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

Abstract

Consider a Boolean model in $\mathbb{R}^{d}$ 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 $Z_{\infty}$ is given by $\theta_{L}(t)=\mathbb{P}\{Z_{\infty}\cap L\ne\varnothing \}$, defined for each compact $L\subset\mathbb{R}^{d}$.

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 $F:=tF_{0}$, 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 $\theta_{L}(\cdot)$ 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 $d\geq3$ with fixed-radius balls, to the case with bounded random radii.