Mathematics & Physics 2016, 9(1), 123–127 УДК 517.9 Non Uniqueness of p-adic Gibbs Distribution for the Ising Model on the Lattice Zd Zohid T.Tugyonov∗ Institute of Mathematics Durmon Yuli, 29, Tashkent, 100125 Uzbekistan Received 05.08.2015, received in revised form 24.09.2015, accepted 28.12.2056 In this paper, we show non uniqueness of p-adic Gibbs distribution for the Ising model on the Zd. <...> Moreover, we prove that a p-adic Gibbs distribution is bounded if and only if p ̸= 2. <...> Introduction 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]). <...> Every rational number x ̸= 0 can be represented in the form x = pr n r,n ∈ Z, m is a positive number, (n,m) = 1, where m and n are not divisible by p. <...> The completion of the set of rational numbers Q under p-adic norm leads to the field of p-adic numbers Qp for every prime p. <...> This p-adic norm satisfies the strong triangle inequality: |x+y|p max{|x|p, |y|p}. <...> This property shows that p-adic norm is a non-Archimedean norm. <...> The p-adic exponent is defined as expp(x) = and the power series converges for x ∈ B(0, p−1/(p−1)). <...> A p-adic measure is a probability measure if µ(X) = 1. <...> The set of all configurations on Mn is denoted by Ωn and the set of all configurations on Zd is denoted by Ω. <...> Conditional Hamiltonian Hn <...>