Nodal length fluctuations for arithmetic random waves

TL;DR

This paper investigates the fluctuations in the length of nodal lines of random eigenfunctions on the torus, revealing that their variance behavior is influenced by the arithmetic properties of lattice points on circles.

Contribution

It introduces a probabilistic model for Laplace eigenfunctions on the torus based on spectral multiplicities and analyzes the non-universal variance asymptotics related to lattice point arithmetic.

Findings

Variance asymptotics depend on lattice point arithmetic

Nodal length fluctuations are non-universal

Connections between spectral multiplicities and lattice points

Abstract

Using the spectral multiplicities of the standard torus, we endow the Laplace eigenspaces with Gaussian probability measures. This induces a notion of random Gaussian Laplace eigenfunctions on the torus ("arithmetic random waves"). We study the distribution of the nodal length of random eigenfunctions for large eigenvalues, and our primary result is that the asymptotics for the variance is non-universal, and is intimately related to the arithmetic of lattice points lying on a circle with radius corresponding to the energy.