A Personal List of Unsolved Problems Concerning Lattice Gases and Antiferromagnetic Potts Models

TL;DR

This paper reviews recent findings and open problems related to lattice gases and antiferromagnetic Potts models, focusing on equilibrium properties and algorithm dynamics across multiple disciplines.

Contribution

It provides a comprehensive overview of current knowledge and highlights unresolved issues in the study of lattice gases and Potts models, connecting physics, probability, combinatorics, and computer science.

Findings

Summary of recent results on lattice gases and Potts models

Identification of key open problems in equilibrium properties

Discussion of algorithmic dynamics and mixing times

Abstract

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.