Subcritical sharpness for fractal Boolean percolation

Abstract

Institute of Mathematical Statistics Rᵈ with radius distribution a Radon measure ν on (0, +∞), d ≥ 2 such that v|[1,+∞) has a finite d-moment. In this level of generality, the subcritical phase is not known to exist. We prove that under some regularity assumption on ν, the subcritical phase of the model, if existing, is sharp. Moreover, we prove that the existence of an unbounded connected component depends only on the fractal part (and not of the balls with radius larger than 1).