Modul: MAT971 Stochastische Prozesse

## Decorrelation of topological events for smooth Gaussian fields

Alejandro Rivera talk

**Date:** 10.10.18 **Time:** 17.15 - 18.15 **Room:** ETH HG G 43

Consider f a planar smooth centered stationary Gaussian field with covariance K(x)=E[f(x)f(0)]. Under mild non-degeneracy assumptions on K, the zero set Z of f is a.s. a random locally finite collection of disjoint smooth loops on the plane and perhaps some smooth curves going to infinity. Assuming that K(x) goes to zero as |x| goes to infinity, we would like to prove that the topologies of the level lines on distant regions become independent from each other. This type of information is useful to derive central limit theorems or to prove percolation estimates for level sets of smooth Gaussian fields. However, if the field is, say, analytic, conditioning on the field inside an open subset determines the field on the whole plane. I will discuss different ways of surpassing this obstacle, including a recent result obtained in collaboration with Hugo Vanneuville, and some applications to component counting and percolation.