a slight extension of the arguments in [2, page 356 ], we see that if sup_{b∈B} |J_{b}| < ∞, then there exists unique (J, h)-Gibbs state μ on Σ^{G}. This in turn implies that

lim_{Λ↑G} sup_{ω∈S^{G}} |μ^{Λ,ω}f − μf| = 0, for all f ∈ C,  (1.2)

(See [5, page 120, (7.10) ], for example).

The stochastic dynamics. We introduce now for the Ising model above, the time evolution called stochastic Ising model or Glauber dynamics. To do this, we now suppose there exist J ≥ 0 and h ≥ 0 such that J ∈ [−J, J]^{B} and h ∈ [−h, h]^{G}. We define for each x ∈ G an operator A_{x} : C → C by A_{x}f = c_{x}∇_{x}f, where the flip rate c_{x} : S^{G} → (0, ∞) is a function on which we assume the following.

(R-1) Boundedness : There exist positive constants \underline{c}(J, h) and \overline{c}(J, h) such that

\underline{c}(J, h) ≤ c_{x}(σ) ≤ \overline{c}(J, h), for all (x, σ) ∈ G × S^{G}.

(R-2) Finite range: There exists r ≥ 0 such that ∇_{y}c_{x} ≡ 0 if |x − y| > r.

(R-3) Detailed Balanced Condition:

∇_{x}{c_{x}(σ) exp −H_{{x},σ}(σ)} ≡ 0, for all x ∈ G.  (1.3)

We introduce the stochastic Ising model in a finite set Λ ⊂⊂ G with the boundary condition ω ∈ Σ^{G}. We define an operator A^{Λ,ω} : C → C by

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

where σ|_{Λ} · ω|_{Λ^{c}} denotes the following configuration:

(σ|_{Λ} · ω|_{Λ^{c}})_{x} = \{ \begin{matrix}σ_{x} & if x ∈ Λ, \\ ω_{x} & if x ∉ Λ. \end{matrix}

We then have

−μ^{Λ,ω}(fA^{Λ,ω}g) = \frac{1}{2} ∑_{x∈Λ} μ^{Λ,ω}(c_{x}∇_{x}f∇_{x}g), \{f, g\} ⊂ C.  (1.4)

We set

T_{t}^{Λ,ω} = exp tA^{Λ,ω} t > 0.

A^{Λ,ω} and T_{t}^{Λ,ω} can also be regarded as oprators from C_{Λ} into itself.

2 Results

We will show the following property for the local specification, which is called complete analyticity or Dobrushin-Shlosmann mixing condition.

Theorem 2.1 For any J ≥ 0, there exists C_{0} ∈ (0, ∞) such that the following holds for any J ∈ [0, J]^{B}, h ∈ R^{G}, Λ ⊂⊂ G y ∉ Λ and f ∈ C_{Λ};

sup_{ω∈S^{G}} |μ^{Λ,ω^{y}}f − μ^{Λ,ω}f| ≤ C_{0}|||f||| exp −\frac{d(Δ_{f}, y)}{C_{0}}.  (2.1)

Starting from Theorem 2.1, we can derive ergodic properties of the stochastic Ising model.

Theorem 2.2 For any J ≥ 0 and h ≥ 0, there exists {C_{1}, C_{2}, C_{3}} ∈ (0, ∞) such that the following hold for any J ∈ [0, J]^{B}, h ∈ [−h, h]^{G}, Λ ⊂⊂ G, x ∉ Λ and f ∈ C_{Λ};

(i)

μ^{Λ,ω}(|f − μ^{Λ,ω}f|^{2}) ≤ C_{1} ∑_{x∈Λ} μ^{Λ,ω}(|∇_{x}f|^{2}).  (2.2)

(ii)

μ^{Λ,ω} (f^{2} log \frac{f^{2}}{μ^{Λ,ω}(f^{2})}) ≤ C_{2} ∑_{x∈Λ} μ^{Λ,ω}(|∇_{x}f|^{2}).  (2.3)

∥T_{t}^{Λ,ω}f − μ^{Λ,ω}f∥ ≤ C(f) exp −\frac{t}{C_{3}} ,  (2.4)

where C(f) > 0 is a function of f ∈ C, which does not depend on the choice of Λ and ω.

