On percolation of two-dimensional hard disks

TL;DR

This paper studies the percolation properties of a random geometric graph formed by non-overlapping disks in the plane, showing high connectivity at large densities and the existence of infinite clusters.

Contribution

It establishes thresholds for percolation in the hard-core disk model and demonstrates the emergence of infinite components in the associated graph.

Findings

High probability of crossing annuli at large densities

Existence of infinite clusters in Gibbs states for high intensity

Connectivity depends on the intensity parameter mbda

Abstract

We consider the hard-core model in $\mathbb{R}^2$, in which a random set of non-intersecting unit disks is sampled with an intensity parameter $\lambda$. Given $\varepsilon>0$ we consider the graph in which two disks are adjacent if they are at distance $\leq \varepsilon$ from each other. We prove that this graph, $G$, is highly connected when $\lambda$ is greater than a certain threshold depending on $\varepsilon$. Namely, given a square annulus with inner radius $L_1$ and outer radius $L_2$, the probability that the annulus is crossed by $G$ is at least $1 - C \exp(-cL_1)$. As a corollary we prove that a Gibbs state admits an infinite component of $G$ if the intensity $\lambda$ is large enough, depending on $\varepsilon$.