Random Lipschitz-Killing curvatures: reduction principles, integration by parts and Wiener chaos

TL;DR

This survey explores the geometric properties of random eigenfunction excursion sets on the torus and sphere, emphasizing the role of integration by parts and Wiener chaos in understanding Lipschitz-Killing curvatures.

Contribution

It highlights the use of integration by parts and Wiener chaos decomposition to derive reduction principles for Lipschitz-Killing curvatures of random waves' excursion sets.

Findings

Dominant Wiener chaos component proportional to integral of H2(f) for u ≠ 0

Integration by parts yields neat expressions for LKCs

High-energy limit simplifies the analysis of geometric functionals

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 $u\ne0$ 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.