Torpid mixing of local Markov chains on 3-colorings of the discrete torus

Authors:
David Galvin;Dana Randall
Affiliations:
University of Pennsylvania, Philadelphia, PA;Georgia Institute of Technology, Atlanta, GA
Venue:
SODA '07 Proceedings of the eighteenth annual ACM-SIAM symposium on Discrete algorithms
Year:
2007

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Abstract

We study local Markov chains for sampling 3-colorings of the discrete torus TL, d = {0, ..., L--1}d. We show that there is a constant ρ ≈ .22 such that for all even L ≥ 4 and d sufficiently large, certain local Markov chains require exponential time to converge to equilibrium. More precisely, if M is a Markov chain on the set of proper 3-colorings of TL, d that updates the color of at most ρLd vertices at each step and whose stationary distribution is uniform, then the convergence to stationarity of M is exponential in Ld-1. Our proof is based on a conductance argument that builds on sensitive new combinatorial enumeration techniques.