Quasi-independence in Gaussian fields

Язык доклада – русский
Smooth Gaussian fields and their level sets appear very naturally in many areas of mathematics and other sciences. In many cases is is very beneficial to have a statement of the following kind: if values of a Gaussian field or a process are pointwise weakly correlated then events that are defined by far away parts of the field/process are also almost independent. In the case of one-dimensional stochastic processes this property is usually stated in terms of strong mixing. This notion is well studied, in particular in 1960 Kolmogorov and Rozanov gave a simple criterion for strong mixing of stationary Gaussian process. Unfortunately, this result does not easily generalize to higher dimensions. Moreover, many important fields are (real) analytic a.s. and such fields are never mixing. In the first part of the talk I will give a brief introduction into the modern study of level sets of smooth Gaussian fields. In the second part we will discuss a particular type of strong mixing/quasi-independence results: under some mild regularity assumptions it is possible to compute covariance of events that are defined in terms of topology and geometry of level sets. It turns out that under a mild assumption on decay of pointwise correlations these topological events de-correlate uniformly.
This talk is based on joint work with S. Muirhead and A. Rivera