On the distribution of the nodal sets of random spherical harmonics

DOI10.1063/1.3056589zbMath1200.58021arXiv0805.2768OpenAlexW3098155800MaRDI QIDQ3650892

Igor Wigman

Publication date: 7 December 2009

Published in: Journal of Mathematical Physics (Search for Journal in Brave)

Full work available at URL: https://arxiv.org/abs/0805.2768

zbMATH Keywords

varianceeigenfunctions of the Laplacianeigenvalues and eigenfunctionsGaussian probability measureLeray measurelength of the nodal set

Mathematics Subject Classification ID

Spectral problems; spectral geometry; scattering theory on manifolds (58J50) Asymptotic distributions of eigenvalues in context of PDEs (35P20) Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation (35J05) Spherical harmonics (33C55)

Related Items (22)

Remarks on nodal volume statistics for regular and chaotic wave functions in various dimensions ⋮ Fluctuations of the nodal length of random spherical harmonics ⋮ No smooth phase transition for the nodal length of band-limited spherical random fields ⋮ Local universality for zeros and critical points of monochromatic random waves ⋮ Nodal length fluctuations for arithmetic random waves ⋮ Nodal lengths in shrinking domains for random eigenfunctions on \(S^2\) ⋮ Metric properties in the mean of polynomials on compact isotropy irreducible homogeneous spaces ⋮ Scalar curvature and \(Q\)-curvature of random metrics ⋮ On the correlation between nodal and nonzero level sets for random spherical harmonics ⋮ A quantitative central limit theorem for the excursion area of random spherical harmonics over subdomains of S2 ⋮ On the expected Betti numbers of the nodal set of random fields ⋮ On the limiting behaviour of needlets polyspectra ⋮ A reduction principle for the critical values of random spherical harmonics ⋮ Expected number and height distribution of critical points of smooth isotropic Gaussian random fields ⋮ Monochromatic Random Waves for General Riemannian Manifolds ⋮ Critical points of multidimensional random Fourier series: Variance estimates ⋮ Limiting behavior for the excursion area of band-limited spherical random fields ⋮ Restriction of 3D arithmetic Laplace eigenfunctions to a plane ⋮ Ergodicity and Gaussianity for spherical random fields ⋮ Asymptotic behaviour of level sets of needlet random fields ⋮ Almost-sure asymptotics for Riemannian random waves ⋮ High-frequency asymptotics for Lipschitz-Killing curvatures of excursion sets on the sphere

Cites Work

• Nodal sets of eigenfunctions on Riemannian manifolds
• Eigenfunctions and nodal sets
• Über Knoten von Eigenfunktionen des Laplace-Beltrami-Operators
• On the volume of nodal sets for eigenfunctions of the Laplacian on the torus
• The Leray measure of nodal sets for random eigenfunctions on the torus
• Über die Länge der Knotenlinien schwingender Membranen. (On the length of the nodal lines of vibrating membranes.)
• Real and complex zeros of Riemannian random waves
• Statistics of nodal lines and points in chaotic quantum billiards: perimeter corrections, fluctuations, curvature

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