Proof: The equivalence of the conditions (2.1), (2.2),(2.3) and (2.4) is well established in cubic lattice case ([13, 10], for example) and almost the same proof works in our setting. So we will just indicate the references. It is well known that (ii) implies (i) ([3, page 224, (6.1.7)]). Using Theorem 2.1, we can prove (ii) just in the same way as [10, Theorem3]. With (ii) in hand, we can follow the arguments in [12, Lemma(2.9)] to conclude (iii).

Q.E.D.

3 Proof of Theorem 2.1

In this section, we will prove Theorem 2.1. So we assume that the coupling constants J satisfy J ∈ [0, J]^{B} for some J ≥ 0. For a moment, we will consider for each Λ ⊂⊂ G, a magnetic field h = (h_{x}^{Λ})_{x∈Λ} ∈ R^{Λ}, which may be Λ-dependent. We define a probability measure μ^{Λ,h} on Σ^{Λ} by

μ^{Λ,h}f = \frac{ρ^{Λ}(f exp −H_{Λ,h})}{ρ^{Λ} exp −H_{Λ,h}}, f ∈ C_{Λ},  (3.1)

where

−H_{Λ,h}(σ) = ∑_{{x,y}∈B_{Λ}} J_{{x,y}}σ_{x}σ_{y} + ∑_{x∈Λ} h_{x}^{Λ}σ_{x}.  (3.2)

When h_{x}^{Λ} ≡ 0, we will omit the index h:

μ^{Λ} = μ^{Λ,h}|_{h_{x}^{Λ}≡0},
and H_{Λ} = H_{Λ,h}|_{h_{x}^{Λ}≡0}.

The measure μ^{Λ,h} defined above will be called the finite volume Gibbs state with respect to the Hamiltonian (3.2), to which we can apply standard correlation inequalities such as FKG, GHS, GKS1 and GKS2, ([4]). Also, the following inequality due to E. Lieb is useful for our purpose.

Lemma 3.1 ([8]) Suppose that X ⊂⊂ G, Y ⊂⊂ G, h_{x}^{X} ≡ 0 and h_{x}^{Y} ≡ 0. Then, for any (x, y) ∈ X\Y × Y \X,

μ^{H_{X}+H_{Y}} (σ_{x}σ_{y}) ≤ ∑_{z∈X∩Y} μ^{X}(σ_{x}σ_{z})μ^{H_{X}+H_{Y}} (σ_{z}σ_{y}),  (3.3)

where

μ^{H_{X}+H_{Y}} f = \frac{ρ^{X∪Y} (f exp(−H_{X} − H_{Y}))}{ρ^{X∪Y} exp(−H_{X} − H_{Y})}.

Proof: This is (20) of [Lie].

Q.E.D.

Although G is not homogeneous object as Z^{d}, the non-homogeneity can be circumvented by the following simple observation.

Lemma 3.2 For each x ∈ G and l ∈ N, there exist l-pair U_{l,x}, (l + 1)-pair V_{l+1,x} and a point q_{l,x} ∈ G such that the following hold.

(i) x ∈ U_{l,x}

(ii) U_{l,x} and V_{l+1,x} have q_{l,x} as the common joint.

Proof: For any l ∈ N, there exists X_{l} ⊂ G such that G = ∪_{x∈X_{l}}(x + Λ_{l}). Let us call each x + Λ_{l} (x ∈ X_{l} ) an l-cell. For each (l + 1)-cell Λ, there exist another three (l + 1)-cells {Λ^{(j)}}_{j=1}^{3} such that {Λ ∪ Λ^{(j)}}_{j=1}^{3} are (l + 1)-pairs. This implies that for any x ∈ Λ, there exists a triple {U_{l,x}, V_{l+1,x}, q_{l,x}}, with desired properties.

Q.E.D.

