Phase coexistence and torpid mixing in the 3-coloring model on Z^d

TL;DR

This paper demonstrates phase coexistence and slow mixing in the 3-coloring model on high-dimensional integer lattices, revealing multiple Gibbs measures and exponential mixing times for local Markov chains.

Contribution

It establishes the existence of multiple maximal-entropy Gibbs measures and proves exponential mixing times for certain local Markov chains in high dimensions.

Findings

Multiple Gibbs measures exist for large d.

Local Markov chains mix exponentially slowly in high dimensions.

Probability of a vertex being color 0 is exponentially small under certain boundary conditions.

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}.