Rapid mixing for lattice colourings with fewer colours

We provide an optimally mixing Markov chain for 6-colourings 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 colourings has strong spatial mixing, so the six-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-colourings of the square lattice.

[1]  D. Aldous.  Random walks on finite groups and rapidly mixing markov chains , 1983 .

[2]  Mark Jerrum,et al.  Approximate Counting, Uniform Generation and Rapidly Mixing Markov Chains , 1987, WG.

[3]  Wang,et al.  Antiferromagnetic Potts models. , 1989, Physical review letters.

[4]  Wang,et al.  Three-state antiferromagnetic Potts models: A Monte Carlo study. , 1990, Physical review. B, Condensed matter.

[5]  P. Diaconis,et al.  Geometric Bounds for Eigenvalues of Markov Chains , 1991 .

[6]  P. Diaconis,et al.  COMPARISON THEOREMS FOR REVERSIBLE MARKOV CHAINS , 1993 .

[7]  Alistair Sinclair,et al.  Algorithms for Random Generation and Counting: A Markov Chain Approach , 1993, Progress in Theoretical Computer Science.

[8]  Dana Randall,et al.  Markov chain algorithms for planar lattice structures , 1995, Proceedings of IEEE 36th Annual Foundations of Computer Science.

[9]  Mark Jerrum,et al.  A Very Simple Algorithm for Estimating the Number of k-Colorings of a Low-Degree Graph , 1995, Random Struct. Algorithms.

[10]  A. Sokal,et al.  Absence of phase transition for antiferromagnetic Potts models via the Dobrushin uniqueness theorem , 1996, cond-mat/9603068.

[11]  Martin E. Dyer,et al.  Path coupling: A technique for proving rapid mixing in Markov chains , 1997, Proceedings 38th Annual Symposium on Foundations of Computer Science.

[12]  Alan D. Sokal,et al.  Antiferromagnetic Potts Models on the Square Lattice: A High-Precision Monte Carlo Study , 1998 .

[13]  Martin E. Dyer,et al.  Beating the 2Δ bound for approximately counting colourings: a computer-assisted proof of rapid mixing , 1998, SODA '98.

[14]  A more rapidly mixing Markov chain for graph colorings , 1998, Random Struct. Algorithms.

[15]  Martin E. Dyer,et al.  On Approximately Counting Colorings of Small Degree Graphs , 1999, SIAM J. Comput..

[16]  C Moore,et al.  Vortex dynamics and entropic forces in antiferromagnets and antiferromagnetic Potts models. , 1999, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[17]  Cristopher Moore,et al.  Internal Diffusion-Limited Aggregation: Parallel Algorithms and Complexity , 1999, cond-mat/9909233.

[18]  Cristopher Moore,et al.  Height Representation, Critical Exponents, and Ergodicity in the Four-State Triangular Potts Antiferromagnet , 1999, cond-mat/9902295.

[19]  Eric Vigoda,et al.  Improved bounds for sampling colorings , 1999, 40th Annual Symposium on Foundations of Computer Science (Cat. No.99CB37039).

[20]  F. Martinelli.  Lectures on Glauber dynamics for discrete spin models , 1999 .

[21]  Catherine S. Greenhill,et al.  Surveys in Combinatorics, 1999: Random Walks on Combinatorial Objects , 1999 .

[22]  P. Tetali,et al.  Analyzing Glauber dynamics by comparison of Markov chains , 2000 .

[23]  Alan D. Sokal.  A Personal List of Unsolved Problems Concerning Lattice Gases and Antiferromagnetic Potts Models , 2000 .

[24]  F. Cesi.  Quasi-factorization of the entropy and logarithmic Sobolev inequalities for Gibbs random fields , 2001 .

[25]  Michael Molloy.  Very rapidly mixing Markov Chains for 2D-colorings and for independent sets in a graph with maximum degree 4 , 2001 .

[26]  Martin E. Dyer,et al.  Mixing in time and space for lattice spin systems: A combinatorial view , 2002, RANDOM.

[27]  Leslie Ann Goldberg,et al.  Random sampling of 3-colorings in Z2 , 2004, Random Struct. Algorithms.

[28]  Leslie Ann Goldberg,et al.  Random sampling of 3-colorings in Z 2 , 2004 .

[29]  Martin E. Dyer,et al.  Mixing in time and space for lattice spin systems: A combinatorial view , 2002, International Workshop Randomization and Approximation Techniques in Computer Science.