Speaker:
David Galvin (University of Notre Dame)
Title:
Proper $q$-colourings of the cube
Abstract:
Does the uniform proper $q$-colouring model on ${\mathbb Z}^d$ admit multiple Gibbs measures? It's generally believed that for fixed $q$ and large $d$, the answer is yes, but it was only recently that the question was positively answered even for $q=3$ (joint work with Kahn, Randall and Sorkin; also independently by Peled). For $q>3$, little is known.
One way to approach $q>3$ is by first dealing with the simpler problem of proper $q$-colourings of the cube $\{0,1\}^d$, replacing ``admits multiple Gibbs measures'' with, for example, ``exhibits long range correlation''. With J. Engbers, we have extended an entropy approach of Kahn and can say a good deal about this problem.
Here's a representative result, that supports the conjecture of multiple Gibbs measures on
${\mathbb Z}^d$: if a $q$-colouring of $\{0,1\}^d$ is chosen uniformly, conditioned on a particular vertex being coloured red, then the probability that another vertex in the same partite set is coloured red is close to $2/q$ (twice its unconditioned probability), but the probability that a vertex in the opposite partite set is coloured red is close to $0$.
I'll discuss our results, and generalizations to tori with fixed side length.