A Personal List of Unsolved Problems Concerning Lattice Gases and Antiferromagnetic Potts Models
Abstract
Description
I review recent results and unsolved problems concerning the hard-core lattice gas and the q-coloring model (antiferromagnetic Potts model at zero temperature). For each model, I consider its equilibrium properties (uniqueness/nonuniqueness of the infinite-volume Gibbs measure, complex zeros of the partition function) and the dynamics of local and nonlocal Monte Carlo algorithms (ergodicity, rapid mixing, mixing at complex fugacity). These problems touch on mathematical physics, probability, combinatorics and theoretical computer science.
19 pages, LaTeX2e. Talk presented at the conference on Inhomogeneous Random Systems, Université de Cergy-Pontoise, 25 January 2000. To appear in Markov Processes and Related Fields. Version 2 corrects some mistakes in Sections 4 and 5 (confusion between different notions of mixing time). Version 3 makes the title more specific, adds discussion of the disagreement-percolation method, and makes a few other small improvements
19 pages, LaTeX2e. Talk presented at the conference on Inhomogeneous Random Systems, Université de Cergy-Pontoise, 25 January 2000. To appear in Markov Processes and Related Fields. Version 2 corrects some mistakes in Sections 4 and 5 (confusion between different notions of mixing time). Version 3 makes the title more specific, adds discussion of the disagreement-percolation method, and makes a few other small improvements