On the Excursion Sets of Spherical Gaussian Eigenfunctions

TL;DR

This paper investigates the high frequency behavior of excursion sets of spherical Gaussian eigenfunctions, establishing a uniform central limit theorem for the empirical measure and revealing the asymptotic degeneracy of the limiting process.

Contribution

It introduces a novel uniform CLT for the empirical measure of spherical Gaussian eigenfunctions' excursion sets, using Hermite polynomial expansion and weak reduction techniques.

Findings

Established a uniform CLT for the empirical measure.

Proved the asymptotic degeneracy of the limiting process.

Applied techniques from stationary long memory process literature.

Abstract

The high frequency behaviour for random eigenfunctions of the spherical Laplacian has been recently the object of considerable interest, also because of strong motivations arising from Physics and Cosmology. In this paper, we are concerned with the high frequency behaviour of excursion sets; in particular, we establish a Uniform Central Limit Theorem for the empirical measure, i.e. the proportion of spherical surface where spherical Gaussian eigenfunctions lie below a level $z$. Our proofs borrows some techniques from the literature on stationary long memory processes; in particular, we expand the empirical measure into Hermite polynomials, and establish a uniform weak reduction principle, entailing that the asymptotic behaviour is asymptotically dominated by a single term in the expansion. As a result, we establish a functional central limit theorem; the limiting process is fully degenerate.