Phase coexistence and torpid mixing in the 3-coloring model on ${\mathbb Z}^d$

Abstract

We show that for all sufficiently large $d$, the uniform proper 3-coloring model (in physics called the 3-state antiferromagnetic Potts model at zero temperature) on ${\mathbb Z}^d$ admits multiple maximal-entropy Gibbs measures. This is a consequence of the following combinatorial result: if a proper 3-coloring is chosen uniformly from a box in ${\mathbb Z}^d$, conditioned on color 0 being given to all the vertices on the boundary of the box which are at an odd distance from a fixed vertex $v$ in the box, then the probability that $v$ gets color 0 is exponentially small in $d$. The proof proceeds through an analysis of a certain type of cutset separating $v$ from the boundary of the box and builds on techniques developed by Galvin and Kahn in their proof of phase transition in the hard-core model on ${\mathbb Z}^d$. Building further on these techniques, we study local Markov chains for sampling proper 3-colorings of the discrete torus ${\mathbb Z}^d_n$. We show that there is a constant $\rho \approx 0.22...