Phase transitions for p-adic Potts model on the Cayley tree of order three

Abstract

In the present paper, we study a phase transition problem for the q-state p-adic Potts model over the Cayley tree of order three. We consider a more general notion of p-adic Gibbs measure which depends on parameter ρ∈Qp. Such a measure is called generalized p-adic quasi Gibbs measure. When ρ equals the p-adic exponent, then it coincides with the p-adic Gibbs measure. When ρ = p, then it coincides with the p-adic quasi Gibbs measure. Therefore, we investigate two regimes with respect to the value of |ρ|p. Namely, in the first regime, one takes ρ = expp(J) for some J∈Qp, in the second one |ρ|p < 1. In each regime, we first find conditions for the existence of generalized p-adic quasi Gibbs measures. Furthermore, in the first regime, we establish the existence of the phase transition under some conditions. In the second regime, when |ρ|p,|q|p ≤ p−2 we prove the existence of a quasi phase transition. It turns out that if and , then one finds the existence of the strong phase transition.