Hard constraints and the bethe lattice: adventures at the interface of combinatorics and statistical physics

TL;DR

This paper explores the interplay between combinatorics and statistical physics through models with hard constraints on the Bethe lattice, highlighting phase transitions, multiple Gibbs measures, and applications to physics and coloring problems.

Contribution

It surveys recent work on homomorphism spaces from infinite graphs to finite graphs, characterizing when multiple Gibbs measures exist on the Bethe lattice, with new combinatorial insights.

Findings

Characterization of constraint graphs with multiple Gibbs measures

Applications to symmetry-breaking phase transitions

Insights into random graph coloring problems

Abstract

Statistical physics models with hard constraints, such as the discrete hard-core gas model (random independent sets in a graph), are inherently combinatorial and present the discrete mathematician with a relatively comfortable setting for the study of phase transition. In this paper we survey recent work (concentrating on joint work of the authors) in which hard-constraint systems are modeled by the space $\hom(G,H)$ of homomorphisms from an infinite graph $G$ to a fixed finite constraint graph $H$. These spaces become sufficiently tractable when $G$ is a regular tree (often called a Cayley tree or Bethe lattice) to permit characterization of the constraint graphs $H$ which admit multiple invariant Gibbs measures. Applications to a physics problem (multiple critical points for symmetry-breaking) and a combinatorics problem (random coloring), as well as some new combinatorial notions, will be presented.