Nonlinear Functionals of Hyperbolic Random Waves: the Wiener Chaos Approach

TL;DR

This paper analyzes the behavior of nonlinear functionals of hyperbolic random waves using Wiener chaos expansions, revealing new fluctuation phenomena and providing asymptotic results with applications to geometric properties.

Contribution

It introduces a Wiener chaos-based method to study integral functionals of hyperbolic random waves, highlighting unique fluctuation behaviors in hyperbolic geometry.

Findings

Variance asymptotics for integral functionals

Central limit theorems in high-frequency and large domain limits

Discrepancy in fluctuations of fourth polyspectra in hyperbolic spaces

Abstract

We consider Gaussian random waves on hyperbolic spaces and establish variance asymptotics and central limit theorems for a large class of their integral functionals, both in the high-frequency and large domain limits. Our strategy of proof relies on a fine analysis of Wiener chaos expansions, which in turn requires us to analytically assess the fluctuations of integrals involving mixed moments of covariance kernels. Our results complement several recent findings on non-linear transforms of planar and arithmetic random waves, as well as of random spherical harmonics. In the particular case of 2-dimensional hyperbolic spaces, our analysis reveals an intriguing discrepancy between the high-frequency and large domain fluctuations of the so-called fourth polyspectra -- a phenomenon that has no counterpart in the Euclidean setting. We develop applications of a geometric flavor, most notably to excursion volumes and occupation densities.