Ellipses Percolation

Abstract

We define a continuum percolation model that provides a collection of random ellipses on the plane and study the behavior of the covered set and the vacant set, the one obtained by removing all ellipses. Our model generalizes a construction that appears implicitly in the Poisson cylinder model of Tykesson and Windisch. The ellipses model has a parameter $α> 0$ associated with the tail decay of the major axis distribution; we only consider distributions $ρ$ satisfying $ρ[r, \infty) \asymp r^{-α}$. We prove that this model presents a double phase transition in $α$. For $α\in (0,1]$ the plane is completely covered by the ellipses, almost surely. For $α\in (1,2)$ the vacant set is not empty but does not percolate for any positive density of ellipses, while the covered set always percolates. For $α\in (2, \infty)$ the vacant set percolates for small densities of ellipses and the covered set percolates for large densities. Moreover, we prove for the critical parameter $α= 2$ that there is a non-degenerate interval of density for which the probability of crossing boxes of a fixed proportion is bounded away from zero and one, a rather unusual phenomenon. In this interval neither the covered set nor the vacant set percolate, a behavior that is similar to critical independent percolation on $\mathbb{Z}^2$.