Nodal intersections of random eigenfunctions against a segment on the 2-dimensional torus

Maffucci, Riccardo W.

doi:10.1007/s00605-016-1001-2

Cited by 16 publications

(18 citation statements)

References 12 publications

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“…[15,38] as m → ∞. Rudnick-Wigman [36] and subsequently Rossi-Wigman [34] and the author [31] investigated Z…”

Section: )mentioning

confidence: 99%

Nodal Intersections for Arithmetic Random Waves Against a Surface

Maffucci¹

2019

Ann. Henri Poincaré

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Given the ensemble of random Gaussian Laplace eigenfunctions on the three-dimensional torus ('3d arithmetic random waves'), we investigate the 1-dimensional Hausdorff measure of the nodal intersection curve against a compact regular toral surface (the 'nodal intersection length'). The expected length is proportional to the square root of the eigenvalue, times the surface area, independent of the geometry.Our main finding is the leading asymptotic of the nodal intersection length variance, against a surface of nonvanishing Gauss-Kronecker curvature. The problem is closely related to the theory of lattice points on spheres: by the equidistribution of the lattice points, the variance asymptotic depends only on the geometry of the surface.

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“…[15,38] as m → ∞. Rudnick-Wigman [36] and subsequently Rossi-Wigman [34] and the author [31] investigated Z…”

Section: )mentioning

confidence: 99%

Nodal Intersections for Arithmetic Random Waves Against a Surface

Maffucci¹

2019

Ann. Henri Poincaré

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“…Several recent papers investigate the nodal volume [37,24] and nodal intersections of arithmetic waves against a fixed curve [38,29,36,39,28].…”

Section: The Arithmetic Wavesmentioning

confidence: 99%

Restriction of 3D arithmetic Laplace eigenfunctions to a plane

Maffucci¹

2020

Electron. J. Probab.

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We consider a random Gaussian ensemble of Laplace eigenfunctions on the 3D torus, and investigate the 1-dimensional Hausdorff measure ('length') of nodal intersections against a smooth 2-dimensional toral sub-manifold ('surface'). A prior result of ours prescribed the expected length, universally proportional to the area of the reference surface, times the wavenumber, independent of the geometry.In this paper, for surfaces contained in a plane, we give an upper bound for the nodal intersection length variance, depending on the arithmetic properties of the plane. The bound is established via estimates on the number of lattice points in specific regions of the sphere.

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“…In particular, it is well‐known that isotropic random fields on the sphere can be decomposed by means of the spectral representation theorem into the sum of orthogonal components, each of them corresponding to a different multipole

ℓ

. The behavior of geometric functionals in the high‐frequency/high‐energy limit for these components has been studied by several authors in recent years, starting from Nazarov and Sodin (, ) for the number of connected components; Wigman () for the nodal length; and then including, among others, Marinucci and Rossi () and Marinucci and Wigman () for the excursion area; Marinucci and Wigman (), Marinucci and Wigman () and Rossi () for the defect; Cammarota and Marinucci () for the Euler‐Poincaré characteristic; Marinucci, Peccati, Rossi, and Wigman (), Marinucci, Rossi, and Wigman (), Rossi () and Wigman () for the distribution of the nodal length; Cammarota, Marinucci, and Wigman () for the critical values; and Cammarota and Wigman () for the total number of critical points (see also Dalmao, Nourdin, Peccati, and Rossi (), Krishnapur, Kurlberg, and Wigman (), Maffucci (), Peccati and Rossi (), Rossi and Wigman (), Rudnick and Wigman () for related works covering also the 2‐dimensional torus; Maffucci () for the 3‐dimensional torus; and Nourdin, Peccati, and Rossi () for planar random waves).…”

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confidence: 99%

A numerical investigation on the high‐frequency geometry of spherical random eigenfunctions

et al. 2019

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A lot of attention has been drawn over the last few years by the investigation of the geometry of spherical random eigenfunctions (random spherical harmonics) in the high‐frequency regime, that is, for diverging eigenvalues. In this paper, we present a review of these results and we collect for the first time a comprehensive numerical investigation, focussing on particular on the behavior of Lipschitz‐Killing curvatures/Minkowski functionals (i.e., the area, the boundary length, and the Euler‐Poincaré characteristic of excursion sets) and on critical points. We show in particular that very accurate analytic predictions exist for their expected values and variances, for the correlation among these functionals, and for the cancellation that occurs for some specific thresholds (the variances becoming an order of magnitude smaller—the so‐called Berry's cancellation phenomenon). Most of these functionals can be used for important statistical applications, for instance, in connection to the analysis of cosmic microwave background data.

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Nodal intersections of random eigenfunctions against a segment on the 2-dimensional torus

Cited by 16 publications

References 12 publications

Nodal Intersections for Arithmetic Random Waves Against a Surface

Nodal Intersections for Arithmetic Random Waves Against a Surface

Restriction of 3D arithmetic Laplace eigenfunctions to a plane

A numerical investigation on the high‐frequency geometry of spherical random eigenfunctions

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