Subcritical regimes in Poisson Boolean percolation on Ahlfors regular spaces

We study the Poisson Boolean percolation model on Ahlfors regular metric measure spaces, extending fundamental results from the Euclidean spaces to more general geometric settings. Ahlfors regular space is a metric measure space that has a polynomial growth rate of metric balls. Our main result establishes that for s-Ahlfors regular spaces, the model exhibits a subcritical regime (no infinite clusters for small intensities) if and only if the radius distribution has a finite s-th moment, generalizing Gouéré’s result for the Euclidean spaces. We prove both directions: when an s-th moment is finite, we show that subcritical behavior exists using geometric properties of Ahlfors regular spaces, particularly the doubling property and the uniform perfectness. Conversely, when an s-th moment diverges, we demonstrate that infinite clusters occur almost surely for any positive intensity. The key technical innovation lies in handling the geometric challenges absent in Euclidean spaces, such as potentially empty annuli between concentric balls. We overcome this using uniform perfectness, which guarantees nonempty annuli under sufficient expansion, combined with doubling properties to control covering numbers. Our results apply broadly to Riemannian manifolds with nonnegative Ricci curvature, ultrametric spaces, unbounded Sierpinski gaskets, and snowflake constructions of Ahlfors regular spaces.