Vol. 129 (2016) ACTA PHYSICA POLONICA A No. 4 5th International Science Congress & Exhibition APMAS2015, Lykia, Oludeniz, April 16–19, 2015 Existence of p-Adic Quasi Gibbs Measures for Mixed Type p-Adic Ising λ-Model M. Dogan a H. Akin a, * and F. Mukhamedov b a Zirve University, Faculty of Education, Department of Mathematics, Gaziantep, 27260, Turkey b Department of Computational and Theoretical Sciences, Faculty of Science, IIUM, 25200 Kuantan, Malaysia We consider nearest-neighbors and next nearest-neighbors p-adic Ising λ-model with spin values {∓1} on a Cayley tree of order two. First we prove that the model satisfies the Kolmogorov consistency condition and then we prove that the nonlinear equation corresponding to the model has at least two solutions in Qp, where p is a prime number p ≥ 3. One of the roots is in εp and the others are in Qp\εp. If the nonlinear equation has more than one non-trivial solutions for the model then we conclude that p-adic quasi Gibbs measure exists for the model. DOI: 10.12693/APhysPolA.129.861 PACS/topics: 05.50.+q, 02.20.Qs 1. Introduction The p-adic numbers were first defined by the German mathematician K. Hensel as a branch of pure mathemat- ics [1–5]. However, a lot of applications of these num- bers in theoretical physics have been proposed (see [1–5]). A number of p-adic models in physics cannot be de- scribed using ordinary probability theory based on the Kolmogorov axioms [2, 5]. New probability model is called “ p-adic probability model” since it takes values in Q p . The non-Archimedean analogy of the Kolmogorov probability theory gives us the possibility to construct wide classes of stochastic processes by using finite dimen- sional probability distributions instead of the infinite one, as was proved before. This gives a possibility to develop the theory of statisti- cal mechanics in the context of the p-adic theory, since it lies on the base of the theory of probability and stochas- tic processes. One of the central problems of the theory of statistical mechanics is the study of infinite-volume Gibbs measures corresponding to a given Hamiltonian. However, a complete analysis of the set of Gibbs mea- sures for a specific Hamiltonian is often a difficult prob- lem. This problem includes the study of phase transition problems. Recall that for a given Hamiltonian there is a phase transition if there exist at least two distinct p-adic Gibbs measures, of which one is bounded and the other is unbounded. In the present work, we first establish the Hamilto- nian for the mixed type p-adic Ising λ-model. After that we prove that the model satisfies the Kolmogorov con- sistency condition. Finally we prove the existence of the p-adic quasi Gibbs measures. * corresponding author; e-mail: hasan.akin@zirve.edu.tr 2. Definitions and preliminaries and the model 2.1. p-adic numbers and measures Let Q be the field of rational numbers. Every rational number x =0 can be represented in the form of x = p r m/n, where r, m ∈ Z, n is a positive integer, (p, m)=1, (p, n)=1 and p is a fixed prime number. The p-adic norm of x is given by |x| p =  p −r for x =0 0 for x =0. It satisfies the following properties: 1) |xy| p = |x| p |y| p ; 2) The strong triangle inequality |x + y| p ≤ max  |x| p , |y| p  ; hence this is a non-Archimedean norm. Existence of two types of the completions of the ra- tional numbers with respect to p-adic norm |x| p and usual norm |x| ∞ was proved by Ostrowsky. p-adic norms are non-equivalent norms on Q. Any p-adic number can be uniquely represented as in the following canonical form x = p γ(x) ( x 0 + x 1 p + x 2 p 2 + ... ) , where γ = γ (x) ∈ Z and x j are integers 0 ≤ x j ≤ p − 1, x 0 > 0 (j =0, 1, 2,...). In this case |x| p = p −γ(x) . Let B(a, r)=  x ∈ Q p : |x − a| p ≤ r  , where a ∈ Q p , r > 0 be a disc. The p-adic logarithm is defined by log p (x) = log p (1 + (x − 1)) = ∞ ∑ n=1 (−1) n+1 (x−1) n n , which converges for x ∈ B(1, 1). The p-adic exponen- tial is defined by exp p (x)= ∞ ∑ n=1 x n n! , which converges for x ∈ B  0,p −(p−1) -1  . Lemma 2.1 For the equation x 2 = a, where 0 = a = p γ(a) (a 0 + a 1 p + ...) [3], 0 ≤ a j ≤ p − 1, a 0 > 0, to have a solution x ∈ Q p , it is necessary and sufficient that the following conditions are fulfilled: 1) γ (a) is even; 2) a 0 is a quadratic residue modulo p if p =2, and moreover a 1 = a 2 =0 if p =2. Lemma 2.2 Let x ∈ B  0,p −(p−1) -1  then | exp p (x)| p =1, | exp p (x)−1| p = |x| p < 1, | log p (1+x)| p = (861)