Limiting Gibbs measures of Potts model with countable set of spin values

Abstract

We consider a nearest-neighbor Potts model, with countable spin values Φ = { 0 , 1 , … } , and nonzero external field, on a Cayley tree of order k (with k + 1 neighbors). We study translation-invariant ‘splitting’ Gibbs measures. The problem is reduced to the description of the solutions of some infinite system of equations. We give full description of the class of probabilistic measures ν on Φ such that our infinite system of equations has unique solution with respect to each element of this class. In particular we describe the Poisson measures which are Gibbsian.