Nodal intersections for random waves on the 3-dimensional torus

Rudnick, Zeév; Wigman, Igor; Yesha, Nadav

Annales de l'Institut Fourier, Tome 66 (2016) no. 6, pp. 2455-2484.

Nous étudions le nombre d’intersections nodales des fonctions propres gaussiennes aléatoires du Laplacien sur le tore plat à trois dimensions avec une courbe régulière de référence fixée de courbure partout non-nulle. Le nombre d’intersections moyen est toujours proportionnel à la longueur de la courbe de référence, multipliée par le nombre d’onde et est indépendant de la géométrie. Notre résultat principal est une borne sur la variance, lorsque la torsion de la courbe est partout non-nulle ou lorsque la courbe est planaire.

We investigate the number of nodal intersections of random Gaussian Laplace eigenfunctions on the standard three-dimensional flat torus with a fixed smooth reference curve, which has nowhere vanishing curvature. The expected intersection number is universally proportional to the length of the reference curve, times the wavenumber, independent of the geometry. Our main result gives a bound for the variance, if either the torsion of the curve is nowhere zero or if the curve is planar.

Classification : 60G15, 11P21
Keywords: Nodal line, torus, Laplace eigenfunctions, variance, test curve, intersection points, curvature, asymptotics
Mot clés : Ligne nodale, tore, fonctions propres du Laplacien, variance, courbe test, points d’intersections, courbure, comportement asymptotique

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