Lemma 3.3 For any J ≥ 0, there exists C_{4} ∈ (0, ∞) such that the following holds for any Λ ⊂⊂ G, (J_{b})_{b} ∈ [0, J]^{B} and (h_{x}^{Λ})_{x} ∈ [0, ∞)^{Λ} ∪ (−∞, 0]^{Λ};

0 ≤ μ^{Λ.h}(σ_{x} ; σ_{y}) ≤ C_{4} exp −\frac{|x − y|}{C_{4}}, for all \{x, y\} ⊂ Λ.  (3.4)

Proof: The left hand side inequality is nothing but the FKG. To prove the right hand side inequality, it is sufficient to consider the follwing special case;

J_{b} ≡ J, ∀b ∈ B,  (3.5)
and h_{x}^{Λ} ≡ 0, ∀Λ ⊂⊂ G, ∀x ∈ Λ,  (3.6)

(3.7)

In fact, μ^{Λ,h}(σ_{x} ; σ_{y}) is an even function of (h_{x}^{Λ})_{x∈Λ} ∈ R^{Λ}, and is non-increasing in [0, ∞)^{Λ} by GHS inequality. When h ∈ [0, ∞)^{Λ}, μ^{Λ,h}(σ_{x}; σ_{y}) is also a non-decreasing function of J by GKS2 inequality. Therefore, if (h_{x}^{Λ})_{x∈Λ} ∈ [0, ∞)^{Λ} ∪ (−∞, 0]^{Λ}, then

μ^{Λ,h}(σ_{x} ; σ_{y}) ≤ μ^{Λ,h}(σ_{x}σ_{y})|_{h_{x}^{Λ}≡0,J_{b}≡J}.

In view of this, we will assume (3.5) and (3.6). We will devide the rest of the proof into three steps, in which we will abbreviate μ^{Λ,h}(σ_{x}σ_{y}) by μ^{Λ}(x, y) and \frac{x_{l}+y_{l}}{2} by z_{l}.

Step 1: We first show that

α_{l} := μ^{Λ_{l}}(0, x_{l}) →^{l→∞} 0.  (3.8)

To see this, we consider the unique infinite volume Gibbs state μ for J_{b} ≡ J and h_{x} ≡ 0. Then, (1.2) implies that

lim_{l→∞} μ(0, x_{l}) = 0,

while (1.2), FKG and GKS2 yield

sup_{Λ⊂⊂G} μ^{Λ}(0, x_{l}) = μ(0, x_{l}).

Combining these two, we obtain (3.8).

Step 2: Define δ_{l} ∈ [0, 1] by

δ_{l} = max{μ^{W_{l+1}}(z, z′) ; z ∈ W_{l}, z′ ∈ ∂_{int}W_{l+1} }.  (3.9)

We next prove that

δ_{l} →^{l→∞} 0.  (3.10)

Set Y_{l} = {x ∈ W_{l+1}; |x| ≥ 2^{l}}. We then have H_{W_{l+1}} = H_{W_{l}} + H_{Y_{l}}. Therefore, for z ∈ W_{l}\∂_{int}W_{l} and z′ ∈ ∂_{int}W_{l+1}, we have by Lieb’s inequality (3.3)

μ^{W_{l+1}}(z, z′) ≤ ∑_{w∈∂_{int}W_{l}} μ^{W_{l}}(z, w)μ^{W_{l+1}}(w, z′),

and thus,

δ_{l} ≤ 4 max{μ^{W_{l+1}}(w, z′) ; w ∈ ∂_{int}W_{l}, z′ ∈ ∂_{int}W_{l+1} }.  (3.11)

We now estimate the right hand side of (3.11). Beacause of of the symmetry which comes from the assumptions (3.5) and (3.6), we have only to consider the following three correlations;

μ^{W_{l+1}}(x_{l}, x_{l+1}), μ^{W_{l+1}}(x_{l}, y_{l+1}) and μ^{W_{l+1}}(x_{l}, x_{l+1}^{∗}).

For μ^{W_{l+1}}(x_{l}, x_{l+1}), we have by Lieb’s inequality that

μ^{W_{l+1}}(x_{l}, x_{l+1}) ≤ μ^{Λ_{l−1}+ \frac{x_{l}+x_{l+1}}{2}} (x_{l+1}, \frac{x_{l} + x_{l+1}}{2} ) μ^{W_{l+1}} (\frac{x_{l} + x_{l+1}}{2} , x_{l})

+μ^{Λ_{l−1}+ \frac{x_{l}+x_{l+1}}{2}} (x_{l+1}, \frac{x_{l+1} + z_{l+1}}{2} ) μ^{W_{l+1}} (\frac{x_{l+1} + z_{l+1}}{2} , x_{l})

