Random Nodal Lengths and Wiener Chaos

TL;DR

This survey reviews recent advances in understanding the asymptotic behavior of nodal lengths of random eigenfunctions on Riemannian surfaces, highlighting phenomena like Berry's cancellation and fluctuations in various models.

Contribution

It synthesizes recent results on the asymptotics and fluctuations of nodal lengths in Gaussian eigenfunctions, including on the torus, sphere, and general monochromatic random waves.

Findings

Nodal length fluctuations are non-Gaussian on the torus.

Nodal length fluctuations are Gaussian on the sphere.

Recent results on the distribution of nodal lengths in high energy limits.

Abstract

In this survey we collect some of the recent results on the "nodal geometry" of random eigenfunctions on Riemannian surfaces. We focus on the asymptotic behavior, for high energy levels, of the nodal length of Gaussian Laplace eigenfunctions on the torus (arithmetic random waves) and on the sphere (random spherical harmonics). We give some insight on both Berry's cancellation phenomenon and the nature of nodal length second order fluctuations (non-Gaussian on the torus and Gaussian on the sphere) in terms of chaotic components. Finally we consider the general case of monochromatic random waves, i.e. Gaussian random linear combination of eigenfunctions of the Laplacian on a compact Riemannian surface with frequencies from a short interval, whose scaling limit is Berry's Random Wave Model. For the latter we present some recent results on the asymptotic distribution of its nodal length in the high energy limit (equivalently, for growing domains).