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Continuum percolation in high dimensions. (English. French summary) Zbl 1417.60080
Summary: 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. We study the asymptotic behaviour, as \(d\) tends to infinity, of the critical covered volume. It appears that, in contrast to what happens in the constant radii case studied by Penrose, geometrical dependencies do not always vanish in high dimension.
60K35 Interacting random processes; statistical mechanics type models; percolation theory
82B43 Percolation
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