Universal Cancellations in Uniform Random Waves

TL;DR

This paper studies the geometric properties of uniform random waves, a non-Gaussian model with fixed L^2 norm, revealing impacts on Berry's cancellation phenomenon and providing new variance estimates and analytical tools.

Contribution

It introduces a systematic chaos-based analysis of non-Gaussian random waves with fixed norm, connecting Hermite and spherical harmonic expansions.

Findings

Deep effects on Berry's cancellation phenomenon due to fixed norm

New high-frequency asymptotic variance estimates for geometric functionals

Explicit relation between Hermite expansions and spherical harmonics

Abstract

A vast literature over the past fifteen years has been devoted to the study of the geometric properties of Gaussian random waves. In this work, we investigate the geometric behavior of \emph{uniform random waves}, a much less studied non-Gaussian model in which the $L^2$ norm is constrained to be exactly equal to one in every realization (a normalization that is natural from the standpoint of quantum mechanics). We show that this norm-constrained formulation has deep consequences for the universality of the so-called \emph{Berry's cancellation phenomenon}, as well as for novel high-frequency asymptotic variance estimates. These effects manifest themselves in both local geometric functionals, such as the Lipschitz--Killing curvatures, and global ones, such as the number of connected components above a fixed threshold. A key byproduct of our analysis is a new explicit relation between Hermite expansions and spherical harmonic decompositions for $0$-homogeneous functionals of Gaussian vectors, which enables a systematic chaos-based analysis of non-Gaussian random waves.