Introduction

Abstract

Let A be a finite subset of ℤ^d, for example, Λ=Λ_L= {−L, −L + 1, …, L}^d. An Ising model on Λ is a family {S _x : x ∈ Λ} of random variables (called spins) taking values +1 or −1, whose joint distribution P ^Λ,β depends on a parameter β ≥ 0 and has the form,

$$P \wedge,\beta \left( {\left\{ {\left. s \right\}} \right.} \right)\, = \,Z_{ \wedge,\beta }^{ - 1}\,\exp \left\{ {\left. { - \beta H \wedge \left( s \right)} \right\}} \right..\,(0.1)$$