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

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