Rapid Mixing for Lattice Colorings with Fewer Colors

TL;DR

This paper introduces an optimally mixing Markov chain for 6-colorings of the square lattice, demonstrating strong spatial mixing and a finite correlation length, advancing understanding of the Potts antiferromagnet at zero temperature.

Contribution

It provides the first known rapidly mixing Markov chain for 6-colorings on the square lattice, filling a key gap in lattice coloring research.

Findings

Markov chain mixes rapidly for 6-colorings

Strong spatial mixing established for these colorings

Implication for the uniqueness of Gibbs measure at zero temperature

Abstract

We provide an optimally mixing Markov chain for 6-colorings of the square lattice on rectangular regions with free, fixed, or toroidal boundary conditions. This implies that the uniform distribution on the set of such colorings has strong spatial mixing, so that the 6-state Potts antiferromagnet has a finite correlation length and a unique Gibbs measure at zero temperature. Four and five are now the only remaining values of q for which it is not known whether there exists a rapidly mixing Markov chain for q-colorings of the square lattice.