On the strong phase transition for the one-dimensional countable state p-adic Potts model

Abstract

In the present paper we consider the countable state p-adic Potts model on . A main aim is to establish the existence of the strong phase transition for the model. In our study, we essentially use one dimensionality of the model. To prove the existence of the phase transition, we reduce the problem to the investigation of an infinite-dimensional nonlinear equation. We find a condition on weights to show that the derived equation has two solutions. We show that measures corresponding to first and second solutions are a p-adic Gibbs and generalized p-adic Gibbs measures, respectively. Moreover, it is proved that the p-adic Gibbs measure is bounded, and the generalized one is not bounded. This implies the existence of the strong phase transition. Note that it turns out that the obtained condition does not depend on values of the prime p and, therefore, an analogous fact is not true when the number of spins is finite. Note that, in the usual real case, if one considers a one-dimensional translation-invariant model with nearest neighbor interaction, then such a model does not exhibit a phase transition. Nevertheless, we should stress that our model exhibits a unique p-adic Gibbs measure.