Recent progress in the understanding of the nodal structure of random functions
The nodal set of a nice function defined on a smooth manifold or the Euclidean space is its zero set. The study of nodal sets of Gaussian random fields has positioned itself in the heart of several disciplines, including probability theory and spectral geometry, and, more recently, it has exhibited connections to number theory. We are interested in the asymptotic topology and geometry of the nodal lines in the high energy limit.
In the first part of the talk I will give an overview of the classical results in this field, and the related methods. In the second part of the talk I will describe the more recent progress, related to percolation properties of the nodal lines, inspired by the beautiful percolation model due to Bogomolny-Schmit. Finally, I will describe the recent results obtained in a joint work with D. Beliaev and S. Muirhead on the relation between the percolation properties of the nodal sets and their connectivity measures, that were defined and whose existence was established in a joint work with P. Sarnak.
Math&Stats Zoom Room
Part of Mathematics Colloquium Series