# Variance bounds for the number of level/excursion sets of a planar Gaussian field

vendredi 15 février 2019, 9h30 - 10h30

Abstract: Planar Gaussian fields are a model of spatial noise, and in many applications it is useful to understand the geometric structure of their level sets. There is a natural classification of geometric functionals of the level sets as either local’ (e.g. length of the level sets, volume of the excursion sets, Euler characteristic of the excursion sets) or ‘non-local’ (e.g. number of connected components of the level and excursion sets) depending on whether there exists an integral representation for the functional. In the case of local’ functionals, first order properties (e.g. asymptotics for the mean) can be derived from the Kac-Rice formula, and second order properties (e.g. asymptotics for the variance, central limit theorems) have also been recently established via Weiner chaos expansions (see, e.g., Estrade–Leon ’16, Marinucci–Rossi–Wigman ’17, Nourdin–Peccati–Rossi ’17 etc). In the case of `non-local’ functionals, the lack of integral representations makes the analysis more challenging, and whereas first order properties were established 10 years ago by Nazarov–Sodin using ergodic theoretical techniques, up until now there have been no second order results. In this talk I will discuss some first steps in this directions, namely proving variance bounds for the number of level and excursion set components. Joint with Dmitry Belyaev and Michael McAuley (University of Oxford).