≤ 2α_{l−1}

Similarly,

μ^{W_{l+1}}(x_{l}, y_{l+1}) ≤ μ^{Λ_{l,2}}(y_{l+1}, y_{l})μ^{W_{l+1}}(y_{l}, x_{l}) + μ^{Λ_{l,2}}(y_{l+1}, z_{l+1})μ^{W_{l+1}}(z_{l+1}, x_{l})
≤ 2α_{l},

and

μ^{W_{l+1}}(x_{l}, x_{l+1}^{∗}) ≤ μ^{Λ_{l+1}^{∗}}(x_{l+1}^{∗}, 0)μ^{W_{l+1}}(0, x_{l})
≤ α_{l+1}.

Therefore, (3.8) implies (3.10).

Step 3: We are now, in position to conclude (3.4). The following argument will be reminiscient of the proof of [11, Theorem 1.2]. Take l ∈ N satisfying

4δ_{l} < exp −1.  (3.12)

We may asuume that |x − y| > 2^{l} and hence that n2^{l} < |x − y| ≤ (n + 1)2^{l} for some n = 1, 2, · · ·. We may also assume that Λ ⊃ {z ∈ G; |x − z| ≤ (n + 1)2^{l}}. For each z ∈ Λ, take a triple {U_{l,z}, V_{l+1,z}, q_{l,z}} in Lemma 3.2, then we obtain by Lieb’s inequality and (3.12) that

μ^{Λ}(x, y) ≤ ∑_{v_{1}∈∂_{int}V_{l+1,x}} μ^{V_{l+1,x}}(x, v_{1})μ^{Λ}(v_{1}, x)

≤ ∑_{v_{1}∈∂_{int}V_{l+1,v_{1}}} ⋯ ∑_{v_{n}∈∂_{int}V_{l+1,vn}} μ^{V_{l+1,x}}(x, v_{1}) ⋯ μ^{V_{l+1,v_{n−1}}}(v_{n−1}, v_{n})μ^{Λ}(v_{n}, y)

≤ (4δ_{l})^{n} ≤ exp −n

≤ exp (1 − \frac{|x − y|}{2^{l}}) ,

which proves (3.4).

Q.E.D.

Proof of Theorem 2.1:

Step1: We will prove that

0 ≤ μ^{Λ,h}(σ_{x}; σ_{y}) ≤ C_{4} exp −\frac{|x − y|}{C_{4}},  (3.13)

for all Λ ⊂⊂ G and h ∈ R^{Λ}, where C_{4} is the constant which appears in (3.4). The left hand side inequality is nothing but the FKG. To show the right hand side inequality, define a bijection

(σ^{1}, σ^{2}) ∈ (S^{Λ})^{2} ↦ (σ^{+}, σ^{−}) ∈ T := {(σ^{+}, σ^{−}) ∈ {0, ±1}^{Λ}; (σ_{x}^{+})^{2} + (σ_{x}^{−})^{2} = 1}

by σ_{x}^{±} = \frac{σ_{x}^{1}±σ_{x}^{2}}{2} . We then have that

−H_{Λ,h}(σ^{1}) − H_{Λ,h}(σ^{2})
= ∑_{{x,y}∈B_{Λ}} J_{{x,y}}(σ_{x}^{1}σ_{y}^{1} + σ_{x}^{2}σ_{y}^{2}) + ∑_{x∈Λ}(h_{x}^{1}σ_{x}^{1} + h_{x}^{1}σ_{x}^{2})
= ∑_{{x,y}∈B_{Λ}} 2J_{{x,y}}(σ_{x}^{+}σ_{y}^{+} + σ_{x}^{−}σ_{y}^{−}) + ∑_{x∈Λ} 2h_{x}^{+}σ_{x}^{+}
= −2H_{Λ,h}(σ^{+}) − 2H_{Λ,0}(σ^{−})  (3.14)

and thus, that for a function f = f(σ^{1}, σ^{2}) on (S^{Λ})^{2},

(μ^{Λ,J,h} ⊗ μ^{Λ,J,h})f = \frac{∑_{σ^{1},σ^{2}} f exp −2\{H_{Λ,h}(σ^{+}) + H_{Λ,0}(σ^{−})\}}{(Ẑ^{Λ,J,h})^{2}},  (3.15)

where μ^{Λ,J,h} = μ^{Λ,h} and Ẑ^{Λ,J,h} = ∑_{σ∈S^{Λ}} exp −H_{Λ,h}(σ). Also, it is easy to see that

