Antiferromagnetic Potts model on the Erdos-Renyi random graph

TL;DR

This paper investigates the antiferromagnetic Potts model on Erdős-Rényi graphs, establishing a phase transition at a positive temperature through variational principles and second-moment analysis.

Contribution

It introduces a novel interpolation structure and variational framework to analyze phase transitions in the model on random graphs.

Findings

Existence of a phase transition at positive temperature

Bounds on the critical temperature from stability analysis

Application of second-moment method for phase transition proof

Abstract

We study the antiferromagnetic Potts model on the Poissonian Erd\"os-R\'enyi random graph. By identifying a suitable interpolation structure and an extended variational principle, together with a positive temperature second-moment analysis we prove the existence of a phase transition at a positive critical temperature. Upper and lower bounds on the temperature critical value are obtained from the stability analysis of the replica symmetric solution (recovered in the framework of Derrida-Ruelle probability cascades)and from a positive entropy argument.