# A Numerical Investigation on the High-Frequency Geometry of Spherical Random Eigenfunctions

## TL;DR

This paper reviews and numerically investigates the high-frequency geometry of spherical random eigenfunctions, focusing on Lipschitz-Killing curvatures and critical points, confirming analytic predictions and exploring statistical applications like CMB analysis.

## Contribution

It provides the first comprehensive numerical study of geometric functionals of spherical eigenfunctions, validating analytic formulas and examining cancellation phenomena.

## Key findings

- Accurate analytic predictions for expected values and variances of geometric functionals.
- Confirmation of Berry's cancellation phenomenon at specific thresholds.
- Potential applications in cosmic microwave background data analysis.

## Abstract

A lot of attention has been drawn over the last few years by the investigation of the geometry of spherical random eigenfunctions (random spherical harmonics) in the high frequency regime, i.e ., for diverging eigenvalues. In this paper, we present a review of these results and we collect for the first time a comprehensive numerical investigation, focussing on particular on the behaviour of Lipschitz-Killing curvatures/Minkowski functionals (i.e., the area, the boundary length and the Euler-Poincar\'e characteristic of excursion sets) and on critical points. We show in particular that very accurate analytic predictions exist for their expected values and variances, for the correlation among these functionals, and for the cancellation that occurs for some specific thresholds (the variances becoming an order of magnitude smaller - the so-called Berry's cancellation phenomenon). Most of these functionals can be used for important statistical applications, for instance in connection to the analysis of Cosmic Microwave Background data.