Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

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

Günter Last, Mathew D. Penrose, Sergei Zuyev

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 $θ_L(t) = P(Z_\infty\cap L \neq \emptyset)$, defined for each compact $L\subset R^d$. We prove for any fixed $L$ and $F_0$ that $θ_L(t)$ 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 $θ_L$ 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 $θ_L(\cdot)$ grows at least linearly at the critical value. This implies that the critical exponent known as $β$ 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.

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

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 with balls of random, bounded radii with distribution , centered at the points of a Poisson process of intensity . The capacity functional of the infinite cluster is given by , defined for each compact . We prove for any fixed and that is infinitely differentiable in , except at the critical value ; we give a Margulis-Russo type formula for the derivatives. More generally, allowing the distribution to vary and viewing as a function of the measure , we show that it is infinitely differentiable in all directions with respect to the measure in the supercritical region of the cone of positive measures on a bounded interval. We also prove that 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 H.Tanemura (1993), on regularity of the supercritical Boolean model in with fixed-radius balls, to the case with bounded random radii.

Paper Structure

This paper contains 10 sections, 9 theorems, 80 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Main results
  4. Smoothness of the capacity functional
  5. Bounds for the capacity functional
  6. Percolation in a slab when $d \geq 3$
  7. A preparatory result
  8. Proof of Theorem \ref{['3diff']}
  9. Proof of Theorem \ref{['texponent']}
  10. Final remarks

Key Result

Proposition 2.2

Let $\mu$ be a locally finite measure and $\nu$ a finite signed measure on $\mathcal{B}({\mathbb X})$. Let $f:{\mathbf N}\rightarrow{\mathbb R}$ be measurable and bounded. If $\mu + a \nu$ is a measure for some $a >0$, then If also $\mu - a \nu$ is a measure, then $\mathbb{E}_{\mu+s\nu} f(\Phi)$ is differentiable in $s$ at $s=0$.

Figures (1)

  • Figure 1: Geometry of the paths (depicted as black curves) connecting $L$ to ${\mathbb R}^d\setminus B_n$. Pivotal grains for $J_L^n$ are coloured white. The last pivotal grain starting from $L$ is denoted $K$.

Theorems & Definitions (21)

  • Remark 2.1
  • Proposition 2.2
  • Theorem 3.1
  • Theorem 3.2
  • proof
  • Remark 3.3
  • Theorem 3.4
  • Remark 3.5
  • Remark 3.6
  • Theorem 3.7
  • ...and 11 more