A Personal List of Unsolved Problems Concerning Lattice Gases and Antiferromagnetic Potts Models
2001, v.7, Issue 1, 21-38
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.
Keywords: hard-core lattice gas,$q$-coloring problem,antiferromagnetic Pottsmodel,independent-set polynomial,chromatic polynomial,Gibbs measure,phase transition,Monte Carlo,rapid mixing