Structure of the set of all proper colourings

Speaker: Roman Kotecky

Affiliation: Warwick Mathematics Institute, UK

Abstract: A possibility that Potts antiferromagnets at zero temperature feature a long range order has been an intriguing conjecture for quite a long time. Mathematically, the existence of this phase transition amounts to an easily formulated claim about a non-trivial structure of the uniform distribution on the set of all proper colourings of a particular graph---usually a regular lattice.

The needed notions from statistical physics including Gibbs states on proper colourings will be introduced and a reformulation of the transition in terms of a non-unicity of Gibbs states will be explained. The proof that the seeked phase transition indeed occurs for the 3-state Potts antiferromagnet (~ 3-colourings) on the diced lattice will be presented. The main idea is to argue for non-unicity from an appropriate evaluation of entropic barriers between distinct Gibbs states. The talk is based on a joint paper with J. Salas and A. Sokal.