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

Consider a Boolean model in .d with balls of random, bounded radii with distribution F0, centered at the points of a Poisson process of intensity t > 0. The capacity functional of the infinite cluster Z. is given by .L(t) = .{Z. . L . .}, defined for each compact L . .d. We prove for any fixed L and F0 that .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 .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 .L(·) grows at least linearly at the critical value. This implies that the critical exponent known as . 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.