Abstract
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 \(\mathbb {Z}^2\).
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Acknowledgements
The authors would like to thank Caio Alves and Serguei Popov for insights on how to find the major axis distribution for the ellipses model derived from Poisson cylinder model. Also, we thank an unknown referee for simplifying our argument that \(\mathcal {E}\) does not percolate for \(\alpha = 2\). Finally, we thank Leandro Cruz for Fig. 1. This work had financial support from CNPq Grants 306348/2012-8 and 478577/2012-5, FAPERJ by Grants 202.231/2015 and 200.195/2015 and also from Capes.
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Teixeira, A., Ungaretti, D. Ellipses Percolation. J Stat Phys 168, 369–393 (2017). https://doi.org/10.1007/s10955-017-1795-x
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DOI: https://doi.org/10.1007/s10955-017-1795-x