# Tree-ansatz percolation of hard spheres

## TL;DR

This paper develops an analytical model for continuum percolation in suspensions of hard spheres, using a treelike approximation valid at small connectivity ranges, and derives an accurate expression for the percolation threshold.

## Contribution

It introduces a novel analytic expression for the percolation threshold of hard spheres based on a treelike network approximation for small connectivity ranges.

## Key findings

- Derived an analytic percolation threshold formula
- Validated the formula's accuracy at small connectivity ranges
- Extended the model's applicability through simple rescaling

## Abstract

Suspensions of hard core spherical particles of diameter $D$ with inter-core connectivity range $\delta$ 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 $\delta$. By exploiting the property that closed loops of connected spheres becomes increasingly rare as the connectivity range diminishes, we study continuum percolation of hard spheres by treating the network of connected particles as having a treelike structure for small $\delta/D$. We derive an analytic expression of the percolation threshold which becomes increasingly accurate as $\delta/D$ diminishes, and whose validity can be extended to a broader range of connectivity distances by a simple rescaling.