On Percolation of Two-Dimensional Hard Disks

Let QL = [−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 QL 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 \lambda_0}\) \({(\epsilon}\)).