Ellipses Percolation

We define a continuum percolation model that provides a collection of random ellipses on the plane and study the connectivity 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 (alpha > 0) associated with the tail decay of the major axis distribution; we only consider distributions (rho ) satisfying (rho [r, infty ) asymp r^{-alpha }). We prove that this model presents a double phase transition in (alpha ). For (alpha in (0,1]) the plane is completely covered by the ellipses, almost surely. For (alpha 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 (alpha 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 (alpha = 2) that there is a non-degenerate interval of densities for which the probability of crossing boxes of a fixed proportion is bounded away from zero and one. 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).