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Remote Access Theory of Probability and Mathematical Statistics

Theory of Probability and Mathematical Statistics

ISSN 1547-7363(online) ISSN 0094-9000(print)



Random Lipschitz–Killing curvatures: Reduction Principles, Integration by Parts and Wiener chaos

Author: A. Vidotto
Journal: Theor. Probability and Math. Statist. 106 (2022), 157-175
MSC (2020): Primary 60G60, 60D05; Secondary 35J05, 60G10, 60G15
Published electronically: May 16, 2022
MathSciNet review: 4438449
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Abstract: In this survey we collect some recent results regarding the Lipschitz–Killing curvatures (LKCs) of the excursion sets of random eigenfunctions on the two-dimensional standard flat torus (arithmetic random waves) and on the two-dimensional unit sphere (random spherical harmonics). In particular, the aim of the present survey is to highlight the key role of integration by parts formulae in order to have an extremely neat expression for the random LKCs. Indeed, the main tool to study local geometric functionals of random waves on manifold is to exploit their Wiener chaos decomposition and show that (often), in the so-called high-energy limit, a single chaotic component dominates their behavior. Moreover, reduction principles show that the dominant Wiener chaotic component of LKCs of random waves’ excursion sets at threshold level $u\ne 0$ is proportional to the integral of $H_2(f)$, $f$ being the random field of interest and $H_2$ the second Hermite polynomial. This will be shown via integration by parts formulae.

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Additional Information

A. Vidotto
Affiliation: Dipartimento di Matematica e Applicazioni, Università degli Studi di Napoli Federico II

Keywords: Lipschitz-Killing curvatures, random eigenfunctions, Wiener chaos expansion, reduction principles
Received by editor(s): July 25, 2021
Accepted for publication: October 18, 2021
Published electronically: May 16, 2022
Article copyright: © Copyright 2022 Taras Shevchenko National University of Kyiv