Subcritical regimes in the Poisson Boolean percolation on Ahlfors regular spaces

TL;DR

This paper extends Gouéré's characterization of subcritical regimes in Poisson Boolean percolation from Euclidean spaces to Ahlfors regular metric measure spaces, broadening the understanding of percolation behavior in more general geometric contexts.

Contribution

It generalizes Gouéré's result on subcritical regimes to Ahlfors regular spaces, providing a broader theoretical framework for percolation models.

Findings

Subcritical regime characterized by finite n-th moment of radius distribution.

Extension of Euclidean results to Ahlfors regular metric measure spaces.

Provides conditions for percolation behavior in more general spaces.

Abstract

The Poisson Boolean percolation on a metric measure space is one of the percolation models. Intuitively, this model is obtained by collecting random balls whose centers form a Poisson point process. In 2008, Gou\'{e}r\'{e} proved that for $n \geq 2$, the Poisson Boolean percolation on $\mathbb{R}^n$ has the subcritical regime if and only if the radius distribution has finite $n$-th moment. In this paper, we extend Gou\'{e}r\'{e}'s result to Ahlfors regular metric measure spaces.