Subcritical regimes in the Poisson Boolean percolation on Ahlfors regular spaces

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, Gouéré proved that for $n\ge 2$, 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 Gouéré’s result to Ahlfors regular metric measure spaces.