On Phase Transitions for $P$-Adic Potts Model with Competing Interactions on a Cayley Tree

TL;DR

This paper investigates a three-state p-adic Potts model on a Cayley tree, revealing phase transitions only occur when p=3 and establishing conditions for the uniqueness of Gibbs measures in the p-adic setting.

Contribution

It provides a complete analysis of phase transitions and Gibbs measure uniqueness for the p-adic Potts model on Cayley trees, highlighting the special role of p=3.

Findings

Phase transition occurs if and only if p=3.

Unique Gibbs measure exists for p ≠ 3.

Recursive equations characterize Gibbs measures.

Abstract

In the paper we considere three state $p$-adic Potts model with competing interactions on a Cayley tree of order two. We reduce a problem of describing of the $p$-adic Gibbs measures to the solution of certain recursive equation, and using it we will prove that a phase transition occurs if and only if $p=3$ for any value (non zero) of interactions. As well, we completely solve the uniqueness problem for the considered model in a $p$-adic context. Namely, if $p\neq 3$ then there is only a unique Gibbs measure the model.