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On the area of excursion sets of spherical Gaussian eigenfunctions

Wigman, Igor and Marinucci, Domenico 2011. On the area of excursion sets of spherical Gaussian eigenfunctions. Journal of Mathematical Physics 52 (9) , 093301. 10.1063/1.3624746

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The high frequency behaviour for random eigenfunctions of the spherical Laplacian has been recently the object of considerable interest, also because of strong motivation 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 borrow 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. © 2011 American Institute of Physics

Item Type: Article
Date Type: Publication
Status: Published
Schools: Mathematics
Subjects: Q Science > QA Mathematics
Additional Information: 21 pages.
Publisher: American Institute of Physics
ISSN: 1089-7658
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Last Modified: 26 Jun 2019 01:57

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