Almost Sharp Sharpness for Boolean Percolation

We consider a Poisson point process on Rd with intensity lambda for d greater than or = 2. On each point, we independently center a ball whose radius is distributed according to some power-law distribution mu. When the distribution mu has a finite d-moment, there exists a non-trivial phase transition in lambda associated to the existence of an infinite connected component of balls. We aim here to prove subcritical sharpness, that is, that the subcritical regime behaves well in some sense. For distribution mu with a finite 5d−3−moment, Duminil-Copin--Raoufi--Tassion proved subcritical sharpness using a randomized algorithm. We prove here using different methods that the subcritical regime is sharp for all but a countable number of power-law distributions. Joint work with Vincent Tassion.