Abstract

The q-state Potts model is a statistical-mechanical model that generalizes the well-known Ising model. It can be defined on an arbitrary finite graph, and its partition function encodes much important information about that graph (including its chromatic polynomial and its reliability polynomial). The complex zeros of the Potts-model partition function are of interest both to statistical mechanicians (in connection with the Lee-Yang picture of phase transitions) and to combinatorists. I begin by giving an introduction to all these problems. I then sketch two recent (and as yet unpublished) results: (a) Proof of a universal upper bound on the q-plane zeros of the chromatic polynomial (or antiferromagnetic Potts-model partition function) in terms of the graph's maximum degree (maximum number of nearest neighbors to any site). (b) Construction of a countable family of planar graphs whose chromatic zeros are dense in the whole complex q-plane except possibly for the disc |q-1| < 1.

This talk is intended to be understandable to both mathematicians and physicists; no prior knowledge of either graph theory or statistical mechanics is required.