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

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 rho is an element of Q(p). Such a measure is called generalized p-adic quasi Gibbs measure. When rho equals the p-adic exponent, then it coincides with the p-adic Gibbs measure. When rho = p, then it coincides with the p-adic quasi Gibbs measure. Therefore, we investigate two regimes with respect to the value of vertical bar rho vertical bar(p). Namely, in the first regime, one takes rho = exp(p)(J) for some J is an element of Q(p), in the second one vertical bar rho vertical bar(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 vertical bar rho vertical bar(p), vertical bar q vertical bar(p) <= p(-2) we prove the existence of a quasi phase transition. It turns out that if vertical bar rho vertical bar(p) < vertical bar q - 1 vertical bar(2)(p) < 1 and root-3 is an element of Q(p), then one finds the existence of the strong phase transition.