Nonlinear Functionals of Hyperbolic Random Waves: the Wiener Chaos Approach

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.