IS21: On Nodal Random Variables

Organizer: Guillaume Poly & Maurizia Rossi (University Rennes 1 & University Milano-Bicocca)

Level set percolation of smooth Gaussian fields: an overview

Franco Severo (Université Lyon 1)

Consider the connected components of the level sets of a smooth Gaussian field on the d-dimensional Euclidean space. For which levels does an unbounded connected component appear? When it exists, is it unique? What is the typical size of large bounded components and how does it depend on the level? What happens at the critical level? How do the answers to these questions depend on the dimension? These natural questions lie at the heart of level-set percolation of smooth gaussian fields, a topic that has seen significant progress over the past decade. In this talk, I will give an overview of several recent results in the field and highlight a number of open problems that remain to be addressed.