Nodal intersections of random eigenfunctions against a segment on the 2-dimensional torus

TL;DR

This paper studies the number of intersections between random eigenfunctions on a 2D torus and a line segment, revealing universal proportionality and variance bounds linked to lattice point theory.

Contribution

It provides new bounds on the variance of nodal intersections and connects the problem to lattice points on circles, highlighting the influence of slope rationality.

Findings

Expected intersection number proportional to segment length and wavenumber

Variance bounds depend on whether the line slope is rational or irrational

Connection established between nodal intersections and lattice points on circles

Abstract

We consider random Gaussian eigenfunctions of the Laplacian on the standard torus, and investigate the number of nodal intersections against a line segment. The expected intersection number, against any smooth curve, is universally proportional to the length of the reference curve, times the wavenumber, independent of the geometry. We found an upper bound for the nodal intersections variance, depending on whether the slope of the straight line is rational or irrational. Our findings exhibit a close relation between this problem and the theory of lattice points on circles.