Solvable random network model for disordered sphere packing

A notorious problem in physics is to create a solvable model for random sequential adsorption of non-overlapping congruent spheres in the $d$-dimensional Euclidean space with $d\geq 2$. Spheres arrive sequentially at uniformly chosen locations in space and are accepted only when there is no overlap with previously deposited spheres. Due to spatial correlations, characterizing the fraction of accepted spheres or the area covered by the deposited spheres remains largely intractable. We study these disordered sphere packings by taking a radically novel approach that compares random sequential adsorption in Euclidean space to the nearest-neighbor blocking on a sequence of clustered random networks with a growing number of vertices. This tractable network model leads to a precise characterization of the coverage and elementary laws that describe the packing fraction as a function of density and dimension. By investigating the spatial dimensions two to five over wide density ranges, we show that these laws are accurate and universal. The model also supports a previously conjectured lower bound on the densest packing in high dimensions