Phase coexistence and torpid mixing in the 3-coloring model on 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 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 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 Z^d. Building further on these techniques, we study local Markov chains for sampling proper 3-colorings of the discrete torus Z^d_n. We show that there is a constant \rho \approx 0.22 such that for all even n \geq 4 and d sufficiently large, if M is a Markov chain on the set of proper 3-colorings of Z^d_n that updates the color of at most \rho n^d vertices at each step and whose stationary distribution is uniform, then the mixing time of M (the time taken for M to reach a distribution that is close to uniform, starting from an arbitrary coloring) is essentially exponential in n^{d-1}.