Galvin, David, Kahn, Jeff, Randall, Dana and Sorkin, Gregory B. ORCID: 0000000349357820
(2015)
Phase coexistence and torpid mixing in the 3coloring model on Z^d.
SIAM Journal on Discrete Mathematics, 29 (3).
pp. 12231244.
ISSN 08954801
Abstract
We show that for all sufficiently large d, the uniform proper 3coloring model (in physics called the 3state antiferromagnetic Potts model at zero temperature) on Z^d admits multiple maximalentropy Gibbs measures. This is a consequence of the following combinatorial result: if a proper 3coloring 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 hardcore model on Z^d. Building further on these techniques, we study local Markov chains for sampling proper 3colorings 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 3colorings 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^{d1}.
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