### 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.