μ^{Λ,J,h}(σ_{x}; σ_{y}) = \frac{1}{2}μ^{Λ,J,h} ⊗ μ^{Λ,J,h}(σ_{x}^{−}σ_{y}^{−}).  (3.16)

By (3.15) and (3.16),

2(Ẑ^{Λ,J,h})^{2}μ^{Λ,J,h}(σ_{x}; σ_{y})
= ∑_{σ^{1},σ^{2}} σ_{x}^{−}σ_{y}^{−} exp −2{H_{Λ,h}(σ^{+}) + H_{Λ,0}(σ^{−})}
= ∑_{(σ^{+},σ^{−})∈T} σ_{x}^{−}σ_{y}^{−} exp −2{H_{Λ,h}(σ^{+}) + H_{Λ,0}(σ^{−})}
= ∑_{Γ⊂Λ} ∑_{(σ^{+}, σ^{−}); \begin{matrix}σ_{x}^{−} = ±1 on Γ, \\ σ_{x}^{+} = ±1 on Λ\Γ \end{matrix}} σ_{x}^{−}σ_{y}^{−} exp −2\{H_{Λ,h}(σ^{+}) + H_{Γ,0}(σ^{−})\} (3.17)

In the forth line, we have decomposed the summation over (σ^{+}, σ^{−}) ∈ T according to the “ support” of σ^{+}. The following trivial observation has also been used;

H_{Λ,0}(σ^{−}) = H_{Γ,0}(σ^{−}), if σ^{−} ≡ 0 on Λ\Γ.

By (3.16) and Lemma 3.3, we have that

∑_{σ^{−}∈{−1,+1}^{Γ}} σ_{x}^{−}σ_{y}^{−} exp −2H_{Γ,0}(σ^{−}) = 2Ẑ^{Γ,2J,0}μ^{Γ,2J,0}(σ_{x}σ_{y})

≤ 2C_{4}Ẑ^{Γ,2J,0} exp −\frac{|x − y|}{C_{4}}.

Plugging this into (3.17), we end up with

(3.17) ≤ 2C_{4} ∑_{Γ⊂Λ} Ẑ^{Γ,2J,0}Ẑ^{Λ\Γ,2J,2h} exp −\frac{|x − y|}{C_{4}}.  (3.18)

Now, by repeating similar computations, we have

(Ẑ^{Λ,J,h})^{2} = ∑_{Γ⊂Λ} Ẑ^{Γ,2J,0}Ẑ^{Λ\Γ,2J,2h}.  (3.19)

Putting (3.17),(3.18) and (3.19) together, we conclude (3.13).

Step2: We will prove (2.1). We first assume that f(σ) = σ_{z} ((z ∈ Λ) ). In this case, we obtain.

sup_{ω∈S^{G}} |μ^{Λ,ω^{y}}(σ_{z}) − μ^{Λ,ω}(σ_{z})| = ∫_{0}^{1} \frac{d}{dθ}μ^{Λ,h(θ)}(σ_{z})dθ
= −2 ∫_{0}^{1} ∑_{v∈Λ:|v−y|=1} J_{{y,v}}ω_{v}μ^{Λ,h(θ)}(σ_{z}; σ_{v}) (3.20)

where we have defined the magnetic field h(θ) by

h_{x}(θ) = h_{x} + ∑_{w∉Λ:|w−x|=1} J_{{x,w}}{θω_{w}^{y} + (1 − θ)ω_{w}}.

But we have by (3.13) that

μ^{Λ,h(θ)}(σ_{z}; σ_{v}) ≤ C_{4} exp −\frac{|v − z|}{C_{4}}
≤ C_{4} exp −\frac{|y − z| − 1}{C_{4}}.

Plugging this into (3.20), we obtain (2.1) for the case f(σ) = σ_{z}. The general case follows from [1, Lemma(2.1)].

Q.E.D.

Acknowledgements. We thank T. Kumagai for variable comments, which encouraged the authors to improve the previous version of our results. We also thank T. Funaki, T. Chiyonobu and M. Sugiura for discussions.

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