Magazine article, 2017

Consider a Boolean model in R-d with balls of random, bounded radii with distribution F-0, centered at the points of a Poisson process of intensity t > 0. The capacity functional of the infinite cluster Z(infinity) is given by theta(L) (t) = P{Z(infinity) boolean AND L not equal empty set L not equal phi}, defined for each compact L subset of R-d. We prove for any fixed L and F-0 that theta(L) (t) is infinitely differentiable in t, except at the critical value t(c); we give a Margulis-Russo-type formula for the derivatives. More generally, allowing the distribution F-0 to vary and viewing theta(L), as a function of the measure F := t F-0, 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 >= 3 with fixed-radius balls, to the case with bounded random radii.

poisson processes

supercritical phase

continuum percolation

Karlsruhe Institute of Technology (KIT)

University of Bath

University of Gothenburg

Chalmers, Mathematical Sciences, Analysis and Probability Theory

1050-5164 (ISSN)

Vol. 27 3 1678-1701Probability Theory and Statistics

10.1214/16-AAP1241