# The Leray measure of nodal sets for random eigenfunctions on the torus

Ferenc Oravecz[1]; Zeév Rudnick[2]; Igor Wigman[3]

• [1] Alfréd Rényi Institute of Mathematics Hungarian Academy of Sciences Reáltanoda utca 13-15 1053 Budapest (Hungary)
• [2] Tel Aviv University School of Mathematical Sciences 69978 Tel Aviv (Israel)
• [3] Université de Montréal Centre de recherches mathématiques (CRM) C.P. 6128, succ. centre-ville Montréal, Québec H3C 3J7 (Canada)
• Volume: 58, Issue: 1, page 299-335
• ISSN: 0373-0956

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## Abstract

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We study nodal sets for typical eigenfunctions of the Laplacian on the standard torus in $d\ge 2$ dimensions. Making use of the multiplicities in the spectrum of the Laplacian, we put a Gaussian measure on the eigenspaces and use it to average over the eigenspace. We consider a sequence of eigenvalues with growing multiplicity $𝒩\to \infty$.The quantity that we study is the Leray, or microcanonical, measure of the nodal set. We show that the expected value of the Leray measure of an eigenfunction is constant, equal to $1/\sqrt{2\pi }$. Our main result is that the variance of Leray measure is asymptotically $1/4\pi 𝒩$, as $𝒩\to \infty$, at least in dimensions $d=2$ and $d\ge 5$

## How to cite

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Oravecz, Ferenc, Rudnick, Zeév, and Wigman, Igor. "The Leray measure of nodal sets for random eigenfunctions on the torus." Annales de l’institut Fourier 58.1 (2008): 299-335. <http://eudml.org/doc/10312>.

@article{Oravecz2008,
abstract = {We study nodal sets for typical eigenfunctions of the Laplacian on the standard torus in $d\ge 2$ dimensions. Making use of the multiplicities in the spectrum of the Laplacian, we put a Gaussian measure on the eigenspaces and use it to average over the eigenspace. We consider a sequence of eigenvalues with growing multiplicity $\mathcal\{N\}\rightarrow \infty$.The quantity that we study is the Leray, or microcanonical, measure of the nodal set. We show that the expected value of the Leray measure of an eigenfunction is constant, equal to $1/\sqrt\{2\pi \}$. Our main result is that the variance of Leray measure is asymptotically $1/4\pi \mathcal\{N\}$, as $\mathcal\{N\}\rightarrow \infty$, at least in dimensions $d=2$ and $d\ge 5$},
affiliation = {Alfréd Rényi Institute of Mathematics Hungarian Academy of Sciences Reáltanoda utca 13-15 1053 Budapest (Hungary); Tel Aviv University School of Mathematical Sciences 69978 Tel Aviv (Israel); Université de Montréal Centre de recherches mathématiques (CRM) C.P. 6128, succ. centre-ville Montréal, Québec H3C 3J7 (Canada)},
author = {Oravecz, Ferenc, Rudnick, Zeév, Wigman, Igor},
journal = {Annales de l’institut Fourier},
keywords = {Nodal sets; Leray measure; eigenfunctions of the Laplacian; trigonometric polynomials; nodal set; random eigenfunction; torus},
language = {eng},
number = {1},
pages = {299-335},
publisher = {Association des Annales de l’institut Fourier},
title = {The Leray measure of nodal sets for random eigenfunctions on the torus},
url = {http://eudml.org/doc/10312},
volume = {58},
year = {2008},
}

TY - JOUR
AU - Oravecz, Ferenc
AU - Rudnick, Zeév
AU - Wigman, Igor
TI - The Leray measure of nodal sets for random eigenfunctions on the torus
JO - Annales de l’institut Fourier
PY - 2008
PB - Association des Annales de l’institut Fourier
VL - 58
IS - 1
SP - 299
EP - 335
AB - We study nodal sets for typical eigenfunctions of the Laplacian on the standard torus in $d\ge 2$ dimensions. Making use of the multiplicities in the spectrum of the Laplacian, we put a Gaussian measure on the eigenspaces and use it to average over the eigenspace. We consider a sequence of eigenvalues with growing multiplicity $\mathcal{N}\rightarrow \infty$.The quantity that we study is the Leray, or microcanonical, measure of the nodal set. We show that the expected value of the Leray measure of an eigenfunction is constant, equal to $1/\sqrt{2\pi }$. Our main result is that the variance of Leray measure is asymptotically $1/4\pi \mathcal{N}$, as $\mathcal{N}\rightarrow \infty$, at least in dimensions $d=2$ and $d\ge 5$
LA - eng
KW - Nodal sets; Leray measure; eigenfunctions of the Laplacian; trigonometric polynomials; nodal set; random eigenfunction; torus
UR - http://eudml.org/doc/10312
ER -

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