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 (EJP 2020): combining the Central Limit Theorem in Wasserstein distance obtained by Marinucci and Rossi (JFA 2015) for Hermite-rank $2$ functionals with new results on the asymptotic behavior of their Malliavin-Sobolev norms, we are able to establish Gaussian fluctuations in this stronger probability metric as soon as the functional is regular enough.
From a joint paper with G. Giorgio and M. Rossi (arXiv:2206.02605).
