On the Potts antiferromagnet on random graphs

TL;DR

This paper calculates the free energy of the Potts antiferromagnet on Erdős-Rényi random graphs across all temperatures for certain average degrees, showing no phase transition occurs in this regime.

Contribution

It extends previous work by precisely determining the free energy for the Potts antiferromagnet on random graphs at all temperatures within a specific degree range.

Findings

Free energy computed for all temperatures in the specified degree regime.

No phase transition occurs for average degrees up to the specified bound.

Generalization of prior results to a broader parameter range.

Abstract

Extending a prior result of Contucci et al (Comm. Math. Phys. 2013), we determine the free energy of the Potts antiferromagnet on the Erd\"os-R\'enyi random graph at all temperatures for average degrees $d \le (2k-1)\ln k - 2 - k^{-1/2}$. In particular, we show that for this regime of $d$ there does not occur a phase transition.