Sharp phase transition and noise sensitivity in continuum percolation via continuous time decision trees.

Abstract: Proofs of sharp phase transition and noise sensitivity in percolation have been significantly simplified by the use of randomized algorithms via the OSSS (O'Donnell, Saks, Schramm and Servedio) variance inequality and the Schramm-Steif inequality. In a joint work with Giovanni Peccati and Guenter Last, we prove analogues of these inequalities for a Poisson point process. This talk will be about these two inequalities and their applications to continuum percolation.

In the first talk, we shall first introduce the Poisson Boolean model and continuum percolation. Then, we will show how sharp phase transition in the Poisson Boolean model is derived via the Poisson OSSS inequality. We shall also indicate the proof of the Poisson OSSS inequality. Time-permitting, we will mention applications to other models such as k-percolation and confetti percolation.

In the second talk, we will see how noise sensitivity and existence of exceptional times at criticality for crossing events in the dynamical Poisson Boolean model are derived via the Schramm-Steif inequality for Poisson processes. We shall also discuss the Schramm-Steif inequality very briefly.

Except some of the basics on Poisson process and continuum percolation, the two talks will be somewhat independent.

probability

Audience: advanced learners

Bangalore Probability Seminar

Series comments: The link to zoom meeting can be found on the seminar's google calendar - www.isibang.ac.in/~d.yogesh/BPS.html