A second moment bound for critical points of planar Gaussian fields in shrinking height windows

Abstract

We consider the number of critical points of a stationary planar Gaussian field, restricted to a large domain, whose heights lie in a certain interval. Asymptotics for the mean of this quantity are simple to establish via the Kac-Rice formula, and recently Estrade and Fournier proved a second moment bound that is optimal in the case that the height interval does not depend on the size of the domain. Here we establish a bound that remains optimal in the more delicate case of height windows that are shrinking with the size of the domain. Mathematics Subject Classification. 60G60 (primary); 60F99 (secondary). Key words and phrases. Gaussian fields, critical points, second moment bound. We would like to thank Dmitry Beliaev and Michael McAuley for helpful discussions, especially with respect to the applications to the number of level and excursion sets that motivated this work, and also Igor Wigman for assisting with references.

Fluctuations of the number of excursion sets of planar Gaussian fields

“…By Cauchy-Schwarz, it is enough to prove the inequality above for each restriction separately. This is precisely the conclusion of [32,Theorem A.1]. □…”

“…Using a bound on the second moment of the number of critical points in a shrinking height window from [32] (proven using the Kac-Rice theorem), we then show in Lemma 3.5 that…”

“…The second input is a moment bound on the number of critical/tangent points of f in D R inside shrinking height windows, which was proven in [32]. Proposition 3.3.…”

“…) and [32,Theorem 1.3] states that the second moment of the latter quantity satisfies the first inequality above.…”

Local repulsion of planar Gaussian critical points

“…In these situations, practitioners are particularly interested in the detection of peaks of the random field under study or in high level asymptotics of maximal points [CS17,TW07,WMNE96]. At the opposite of these Extremes Theory results, some situations require the topological study of excursion sets over moderate levels [AT09,CX16] or the location study of critical points (not only extremal ones) [Mui20].…”