Abstract

Consider a circle packing in R^2 (a collection of disjoint discs of radius 1) which covers t fraction of the space. While it is well known there is some maximum fraction t’ < 1 of the space that such a circle packing can cover, which comes from a lattice packing, this talk concerns a sort of “inverse” version of this. Suppose we sample a random circle packing at some density t < t’. When t is small enough, this looks like a “disordered” collection of circles. When t is close enough to t’, one might hope that a global structure emerges and perhaps that it looks like a perturbation of a lattice packing. These problems, along with their higher dimensional analogues, make up some of the central problems on the so called “hard sphere model” where vast gaps in our understanding remain. In this talk I will give a gentle introduction to these problems and discuss some recent progress in a non-Euclidean setting joint with Lewis Bowen and Will Perkins.