CENTRAL LIMIT THEOREM FOR PLANCK-SCALE MASS DISTRIBUTION OF TORAL LAPLACE EIGENFUNCTIONS

Igor Wigman; Nadav Yesha

doi:10.1112/S0025579319000093

CENTRAL LIMIT THEOREM FOR PLANCK-SCALE MASS DISTRIBUTION OF TORAL LAPLACE EIGENFUNCTIONS

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Abstract

We study the fine-scale (Formula presented.) -mass distribution of toral Laplace eigenfunctions with respect to random position in two and three dimensions. In two dimensions, under certain flatness assumptions on the Fourier coefficients and generic restrictions on energy levels, both the asymptotic shape of the variance is determined and the limiting Gaussian law is established in the optimal Planck-scale regime. In three dimensions the asymptotic behaviour of the variance is analysed in a more restrictive scenario (“Bourgain's eigenfunctions”). Other than the said precise results, lower and upper bounds are proved for the variance under more general flatness assumptions on the Fourier coefficients.

Original language	English
Pages (from-to)	643-676
Number of pages	34
Journal	Mathematika
Volume	65
Issue number	3
DOIs	https://doi.org/10.1112/S0025579319000093
State	Published - 2019
Externally published	Yes

Bibliographical note

Funding Information:
The authors of this paper wish to express their gratitude to J. Benatar, A. Granville, P. Kurlberg, Z. Rudnick, P. Sarnak and M. Sodin for numerous stimulating and fruitful discussions concerning various aspects of our work, and their interest in our research. It is a pleasure to thank the anonymous referee for his comments on an earlier version of the paper. The research leading to these results has received funding from the European Research Council under the European Union's Seventh Framework Programme (FP7/2007–2013), ERC grant agreement no. 335141.

Publisher Copyright:
© 2019 University College London

Keywords

11H06
11Z05
35P20 (primary)

ASJC Scopus subject areas

Mathematics (all)

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10.1112/S0025579319000093

Cite this

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title = "CENTRAL LIMIT THEOREM FOR PLANCK-SCALE MASS DISTRIBUTION OF TORAL LAPLACE EIGENFUNCTIONS",

abstract = "We study the fine-scale (Formula presented.) -mass distribution of toral Laplace eigenfunctions with respect to random position in two and three dimensions. In two dimensions, under certain flatness assumptions on the Fourier coefficients and generic restrictions on energy levels, both the asymptotic shape of the variance is determined and the limiting Gaussian law is established in the optimal Planck-scale regime. In three dimensions the asymptotic behaviour of the variance is analysed in a more restrictive scenario (“Bourgain's eigenfunctions”). Other than the said precise results, lower and upper bounds are proved for the variance under more general flatness assumptions on the Fourier coefficients.",

keywords = "11H06, 11Z05, 35P20 (primary)",

author = "Igor Wigman and Nadav Yesha",

note = "Funding Information: The authors of this paper wish to express their gratitude to J. Benatar, A. Granville, P. Kurlberg, Z. Rudnick, P. Sarnak and M. Sodin for numerous stimulating and fruitful discussions concerning various aspects of our work, and their interest in our research. It is a pleasure to thank the anonymous referee for his comments on an earlier version of the paper. The research leading to these results has received funding from the European Research Council under the European Union's Seventh Framework Programme (FP7/2007–2013), ERC grant agreement no. 335141. Publisher Copyright: {\textcopyright} 2019 University College London",

year = "2019",

doi = "10.1112/S0025579319000093",

language = "English",

volume = "65",

pages = "643--676",

journal = "Mathematika",

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T1 - CENTRAL LIMIT THEOREM FOR PLANCK-SCALE MASS DISTRIBUTION OF TORAL LAPLACE EIGENFUNCTIONS

AU - Wigman, Igor

AU - Yesha, Nadav

N1 - Funding Information: The authors of this paper wish to express their gratitude to J. Benatar, A. Granville, P. Kurlberg, Z. Rudnick, P. Sarnak and M. Sodin for numerous stimulating and fruitful discussions concerning various aspects of our work, and their interest in our research. It is a pleasure to thank the anonymous referee for his comments on an earlier version of the paper. The research leading to these results has received funding from the European Research Council under the European Union's Seventh Framework Programme (FP7/2007–2013), ERC grant agreement no. 335141. Publisher Copyright: © 2019 University College London

PY - 2019

Y1 - 2019

N2 - We study the fine-scale (Formula presented.) -mass distribution of toral Laplace eigenfunctions with respect to random position in two and three dimensions. In two dimensions, under certain flatness assumptions on the Fourier coefficients and generic restrictions on energy levels, both the asymptotic shape of the variance is determined and the limiting Gaussian law is established in the optimal Planck-scale regime. In three dimensions the asymptotic behaviour of the variance is analysed in a more restrictive scenario (“Bourgain's eigenfunctions”). Other than the said precise results, lower and upper bounds are proved for the variance under more general flatness assumptions on the Fourier coefficients.

AB - We study the fine-scale (Formula presented.) -mass distribution of toral Laplace eigenfunctions with respect to random position in two and three dimensions. In two dimensions, under certain flatness assumptions on the Fourier coefficients and generic restrictions on energy levels, both the asymptotic shape of the variance is determined and the limiting Gaussian law is established in the optimal Planck-scale regime. In three dimensions the asymptotic behaviour of the variance is analysed in a more restrictive scenario (“Bourgain's eigenfunctions”). Other than the said precise results, lower and upper bounds are proved for the variance under more general flatness assumptions on the Fourier coefficients.

KW - 11H06

KW - 11Z05

KW - 35P20 (primary)

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U2 - 10.1112/S0025579319000093

DO - 10.1112/S0025579319000093

M3 - Article

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SN - 0025-5793

VL - 65

SP - 643

EP - 676

JO - Mathematika

JF - Mathematika

IS - 3

ER -

CENTRAL LIMIT THEOREM FOR PLANCK-SCALE MASS DISTRIBUTION OF TORAL LAPLACE EIGENFUNCTIONS

Abstract

Bibliographical note

Keywords

ASJC Scopus subject areas

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