Fluctuations of the Euler-Poincar\'e characteristic for random spherical harmonics

TL;DR

This paper derives a precise asymptotic variance expression for the Euler-Poincaré characteristic of excursion sets of Gaussian eigenfunctions on the sphere, advancing understanding of their topological fluctuations.

Contribution

It provides a new, exact formula for the asymptotic variance of the Euler-Poincaré characteristic for Gaussian spherical harmonics.

Findings

Derived a precise asymptotic variance formula

Enhanced understanding of topological fluctuations on the sphere

Builds on recent theoretical results

Abstract

In this short note, we build upon some recent results to present a precise expression for the asymptotic variance of the Euler-Poincar\'e characteristic for the excursion sets of Gaussian eigenfunctions on ${\cal S}^2$.