On percolation of two-dimensional hard disks

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 $\epsilon >0$ we consider the graph in which two disks are adjacent if they are at distance $\le \epsilon$ from each other. We prove that this graph, $G$, is highly connected when $\lambda$ is greater than a certain threshold depending on $\epsilon$. 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 $1-C\exp (-c{L}_1)$. As a corollary we prove that a Gibbs state admits an infinite component of $G$ if the intensity $\lambda$ is large enough, depending on $\epsilon$.

49 pages