Hard Constraints and the Bethe Lattice: Adventures at the Interface of Combinatorics and Statistical Physics

Statistical physics models with hard constraints, such as the discrete hardcore 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 symmetrybreaking) and a combinatorics problem (random coloring), as well as some new combinatorial notions, will be presented. 2000 Mathematics Subject Classification: 82B20, 68R10.