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 $T_{L,d}={0,…, L-1}^d$. We show that there is a constant $\rho \approx .22$ such that for all even $L \geq 4$ and $d$ sufficiently large, certain local Markov chains require exponential time to converge to equilibrium. More precisely, 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 step and whose stationary distribution is uniform, then the convergence to stationarity of $\cM$ is exponential in $L^{d-1}$. Our proof is based on a conductance argument that builds on sensitive new combinatorial enumeration techniques.

💡 Research Summary

The paper investigates the mixing time of local Markov chains that sample proper 3‑colorings of the discrete torus (T_{L,d} = {0,\dots ,L-1}^{d}). A proper 3‑coloring assigns one of three colors to each vertex such that adjacent vertices never share the same color. The state space (\Omega) consists of all such colorings, and the target distribution is the uniform distribution over (\Omega).

Main Result.
For any even side length (L \ge 4) and sufficiently large dimension (d), consider a Markov chain (\mathcal M) that satisfies two natural constraints:

1. Locality. In each transition the chain may recolor at most a fraction (\rho) of the vertices, where (\rho \approx 0.22) is a universal constant. Formally, if (V) denotes the vertex set, then the set of vertices whose colors are changed in a single step has size at most (\rho |V| = \rho L^{d}).
2. Stationarity. The chain is reversible with respect to the uniform distribution on (\Omega).

Under these conditions the total‑variation mixing time (\tau_{\text{mix}}) satisfies \