no. 6
Nodal intersections for random waves on the 3-dimensional torus
[Intersections nodales d’ondes aléatoires sur le tore tri-dimensionnel]
Rudnick, Zeév1 ; Wigman, Igor2 ; Yesha, Nadav1
1 School of Mathematical Sciences Tel Aviv University Tel Aviv (Israel)
2 Department of Mathematics King’s College London Strand London WC2R 2LS (UK)
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.

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DOI : 10.5802/aif.3068
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

Rudnick, Zeév 1 ; Wigman, Igor 2 ; Yesha, Nadav 1

1 School of Mathematical Sciences Tel Aviv University Tel Aviv (Israel)
2 Department of Mathematics King’s College London Strand London WC2R 2LS (UK) 
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Rudnick, Zeév; Wigman, Igor; Yesha, Nadav. Nodal intersections for random waves on the 3-dimensional torus. Annales de l'Institut Fourier, Tome 66 (2016) no. 6, pp. 2455-2484. doi : 10.5802/aif.3068. https://aif.centre-mersenne.org/articles/10.5802/aif.3068/

[1] Azaïs, Jean-Marc; Wschebor, Mario Level sets and extrema of random processes and fields, John Wiley & Sons Inc., Hoboken, NJ, 2009, xi+393 pages

[2] Bourgain, J.; Rudnick, Z. On the nodal sets of toral eigenfunctions, Inventiones Math., Volume 185 (2011) no. 1, pp. 199-237 | DOI

[3] Bourgain, J.; Rudnick, Z. Restriction of toral eigenfunctions to hypersurfaces and nodal sets, Geometric and Functional Analysis, Volume 22 (2012), pp. 878-937 | DOI

[4] Bourgain, J.; Rudnick, Z. Nodal intersections and L p restriction theorems on the torus, Israel J. Math., Volume 207 (2015), pp. 479-505 | DOI

[5] Bourgain, J.; Rudnick, Z.; Sarnak, P. manuscript

[6] Bourgain, J.; Rudnick, Z.; Sarnak, P. Local statistics of lattice points on the sphere (2014) (to appear in Contemporary Mathematics, proceedings of Constructive Functions)

[7] Cammarota, V.; Marinucci, D.; Wigman, I. On the distribution of the critical values of random spherical harmonics (to appear in J. Geom. Anal.)

[8] Cramer, H.; Leadbetter, M.R. Stationary and related stochastic processes. Sample function properties and their applications, John Wiley & Sons, Inc., New York-London-Sydney, 1967, xii+348 pages

[9] Duke, W. Hyperbolic distribution problems and half-integral weight Maass forms, Invent. Math., Volume 92 (1988) no. 1, pp. 73-90 | DOI

[10] Golubeva, E. P.; Fomenko, O. M. Asymptotic distribution of lattice points on the three-dimensional sphere, