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We study local Markov chains for sampling 3-colorings of the discrete torus$T_{L,d}={0,..., L-1}^d$. We show that there is a constant $\rho \approx .22$such that for all even $L \geq 4$ and $d$ sufficiently large, certain localMarkov chains require exponential time to converge to equilibrium. Moreprecisely, if $\cM$ is a Markov chain on the set of proper 3-colorings of$T_{L,d}$ that updates the color of at most $\rho L^d$ vertices at each stepand whose stationary distribution is uniform, then the convergence tostationarity of $\cM$ is exponential in $L^{d-1}$. Our proof is based on aconductance argument that builds on sensitive new combinatorial enumerationtechniques.

Torpid Mixing of Local Markov Chains on 3-Colorings of the Discrete Torus