On the Excursion Sets of Spherical Gaussian Eigenfunctions

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.

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Cited by in corpus (9)

• The defect variance of random spherical harmonics
• Fluctuations of the Euler-Poincaré characteristic for random spherical harmonics
• On the Limiting Behaviour of Needlets Polyspectra
• Fluctuations of the number of excursion sets of planar Gaussian fields
• A Quantitative Central Limit Theorem for the Excursion Area of Random Spherical Harmonics over Subdomains of ${\mathbb{S}}^2$
• The Probability of Intransitivity in Dice and Close Elections
• Functional Convergence of Berry's Nodal Lengths: Approximate Tightness and Total Disorder
• Expected Lipschitz-Killing curvatures for spin random fields and other non-isotropic fields
• Three central limit theorems for the unbounded excursion component of a Gaussian field