Concentration of the number of intersections of random eigenfunctions on flat tori

TL;DR

This paper demonstrates that in two-dimensional flat tori, the number of intersections between random eigenfunctions and a smooth curve is highly concentrated around the mean, regardless of the eigenfunctions' distribution being Gaussian or not.

Contribution

It establishes exponential concentration results for intersection counts of random eigenfunctions with curves on flat tori, extending previous Gaussian-based findings to more general randomness.

Findings

Intersection counts are almost exponentially concentrated around the mean.

Concentration holds even for non-Gaussian eigenfunction randomness.

Results apply to general eigenvalues on flat tori.

Abstract

We show that in two dimensional flat tori the number of intersections between random eigenfunctions of general eigenvalues and a given smooth curve is almost exponentially concentrated around its mean, even when the randomness is not gaussian.