Sampling Colourings of the Triangular Lattice

TL;DR

This paper proves rapid mixing of Glauber dynamics for proper 9-colourings of the triangular lattice, enabling efficient sampling and counting, and establishes strong spatial mixing and uniqueness of the Gibbs distribution.

Contribution

It extends previous results by showing rapid mixing and strong spatial mixing for 9 colours, using computational assistance and heuristics to achieve rigorous proofs.

Findings

Rapid mixing of Glauber dynamics for 9-colourings

Existence of a fully polynomial randomized approximation scheme (FPRAS)

Strong spatial mixing and unique Gibbs distribution for the 9-colour case

Abstract

We show that the Glauber dynamics on proper 9-colourings of the triangular lattice is rapidly mixing, which allows for efficient sampling. Consequently, there is a fully polynomial randomised approximation scheme (FPRAS) for counting proper 9-colourings of the triangular lattice. Proper colourings correspond to configurations in the zero-temperature anti-ferromagnetic Potts model. We show that the spin system consisting of proper 9-colourings of the triangular lattice has strong spatial mixing. This implies that there is a unique infinite-volume Gibbs distribution, which is an important property studied in statistical physics. Our results build on previous work by Goldberg, Martin and Paterson, who showed similar results for 10 colours on the triangular lattice. Their work was preceded by Salas and Sokal's 11-colour result. Both proofs rely on computational assistance, and so does our 9-colour proof. We have used a randomised heuristic to guide us towards rigourous results.