Probability Seminar

The hard sphere model is a classical model in statistical physics for particles interacting in the continuum. Given a bounded set S in R^d and an activity parameter lambda, one samples a random sphere packing X in S with density proportional to lambda^{|X|}. A phase transition occurs if for large S, the behavior of the random packing changes as lambda varies from being small to large. In particular, we say that the model is in the fluid phase if correlations decay rapidly, and in the non-fluid phase otherwise. We will provide a gentle introduction to this model and describe recent work that establishes the fluid phase for this model and various related models. In high dimensions, this yields algorithms for approximately sampling uniformly random sphere packings of density C*d / 2^d, matching a prediction of Parisi-Zamponi up to the constant C. Our result also shows that certain closely related models provably have no phase transition despite having unique ground states at each density. Time permitting, we will also discuss recent work demonstrating a phase transition for the hard sphere model on the hyperbolic plane. This is based on joint work with Gobel, Jenssen, Pappik, Perkins, and Schiller as well as Bowen and Perkins.