CLT for Planck scale mass distribution of toral Laplace eigenfunctions

TL;DR

This paper investigates the distribution of mass of toral Laplace eigenfunctions at Planck scale in 2D and 3D, establishing variance asymptotics and Gaussian limits under certain conditions.

Contribution

It provides the first detailed analysis of Planck-scale mass distribution for toral eigenfunctions, including variance asymptotics and Gaussian laws in 2D and 3D.

Findings

Variance asymptotics determined in 2D under flatness assumptions.

Gaussian limiting distribution established in 2D.

Variance bounds proved under general flatness assumptions.

Abstract

We study the fine scale $L^2$-mass distribution of toral Laplace eigenfunctions with respect to random position, in 2 and 3 dimensions. In 2d, under certain flatness assumptions on the Fourier coefficients and generic restrictions on energy levels, both the asymptotic shape of the variance is determined and the limiting Gaussian law is established, in the optimal Planck-scale regime. In 3d the asymptotic behaviour of the variance is analysed in a more restrictive scenario ("Bourgain's eigenfunctions"). Other than the said precise results, lower and upper bounds are proved for the variance, under more general flatness assumptions on the Fourier coefficients.