On the Capacity Functional of the Infinite Cluster of a Boolean Model

Journal 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.