Sharpness of the phase transition for continuum percolation on \(\mathbb{R}^2\)

Many complex systems involving a large number of independent variables have come to be very well understood. One such example is Bernoulli percolation on a planar lattice. However, how to adapt the techniques to closely related models, such as continuum percolation in \(\mathbb{R}^2\), may be far from obvious. We will describe some techniques of this kind that recently has been developed for Poisson Boolean percolation. We will focus on a certain two-stage construction that allows for a reduction to the discrete setting, where a larger arsenal of techniques is available for the study of phenomena such as sharp thresholds and noise sensitivity. Joint work with Vincent Tassion and Augusto Teixeira.