Equivalence of some subcritical properties in continuum percolation

TL;DR

This paper establishes the equivalence between finiteness of the expected volume of the origin's connected component and exponential decay of the chain length probability in the subcritical regime of continuum percolation.

Contribution

It proves a new equivalence between subcritical properties in the Boolean model, linking geometric and probabilistic characteristics under optimal conditions.

Findings

Finiteness of expected component volume is equivalent to exponential tail decay of chain length.

Provides conditions under which these properties are equivalent in continuum percolation.

Enhances understanding of phase transition behavior in the Boolean model.

Abstract

We consider the Boolean model on $\R^d$. We prove some equivalences between subcritical percolation properties. Let us introduce some notations to state one of these equivalences. Let $C$ denote the connected component of the origin in the Boolean model. Let $|C|$ denotes its volume. Let $\ell$ denote the maximal length of a chain of random balls from the origin. Under optimal integrability conditions on the radii, we prove that $E(|C|)$ is finite if and only if there exists $A,B >0$ such that $\P(\ell \ge n) \le Ae^{-Bn}$ for all $n \ge 1$.