Characterizing the limiting critical Potts measures on locally regular-tree-like expander graphs

TL;DR

This paper analyzes the behavior of the ferromagnetic Potts model on expander graphs converging to a regular tree, revealing that at criticality, measures are mixtures of free and wired Gibbs states, with implications for phase coexistence.

Contribution

It extends prior results by characterizing the local weak limits of Potts measures on expander graphs at criticality, including arbitrary phase coexistence and non-integer cluster parameters.

Findings

Subsequential limits are mixtures of free and wired measures.

Existence of graphs with arbitrary phase coexistence extent.

Characterization of local weak limits for random cluster measures.

Abstract

For any integers $d,q\ge 3$, we consider the $q$-state ferromagnetic Potts model with an external field on a sequence of expander graphs that converges to the $d$-regular tree $\mathtt{T}_d$ in the Benjamini-Schramm sense. We show that along the critical line, any subsequential local weak limit of the Potts measures is a mixture of the free and wired Potts Gibbs measures on $\mathtt{T}_d$. Furthermore, we show the possibility of an arbitrary extent of strong phase coexistence: for any $\alpha\in [0,1]$, there exists a sequence of locally $\mathtt{T}_d$-like expander graphs $\{G_n\}$, such that the Potts measures on $\{G_n\}$ locally weakly converges to the $(\alpha,1-\alpha)$-mixture of the free and wired Potts Gibbs measures. Our result extends results of \cite{HJP23} which restrict to the zero-field case and also require $q$ to be sufficiently large relative to $d$, and results of \cite{BDS23} which restrict to the even $d$ case. We also confirm the phase coexistence prediction of \cite{BDS23}, asserting that the Potts local weak limit is a genuine mixture of the free and wired states in a generic setting. We further characterize the subsequential local weak limits of random cluster measures on such graph sequences, for any cluster parameter $q>2$ (not necessarily integer).