On Percolation of Two-Dimensional Hard Disks

Communications in Mathematical Physics. 2018. Vol. 364. No. 1. P. 1–43.

Magazinov A.

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 intensity parameter $\lambda$ (in the same way the intensity of a Poisson point process controls its density).

Given $\varepsilon > 0$, we consider the random graph $G_{\varepsilon}$ in which disks (the vertices) are connected by an edge if they are at distance $\leq \varepsilon$ from each other. We prove that $G$ is highly connected when $\lambda$ is greater than a certain threshold $\lambda_0 = \lambda_0(\varepsilon)$. Namely, given a square annulus with inner radius $L_1$ and outer radius $L_2$ ($L_1 < L_2 < L$), the probability that the annulus is crossed by $G_{\varepsilon}$ is at least $1 - C \exp(-cL_1)$.

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 $G_{\varepsilon}$ corresponding to any Gibbs state almost surely has an infinite connected component whenever the intensity parameter $\lambda$ satisfies $\lambda > \lambda_0(\varepsilon)$.

Keywords: percolation, hard-core model