Random nodal lengths and Wiener chaos

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).

On the Correlation Between Nodal and Nonzero Level Sets for Random Spherical Harmonics

“…Concerning second chaotic components, thanks to Green's formula (for details see again [23,25]), we can write where the second equality is just formal; by this we mean that the first series converges in L 2 (P) for every fixed x, while the second does not. Before we proceed, we need to introduce some more notation: let us fix x = (0, 0) to be the "north pole" and y(θ) = (0, θ) to be points on the meridian where ϕ = 0.…”

On 3-dimensional Berry’s model

“…Canzani and Hanin (2020) studied the universality phenomenon in general Riemannian manifolds. The reader can find results on arithmetic random waves defined on the flat torus (Cammarota, 2019;Dalmao et al, 2019) and on random spherical harmonics in Cammarota and Marinucci (2019); Fantaye et al (2019); Marinucci and Rossi (2021) and references therein, see also Rossi (2019) for a survey on both subjects. The nodal sets of Berry's planar random waves, i.e.…”

Variance of the number of zeros of dependent Gaussian trigonometric polynomials

“…Since then, the variances of numerous models have been studied: for instance, see [4] for the analogous model a k cos(kt), see [6,7] in the independent framework with non Gaussian coefficients, or more recently [10] for random orthogonal polynomials on the real line. In greater dimension, the asymptotic behavior of the variance of random nodal volume has been established in several kinds of random waves models, see for instance the survey [12] and the references therein.…”