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On the capacity functional of the infinite cluster of a boolean model

Journal article
Authors Günter Last
Mathew D. Penrose
Sergei Zuyev
Published in Annals of Applied Probability
Volume 27
Pages 1678-1701
ISSN 1050-5164
Publication year 2017
Published at Department of Mathematical Sciences
Pages 1678-1701
Language en
Keywords Boolean model, Capacity functional, Continuum percolation, Infinite cluster, Margulis-Russo-type formula, Percolation function, REIMER inequality
Subject categories Probability Theory and Statistics


© 2017 Institute of Mathematical Statistics. Consider a Boolean model in Rd with balls of random, bounded radii with distribution F0, centered at the points of a Poisson process of intensity t & lt; 0. The capacity functional of the infinite cluster Z & infin; is given by & theta;L(t) = P{Z & infin; & cap; L & ne; & oslash;}, defined for each compact L & sub; Rd. We prove for any fixed L and F0 that & theta;L(t) is infinitely differentiable in t , except at the critical value tc; we give a Margulis-Russo-type formula for the derivatives. More generally, allowing the distribution F0 to vary and viewing & theta;L as a function of the measure F := tF0, we show that it is infinitely differentiable in all directions with respect to the measure F in the supercritical region of the cone of positive measures on a bounded interval. We also prove that & theta;L grows at least linearly at the critical value. This implies that the critical exponent known as & beta; is at most 1 (if it exists) for this model. Along the way, we extend a result of Tanemura [J. Appl. Probab. 30 (1993) 382-396], on regularity of the supercritical Boolean model in d & ge; 3 with fixed-radius balls, to the case with bounded random radii.

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