Brightwell, Graham and Winkler, Peter
(2002)
*Hard constraints and the Bethe Lattice: adventures at the interface of combinatorics and statistical physics.*
In:
Proceedings of the International Congress of Mathematicians: Beijing 2002, August 20-28 (International Congress of Mathematician.
Higher Education Pr, pp. 605-624.
ISBN 9787040086904

## 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.

Item Type: | Book Section |
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Official URL: | http://www.mathunion.org/activities/icm/previous/2... |

Additional Information: | © 2002 The author |

Library of Congress subject classification: | Q Science > QA Mathematics |

Date Deposited: | 06 Feb 2012 16:36 |

URL: | http://eprints.lse.ac.uk/10602/ |

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