On Critical Points of Random Fields

Cammarota Valentina, Sapienza University of Rome

Critical points of random spherical harmonics and isotropic stationary Gaussian fields are examples of points process showing a regular structure. In this talk we will present some recent results aimed at quantifying how critical points differ from independently picked points. We will focus in particular on the following results. The limiting distribution of critical points and extrema of random spherical harmonics, in the high energy limit: in particular, we derive the density functions of extrema and saddles, we then provide analytic expressions for the variances. By computing the main term in the asymptotic expansion of the two-point correlation function near the diagonal, we obtain that the critical points neither repel nor attract each other. Our analysis also allows to study how the short-range behaviour of critical points depends on their index. Finally we discuss the full correlation in the high frequency limit between critical points and other geometric functionals of random spherical harmonics like the excursion area, level curves, and Euler characteristic of excursion sets.

Area: CS26 - Geometry of random fields (Michele Stecconi)

Keywords: Gaussian fields, critical points, point processes, repulsion

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