Ellipses Percolation

TL;DR

This paper introduces a continuum percolation model with random ellipses, revealing a double phase transition in the plane's coverage and percolation properties depending on the tail decay parameter of the ellipse size distribution.

Contribution

It generalizes the Poisson cylinder model to ellipses, analyzing phase transitions and percolation behavior across different tail decay regimes of the major axis distribution.

Findings

Complete coverage for  

Vacant set non-empty but non-percolating for   2

Both covered and vacant sets percolate for  2  

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 $\alpha > 0$ 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 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 $\mathbb{Z}^2$.