Improved mixing bounds for the anti-ferromagnetic Potts model on Z^2

TL;DR

This paper proves strong spatial mixing for the anti-ferromagnetic Potts model on Z^2 for q ≥ 6 and all λ, extending known results and implying rapid mixing of Glauber dynamics and uniqueness of Gibbs states.

Contribution

It establishes strong spatial mixing for q ≥ 6 and any λ, improving previous bounds and enabling efficient approximation algorithms and uniqueness results.

Findings

Strong spatial mixing for q ≥ 6 and all λ.

Rapid mixing of Glauber dynamics.

Uniqueness of the infinite-volume Gibbs state.

Abstract

We consider the anti-ferromagnetic Potts model on the the integer lattice Z^2. The model has two parameters, q, the number of spins, and \lambda=\exp(-\beta), where \beta is ``inverse temperature''. It is known that the model has strong spatial mixing if q>7 or if q=7 and \lambda=0 or \lambda > 1/8 or if q=6 and \lambda=0 or \lambda > 1/4. The lambda=0 case corresponds to the model in which configurations are proper q-colourings of Z^2. We show that the system has strong spatial mixing for q >= 6 and any \lambda. This implies that Glauber dynamics is rapidly mixing (so there is a fully-polynomial randomised approximation scheme for the partition function) and also that there is a unique infinite-volume Gibbs state. We also show that strong spatial mixing occurs for a larger range of \lambda than was previously known for q = 3, 4 and 5.