Strong Spatial Mixing and Rapid Mixing with Five Colours for the Kagome Lattice

TL;DR

This paper proves strong spatial mixing and rapid mixing of Glauber dynamics for proper 5-colourings of the kagome lattice at zero temperature, advancing understanding of phase uniqueness and sampling efficiency in statistical physics models.

Contribution

It provides the first computer-assisted proof of strong spatial mixing for q=5 colours on the kagome lattice at zero temperature, and establishes rapid mixing of Glauber dynamics for sampling.

Findings

Strong spatial mixing holds for q=5 on the kagome lattice at zero temperature.

Glauber dynamics mixes rapidly with free boundary conditions.

Implication for efficient sampling and counting of proper 5-colourings.

Abstract

We consider proper 5-colourings of the kagome lattice. Proper q-colourings correspond to configurations in the zero-temperature q-state anti-ferromagnetic Potts model. Salas and Sokal have given a computer assisted proof of strong spatial mixing on the kagome lattice for q>=6 under any temperature, including zero temperature. It is believed that there is strong spatial mixing for q>=4. Here we give a computer assisted proof of strong spatial mixing for q=5 and zero temperature. It is commonly known that strong spatial mixing implies that there is a unique infinite-volume Gibbs measure and that the Glauber dynamics is rapidly mixing. We give a proof of rapid mixing of the Glauber dynamics on any finite subset of the vertices of the kagome lattice, provided that the boundary is free (not coloured). The Glauber dynamics is not necessarily irreducible if the boundary is chosen arbitrarily for q=5 colours. The Glauber dynamics can be used to uniformly sample proper 5-colourings. Thus, a consequence of rapidly mixing Glauber dynamics is that there is fully polynomial randomised approximation scheme for counting the number of proper 5-colourings.