# Nonoptimality of constant radii in high dimensional continuum percolation

Abstract : Consider a Boolean model $\Sigma$ in $\R^d$. The centers are given by a homogeneous Poisson point process with intensity $\lambda$ and the radii of distinct balls are i.i.d.\ with common distribution $\nu$. The critical covered volume is the proportion of space covered by $\Sigma$ when the intensity $\lambda$ is critical for percolation. Previous numerical simulations and heuristic arguments suggest that the critical covered volume may be minimal when $\nu$ is a Dirac measure. In this paper, we prove that it is not the case in sufficiently high dimension.
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https://hal.archives-ouvertes.fr/hal-01068557
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Submitted on : Thursday, September 25, 2014 - 6:20:01 PM
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Jean-Baptiste Gouéré, Régine Marchand. Nonoptimality of constant radii in high dimensional continuum percolation. Annals of Probability, Institute of Mathematical Statistics, 2016, 44 (1), pp.307-323. ⟨10.1214/14-AOP974⟩. ⟨hal-01068557⟩

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