Mass distribution for toral eigenfunctions via Bourgain's de-randomisation

TL;DR

This paper investigates how Laplacian eigenfunctions on a flat torus distribute their mass in small regions, using Bourgain's de-randomisation to compare with random waves and classify possible limiting behaviors.

Contribution

It introduces a novel application of Bourgain's de-randomisation to analyze mass distribution of toral eigenfunctions at small scales, providing classification and conditions for equidistribution.

Findings

Classified all possible limiting mass distributions.

Identified conditions for eigenfunction equidistribution at small scales.

Compared toral eigenfunctions with random wave models.

Abstract

We study the problem of mass distribution of Laplacian eigenfunctions in shrinking balls for the standard flat torus $\mathbb{T}^2=\mathbb{R}^2/\mathbb{Z}^2$. By averaging over the centre of the ball we use Bourgain's de-randomisation to compare the mass-distribution of toral eigenfunctions at Plank scale to the mass distribution of random waves in growing balls around the origin. We are then able to classify all possible limiting distributions and variances. Finally we give sufficient and necessary conditions so that the mass of \textquotedblleft generic \textquotedblright eigenfunction equidistributes at Plank scale in almost all balls.