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

Oravecz, Ferenc; Rudnick, Zeév; Wigman, Igor

Annales de l'Institut Fourier, Tome 58 (2008) no. 1, pp. 299-335.

Nous étudions les ensembles nodaux des fonctions propres du Laplacien sur le tore standard de dimension $d\ge 2$. En utilisant la multiplicité du spectre du Laplacien et en introduisant une mesure gaussienne sur l’espace propre, nous nous servons de cette dernière afin d’évaluer des moyennes dans l’espace. Nous considérons une suite de valeurs propres ayant une multiplicité croissante $𝒩\to \infty$.

La quantité que nous étudions est la mesure de Leray (mesure microcanonique). Nous montrons que la moyenne de la mesure de Leray pour une fonction propre est constante et qu’elle vaut $1/\sqrt{2\pi }$. Notre résultat principal précise que la variance de la mesure de Leray est asymptotiquement $1/4\pi 𝒩$ lorsque $𝒩\to \infty$ pour $d=2$ et $d\ge 5$.

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 $d\ge 5$

Classification : 35P20
Keywords: Nodal sets, Leray measure, eigenfunctions of the Laplacian, trigonometric polynomials
Mot clés : ensembles nodaux, mesure de Leray, fonctions propres du Laplacien, polynômes trigonométriques

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