Abstract
Consider a Boolean model $\Sigma$ in $\mathbb{R}^{d}$. 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.
Citation
Jean-Baptiste Gouéré. Régine Marchand. "Nonoptimality of constant radii in high dimensional continuum percolation." Ann. Probab. 44 (1) 307 - 323, January 2016. https://doi.org/10.1214/14-AOP974
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