Tree-ansatz percolation of hard spheres

Suspensions of hard core spherical particles of diameter D with inter-core connectivity range $δ$ can be described in terms of random geometric graphs, where nodes represent the sphere centers and edges are assigned to any two particles separated by a distance smaller than $δ$⁠. By exploiting the property that closed loops of connected spheres become increasingly rare as the connectivity range diminishes, we study continuum percolation of hard spheres by treating the network of connected particles as having a tree-like structure for small $δ/D$⁠. We derive an analytic expression of the percolation threshold which becomes increasingly accurate as $δ/D$ diminishes and whose validity can be extended to a broader range of connectivity distances by a simple rescaling.