On Inhomogeneous $p$-Adic Potts Model on a Cayley Tree

TL;DR

This paper investigates the behavior of an inhomogeneous $p$-adic Potts model on a Cayley tree, revealing conditions for uniqueness or multiplicity of Gibbs measures based on the parameters $q$, $p$, and the interaction couplings.

Contribution

It provides a rigorous analysis of Gibbs measures for the inhomogeneous $p$-adic Potts model, identifying precise conditions for uniqueness and phase coexistence.

Findings

Unique Gibbs measure when $q otin p ext{N}$ for certain $J_{xy}$

Existence of multiple Gibbs measures when $q ext{in} p ext{N}$ and $p ext{geq}3$

Conditions depend on the relationship between $q$, $p$, and the interaction couplings

Abstract

We consider a nearest-neighbor inhomogeneous $p$-adic Potts (with $q\geq 2$ spin values) model on the Cayley tree of order $k\geq 1$. The inhomogeneity means that the interaction $J_{xy}$ couplings depend on nearest-neighbors points $x, y $ of the Cayley tree. We study ($p-$ adic) Gibbs measures of the model. We show that (i) if $q\notin p\mathbb{N}$ then there is unique Gibbs measure for any $k\geq 1$ and $\forall J_{xy}$ with $|J_{xy}|<p^{-1/(p-1)}$. (ii) For $q\in p\mathbb{N}, p\geq 3$ one can choose $J_{xy}$ and $k\geq 1$ such that there exist at least two Gibbs measures which are translation-invariant.