Some Recent Developments on the Geometry of Random Spherical Eigenfunctions

TL;DR

This survey reviews recent advances in understanding the high-frequency geometric properties of random spherical eigenfunctions, focusing on variance asymptotics, phase transitions, fluctuation distributions, and connections to advanced probabilistic tools.

Contribution

It synthesizes recent research on the asymptotic behavior of geometric functionals of random spherical harmonics, highlighting new results and methodological connections.

Findings

Analysis of variance asymptotics for spherical eigenfunctions

Identification of phase transitions in nodal sets

Development of quantitative central limit theorems

Abstract

A lot of efforts have been devoted in the last decade to the investigation of the high-frequency behaviour of geometric functionals for the excursion sets of random spherical harmonics, i.e., Gaussian eigenfunctions for the spherical Laplacian $\Delta_{\mathbf{S}^2}$. In this survey we shall review some of these results, with particular reference to the asymptotic behaviour of variances, phase transitions in the nodal case (the \emph{Berry's Cancellation Phenomenon}), the distribution of the fluctuations around the expected values, and the asymptotic correlation among different functionals. We shall also discuss some connections with the Gaussian Kinematic Formula, with Wiener-Chaos expansions and with recent developments in the derivation of Quantitative Central Limit Theorems (the so-called Stein-Malliavin approach).