Fluctuations of the nodal length of random spherical harmonics, erratum

Abstract

Using the multiplicities of the Laplace eigenspace on the sphere (the space of spherical harmonics) we endow the space with Gaussian probability measure. This induces a notion of random Gaussian spherical harmonics of degree $n$ having Laplace eigenvalue $E=n(n+1)$. We study the length distribution of the nodal lines of random spherical harmonics. It is known that the expected length is of order $n$. It is natural to conjecture that the variance should be of order $n$, due to the natural scaling. Our principal result is that, due to an unexpected cancelation, the variance of the nodal length of random spherical harmonics is of order $\log n$. This behaviour is consistent with the one predicted by Berry for nodal lines on chaotic billiards (Random Wave Model). In addition we find that a similar result is applicable for "generic" linear statistics of the nodal lines.

This is to correct a sign mistake that has been made in the previous version (that was published in Comm. Math. Phys.). As a result the leading constant in all the theorems was wrong, and the constants are now consistent with the one predicted by Berry. A corrected manuscript plus a detailed erratum with all the corrections that were made relatively to the version published is attached

References in corpus (2)

• On the volume of nodal sets for eigenfunctions of the Laplacian on the torus
• On the distribution of the nodal sets of random spherical harmonics

Cited by in corpus (20)

• Nodal portraits of quantum billiards: Domains, lines, and statistics
• On Nonlinear Functionals of Random Spherical Eigenfunctions
• On the distribution of the critical values of random spherical harmonics
• The defect variance of random spherical harmonics
• On the Excursion Sets of Spherical Gaussian Eigenfunctions
• Phase Singularities in Complex Arithmetic Random Waves
• Fluctuations of the Euler-Poincaré characteristic for random spherical harmonics
• Nodal Lengths in Shrinking Domains for Random Eigenfunctions on ${\mathbb{S}}^2$
• Gaussian Random Measures Generated by Berry's Nodal Sets
• On the Limiting Behaviour of Needlets Polyspectra
• Fluctuations of the number of excursion sets of planar Gaussian fields
• High-frequency asymptotics for Lipschitz-Killing curvatures of excursion sets on the sphere
• Nodal Statistics of Planar Random Waves
• The Defect of Random Hyperspherical Harmonics
• A Quantitative Central Limit Theorem for the Excursion Area of Random Spherical Harmonics over Subdomains of ${\mathbb{S}}^2$
• On the expected Betti numbers of the nodal set of random fields
• Remarks on nodal volume statistics for regular and chaotic wave functions in various dimensions
• Functional Convergence of Berry's Nodal Lengths: Approximate Tightness and Total Disorder
• Critical points of multidimensional random Fourier series: central limits
• Nodal deficiency of random spherical harmonics in presence of boundary