Description of all translation-invariant $p$-adic Gibbs measures for the Potts model on a Cayley tree

TL;DR

This paper characterizes all translation-invariant $p$-adic Gibbs measures for the Potts model on Cayley trees, providing exact counts for order two trees and criteria for measure boundedness, extending classical results to the $p$-adic setting.

Contribution

It offers a complete description of translation-invariant $p$-adic Gibbs measures for the Potts model, including explicit enumeration and boundedness criteria, which was previously unexplored.

Findings

Exact number of TIpGMs for Cayley tree of order two.

Criteria for boundedness of TIpGMs.

Extension of classical Gibbs measure results to $p$-adic context.

Abstract

Recently it was proved that usual (real) Potts model on a Cayley tree has up to $2^q-1$ translation-invariant Gibbs measures. This paper is devoted to description of translation- invariant $p$-adic Gibbs measures (TIpGMs) of the $p$-adic Potts model. In particular, for the Cayley tree of order two we give exact number of such measures. Mereover we give criterion of boundedness of TIpGMs.