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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.
##### MSC:
 60K35 Interacting random processes; statistical mechanics type models; percolation theory 82B43 Percolation
##### Keywords:
percolation; continuum percolation; Boolean model
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##### References:
 [1] K. B. Athreya and P. E. Ney. Branching Processes. Die Grundlehren der mathematischen Wissenschaften196 Springer-Verlag, New York, 1972. [2] A. Balram and D. Dhar. Scaling relation for determining the critical threshold for continuum percolation of overlapping discs of two sizes. Pramāna74 (2010) 109–114. [3] R. Consiglio, D. R. Baker, G. Paul and H. E. Stanley. Continuum percolation thresholds for mixtures of spheres of different sizes. Phys. A319 (2003) 49–55. · Zbl 1008.60104 [4] D. Dhar. On the critical density for continuum percolation of spheres of variable radii. Phys. A242 (3–4) (1997) 341–346. [5] R. Durrett. Oriented percolation in two dimensions. Ann. Probab.12 (4) (1984) 999–1040. · Zbl 0567.60095 [6] J.-B. Gouéré. Subcritical regimes in the Poisson Boolean model of continuum percolation. Ann. Probab.36 (4) (2008) 1209–1220. · Zbl 1148.60077 [7] J.-B. Gouéré. Subcritical regimes in some models of continuum percolation. Ann. Appl. Probab.19 (4) (2009) 1292–1318. · Zbl 1172.60335 [8] J.-B. Gouéré and R. Marchand. Nonoptimality of constant radii in high dimensional continuum percolation. Ann. Probab.44 (1) (2016) 307–323. · Zbl 1342.60167 [9] R. Meester and R. Roy. Continuum Percolation. Cambridge Tracts in Mathematics119. Cambridge University Press, Cambridge, 1996. · Zbl 0858.60092 [10] R. Meester, R. Roy and A. Sarkar. Nonuniversality and continuity of the critical covered volume fraction in continuum percolation. J. Stat. Phys.75 (1–2) (1994) 123–134. · Zbl 0828.60083 [11] J. Møller. Lectures on Random Voronoĭ Tessellations. Lecture Notes in Statistics87. Springer-Verlag, New York, 1994. [12] M. D. Penrose. Continuum percolation and Euclidean minimal spanning trees in high dimensions. Ann. Appl. Probab.6 (2) (1996) 528–544. · Zbl 0855.60096 [13] J. A. Quintanilla and R. M. Ziff. Asymmetry in the percolation thresholds of fully penetrable disks with two different radii. Phys. Rev. E76 (5) (2007) 051115. [14] J. van den Berg. A note on disjoint-occurrence inequalities for marked Poisson point processes. J. Appl. Probab.33 (2) (1996) 420–426. · Zbl 0860.60013
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