Percolation of hard disks

TL;DR

This paper investigates how, in random arrangements of points with hard core exclusion, increasing density leads to the emergence of an infinite connected cluster, demonstrating a phase transition in percolation behavior.

Contribution

It proves that at high density, a percolation phase transition occurs with the formation of an infinite cluster in hard disk arrangements.

Findings

Infinite cluster appears at high density

Percolation transition is proven mathematically

Connectivity emerges in dense hard disk systems

Abstract

Random arrangements of points in the plane, interacting only through a simple hard core exclusion, are considered. An intensity parameter controls the average density of arrangements, in analogy with the Poisson point process. It is proved that at high intensity, an infinite connected cluster of excluded volume appears with positive probability.