Fluctuations of the total number of critical points of random spherical harmonics

TL;DR

This paper investigates the asymptotic behavior of the total number of critical points in random Gaussian spherical harmonics as the degree increases, revealing insights into their fluctuation patterns and implications for modeling nodal domains.

Contribution

It provides the first detailed analysis of the fluctuation law for critical points of high-degree random spherical harmonics, connecting to percolation models for eigenfunction nodal domains.

Findings

Asymptotic fluctuation law established for critical points

Implications for percolation models of nodal domains

Enhanced understanding of eigenfunction topology at high degrees

Abstract

We determine the asymptotic law for the fluctuations of the total number of critical points of random Gaussian spherical harmonics in the high degree limit. Our results have implications on the sophistication degree of an appropriate percolation process for modelling nodal domains of eigenfunctions on generic compact surfaces or billiards.