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

Galvin, David, Kahn, Jeff, Randall, Dana and Sorkin, Gregory B. ORCID: 0000-0003-4935-7820 (2015) Phase coexistence and torpid mixing in the 3-coloring model on Z^d. SIAM Journal on Discrete Mathematics, 29 (3). pp. 1223-1244. ISSN 0895-4801

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Identification Number: 10.1137/12089538X


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}.

Item Type: Article
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Additional Information: © 2015 Society for Industrial and Applied Mathematics
Divisions: LSE
Subjects: Q Science > QA Mathematics
Date Deposited: 06 May 2015 08:41
Last Modified: 17 Jul 2024 21:24

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