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

TL;DR

This paper proves the infinite differentiability of the capacity functional of the infinite cluster in a Boolean model with random radii, except at the critical point, and explores its behavior near criticality.

Contribution

It establishes the differentiability of the capacity functional with respect to the intensity and radius distribution, extending previous results to models with random radii.

Findings

Capacity functional is infinitely differentiable in supercritical region.

Growth at critical point is at least linear.

Critical exponent β is at most 1.

Abstract

The original 2017 version of this paper, published in Ann. Appl. Probab., 27, 1678--1801, contains a major gap in the proofs. In the subsequent publication in Ann. Appl. Probab., 34, 3370--3374, 2024, we indicated how to fix this. For convenience of the reader, we here update the original paper to incorporate the suggested fix. 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 $Z_\infty$ is given by $\theta_L(t) = P(Z_\infty\cap L \neq \emptyset)$, defined for each compact $L\subset R^d$. 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 H.Tanemura (1993), on regularity of the supercritical Boolean model in $d \geq 3$ with fixed-radius balls, to the case with bounded random radii.