Convergence in Total Variation for nonlinear functionals of random hyperspherical harmonics

TL;DR

This paper proves that nonlinear functionals of random hyperspherical harmonics converge to a Gaussian distribution in total variation distance in the high energy limit, using advanced probabilistic and combinatorial techniques.

Contribution

It establishes second order Gaussian fluctuations in total variation for nonlinear hyperspherical harmonic functionals, extending previous CLT results with new moment estimates and graph-theoretic methods.

Findings

Proves total variation convergence to Gaussian in high energy limit.

Develops novel moment estimates for Gegenbauer polynomial products.

Links graph theory with diagram formulas for probabilistic analysis.

Abstract

Random hyperspherical harmonics are Gaussian Laplace eigenfunctions on the unit $d$-dimensional sphere ($d\ge 2$). We study the convergence in Total Variation distance for their nonlinear statistics in the high energy limit, i.e., for diverging sequences of Laplace eigenvalues. Our approach takes advantage of a recent result by Bally, Caramellino and Poly (2020): combining the Central Limit Theorem in Wasserstein distance obtained by Marinucci and Rossi (2015) for Hermite-rank $2$ functionals with new results on the asymptotic behavior of their Malliavin-Sobolev norms, we are able to establish second order Gaussian fluctuations in this stronger probability metric as soon as the functional is regular enough. Our argument requires some novel estimates on moments of products of Gegenbauer polynomials that may be of independent interest, which we prove via the link between graph theory and diagram formulas.