Small-scale equidistribution for random spherical harmonics

Courcy-Ireland, Matthew de

doi:10.48550/arxiv.1711.01317

Cited by 4 publications

(5 citation statements)

References 22 publications

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“…Berry's random wave conjectures [Berr] suggest that the eigenfunctions of eigenvalue λ 2 behave like random waves with frequency λ. Recent results about equidistribution at various polynomial scales of random waves on manifolds were proved in [Han2,HT,CI]. In comparison, we see that the logarithmical scales in [Han1,HR1] are at much weaker scales.…”

Section: Introductionmentioning

confidence: 67%

Small scale quantum ergodicity in negatively curved manifolds

Han

2015

Nonlinearity

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In this series, we investigate quantum ergodicity at small scales for linear hyperbolic maps of the torus ("cat maps"). In Part I of the series, we prove quantum ergodicity at various scales. Let N = 1/h, in which h is the Planck constant. First, for all integers N ∈ N, we show quantum ergodicity at logarithmical scales | log h| −α for some α > 0. Second, we show quantum ergodicity at polynomial scales h α for some α > 0, in two special cases: N ∈ S(N) of a full density subset S(N) of integers and Hecke eigenbasis for all integers.2010 Mathematics Subject Classification. 35P20, 58G25, 81Q50, 37D20, 11F25. Key words and phrases. Hyperbolic linear maps of the torus, quantum ergodicity, small scale.

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Section: Introductionmentioning

confidence: 67%

Small scale quantum ergodicity in negatively curved manifolds

Han

2015

Nonlinearity

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“…The interest in studying Laplace eigenfunctions on such geodesic balls comes from the semiclassical eigenfunction hypothesis of Berry [2,3]. Study of quantities including the L 2 mass, volume of the nodal set and the nodal component count on such geodesic balls for Gaussian Laplace eigenfunctions have been carried out in [8,11,12,19,25]. 1.3.…”

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confidence: 99%

Concentration for nodal component count of Gaussian Laplace eigenfunctions

Priya¹

2020

Preprint

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We study nodal component count of the following Gaussian Laplace eigenfunctions: monochromatic random waves (MRW) on R 2 , arithmetic random waves (ARW) on T 2 and random spherical harmonics (RSH) on S 2 . Exponential concentration for nodal component count of RSH on S 2 and ARW on T 2 were established in [22] and [23] respectively. We prove exponential concentration for nodal component count in the following three cases: MRW on growing Euclidean balls in R 2 ; RSH and ARW on geodesic balls, in S 2 and T 2 respectively, whose radius is slightly larger than the wavelength scale.

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“…This was used in [10] to estimate the λ j as follows. There are three cases, which we think of as "bulk", "edge", and "tail".…”

Section: Two-dimensional Casementioning

confidence: 99%

A central limit theorem for integrals of random waves

Courcy-Ireland¹,

Lemm²

2019

Preprint

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We derive a central limit theorem for the mean-square of random waves in the high-frequency limit over shrinking sets. Our proof applies to any compact Riemannian manifold of arbitrary dimension, thanks to the universality of the local Weyl law. The key technical step is an estimate capturing some cancellation in a triple integral of Bessel functions, which we achieve using Gegenbauer's addition formula.

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Small-scale equidistribution for random spherical harmonics

Abstract: We study random spherical harmonics at shrinking scales. We compare the mass assigned to a small spherical cap with its area, and find the smallest possible scale at which, with high probability, the discrepancy between them is small simultaneously at every point on the sphere.

Cited by 4 publications

References 22 publications

Small scale quantum ergodicity in negatively curved manifolds

Small scale quantum ergodicity in negatively curved manifolds

Concentration for nodal component count of Gaussian Laplace eigenfunctions

A central limit theorem for integrals of random waves

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