Extremes of the internal energy of the Potts model on cubic graphs

TL;DR

This paper establishes precise bounds on the internal energy of the anti-ferromagnetic Potts model on cubic graphs, with implications for graph colorings and confirming a conjecture in extremal combinatorics.

Contribution

It provides the first tight bounds on the internal energy for the Potts model on cubic graphs across all temperatures and all q, and proves a conjecture regarding maximum q-colorings.

Findings

Tight bounds on the internal energy per particle for the Potts model.

Maximization of q-colorings by unions of K_{3,3} graphs.

Confirmation of a conjecture by Galvin and Tetali for d=3.

Abstract

We prove tight upper and lower bounds on the internal energy per particle (expected number of monochromatic edges per vertex) in the anti-ferromagnetic Potts model on cubic graphs at every temperature and for all $q \ge 2$. This immediately implies corresponding tight bounds on the anti-ferromagnetic Potts partition function. Taking the zero-temperature limit gives new results in extremal combinatorics: the number of $q$-colorings of a $3$-regular graph, for any $q \ge 2$, is maximized by a union of $K_{3,3}$'s. This proves the $d=3$ case of a conjecture of Galvin and Tetali.