The defect of toral Laplace eigenfunctions and Arithmetic Random Waves

TL;DR

This paper investigates the distribution of the signed area (defect) of toral Laplace eigenfunctions on shrinking balls, analyzing its variance at high energies in both random and deterministic settings.

Contribution

It provides new insights into the high energy behavior of defect variance for toral eigenfunctions, leveraging symmetry properties in both random and deterministic cases.

Findings

Expectation of defect vanishes in both scenarios.

Variance of defect exhibits specific asymptotic behavior at high energies.

Symmetry of eigenfunctions is key to the analysis.

Abstract

We study the defect (or "signed area") distribution of toral Laplace eigenfunctions restricted to shrinking balls of radius above the Planck scale, in either random Gaussian scenario ("Arithmetic Random Waves"), or deterministic eigenfunctions averaged w.r.t. the spatial variable. In either scenario we exploit the associated symmetry of the eigenfunctions to show that the expectation (Gaussian or spatial) vanishes. Our principal results concern the high energy limit behaviour of the defect variance.