On Percolation of Two-Dimensional Hard Disks

Abstract

Let Q_L =  [−L, L]² be a square in the plane ${\mathbb{R}^{2}}$ . We consider the hard-core model with arbitrary boundary conditions in which a random set of non-intersecting unit disks (i.e., a packing) with centers in Q_L is sampled. The density of the packing is controlled by the an intensity parameter ${\lambda}$ similarly to the Poisson point process. Given ${\epsilon}$ > 0, we consider the random graph ${{G}_{\epsilon}}$ in which disks (the vertices) are connected by an edge if they are at distance ≤ ${\epsilon}$ from each other.We prove that G is highly connected when ${\lambda}$ is greater than a certain threshold λ0 =  λ₀(${\epsilon}$). Namely, given a square annulus with inner radius L₁ and outer radius L₂ (L₁ < L₂ < L), the probability that the annulus is crossed by ${{G}_{\epsilon}}$ is at least 1 − C exp(−cL₁). We also extend our results to random packings of disks in the entire plane using the well-known notion of a Gibbs state. We show that a random graph ${{G}_{\epsilon}}$ corresponding to any Gibbs state almost surely has an infinite connected component whenever the intensity parameter ${\lambda}$ satisfies ${\lambda > \lambda_0}$ ${(\epsilon}$).