Sampling colourings of the triangular lattice

• Authors: Markus Jalsenius
• Venue: Random Structures & Algorithms
• Year: 2012

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. © 2011 Wiley Periodicals, Inc. Random Struct. Alg., 40, 501–533, 2012 (In (*), Goldberg, Martin and Paterson define \documentclass{article} \usepackage{mathrsfs} \usepackage{amsmath, amssymb} \pagestyle{empty} \begin{document} \begin{align*}\nu(X)= {\mathbb{E}\lbrack 1_{\Psi_X,v_X}\rbrack }\end{align*} \end{document} **image** , which is the definition of μ(X) in this article. They give an alterative definition of μ(X), however, as pointed out in the proof of Lemma 13 in (*), their μ(X) =ν (X) indeed. © 2012 Wiley Periodicals, Inc.)