It has long been conjectured that on the square lattice ($\mathbb{Z}^2$), the hard lattice gas model has a critical value $\lambda_c = 3.796…$ with the property: if $\lambda < \lambda_c$, then it exhibits uniqueness of phase, while if $\lambda > \lambda_c$ then there is phase coexistence – existence of multiple Gibbs measures.
The speaker will first review the basics of this model of independent interest in combinatorics, probability, statistical physics and theoretical computer science. Then he will give an update on the status of the problem on the square lattice, highlighting recent efforts that have rigorously established that $\lambda_c$ belongs to the interval [2.538, 5.3506], as well as mentioning related open problems of interest.