Existence of $P$-adic quasi Gibbs measure for countable state Potts model on the Cayley tree

TL;DR

This paper introduces a new construction of $p$-adic quasi Gibbs measures for the countable state Potts model on Cayley trees, establishing their existence under certain conditions independent of the prime $p$.

Contribution

It provides a novel measure construction for the $p$-adic Potts model and proves the existence of $p$-adic quasi Gibbs measures under conditions not dependent on the prime $p$.

Findings

Construction of $p$-adic quasi Gibbs measures for the model.

Existence of measures under conditions independent of $p$.

Coincidence with $p$-adic Gibbs measure when $rak{p}= ext{exp}_p$.

Abstract

In the present paper we provide a new construction of measure, called $p$-adic quasi Gibbs measure, for countable state of $p$-adic Potts model on the Cayley tree. Such a construction depends on a parameter $\frak{p}$ and wights. In particular case, i.e. if $\frak{p}=\exp_p$, the defined measure coincides with $p$-adic Gibbs measure. In this paper, under some condition on weights we establish the existence of $p$-adic quasi Gibbs measures associated with the model. Note that this condition does not depend on values of the prime $p$. An analogues fact is not valid when the number of spins is finite.