The phase transition for Boolean percolation

We consider Boolean percolation in dimension d. Around every point of a Poisson point process of intensity lambda, draw a ball of random radius, independently for different points. We investigate the connection probabilities in the subcritical regime and use the randomized algorithm method to prove that the phase transition in lambda is sharp. Interestingly, for this process, sharpness of the phase transition does not imply exponential decay of connection probabilities in the subcritical regime, and its meaning depends on the  law of the radii. In this talk, we will focus on this specific feature of Boolean percolation. This talk is based on a joint work with H. Duminil-Copin and A. Raoufi.