On Gibbs Measures of $P$-Adic Potts Model on the Cayley Tree

TL;DR

This paper investigates phase transitions in a $p$-adic Potts model on Cayley trees, identifying specific conditions under which phase transitions occur based on the parameters $p$, $q$, and the tree order.

Contribution

It provides a rigorous analysis of phase transition phenomena for the $p$-adic Potts model on Cayley trees, highlighting the influence of $p$, $q$, and the tree order on the existence of phase transitions.

Findings

Phase transition occurs at $k=2$, $q ext{ in } p ext{N}$, and $p ext{ at least } 3$.

For $k ext{ at least } 3$, phase transition may only occur if $q ext{ in } p ext{N}$ for } p ext{ at least } 3$.

Phase transition conditions depend on the divisibility properties of $q$ relative to $p$ and the tree order.

Abstract

We consider a nearest-neighbor $p$-adic Potts (with $q\geq 2$ spin values and coupling constant $J\in \Q_p$) model on the Cayley tree of order $k\geq 1$. It is proved that a phase transition occurs at $k=2$, $q\in p\mathbb{N}$ and $p\geq 3$ (resp. $q\in 2^2\mathbb{N}$, $p=2$). It is established that for $p$-adic Potts model at $k\geq 3$ a phase transition may occur only at $q\in p\mathbb{N}$ if $p\geq 3$ and $q\in 2^2\mathbb{N}$ if $p=2$.