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Hard constraints and the Bethe Lattice: adventures at the interface of combinatorics and statistical physics

By Graham Brightwell and Peter Winkler

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

Topics: QA Mathematics
Publisher: Higher Education Pr
Year: 2002
OAI identifier: oai:eprints.lse.ac.uk:10602
Provided by: LSE Research Online
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