Non Uniqueness of p-adic Gibbs Distribution for the Ising Model on the Lattice Zd

Abstract

Real Gibbs measures arise in many problems of probability theory and statistical mechanics. This measure, related to the Boltzmann distribution, generalizes the notion of canonical ensemble. In addition, Gibbs measure is unique measure that maximizes the entropy of the expected energy. But non-archimedean (p-adic) analogue of Gibbs measures have been little studied. It is known that in the case of real numbers concepts of Gibbs measure and Markov random field are identical. But in the p-adic case, the class of p-adic Markov random fields is wider than the class of p-adic Gibbs measures [1]. One of the main problems of physics is to study the set of all p-adic Gibbs measures (see e.g. [1, 2]). Let us present some main definitions from the theory of p-adic numbers (see [3–5]). Let p be a prime number. Every rational number x = 0 can be represented in the form x = p n m , where r, n ∈ Z, m is a positive number, (n,m) = 1, where m and n are not divisible by p. A p-adic norm of rational number x = p n m is defined as follows