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

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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∈ Q p . 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 = exp p ( J ) for some J ∈ Q p , 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.