# Centre for Discrete and Applicable Mathematics

## CDAM Research Report, LSE-CDAM-97-01

October 1997

Graph Homomorphisms and Phase Transitions

Graham R. Brightwell and Peter Winkler

Abstract

We model physical systems with hard constraints'' by the space Hom(G,H) of homomorphisms from a locally finite graph G to a fixed finite constraint graph H. For any assignment $\lambda$ of positive real activities to the nodes of H, there is at least one Gibbs measure on Hom(G,H); when G is infinite, there may be more than one.

When G is a regular tree, the simple, invariant Gibbs measures on Hom(G,H) correspond to node-weighted branching random walks on H. We show that such walks exist for every H and $\lambda$, and characterize those H which, by admitting more than one such construction, exhibit phase transition behavior.

If you would like a free copy of this report, please send the number of this report, LSE-CDAM-97-12, together with your name and postal address to:
 CDAM Research Reports Series Centre for Discrete and Applicable Mathematics London School of Economics Houghton Street London WC2A 2AE, U.K. Phone: +44(0)-171-955 7732. Fax: +44(0)-171-955 6877. Email: info@maths.lse.ac.uk

 Introduction to the CDAM Research Report Series. CDAM Homepage.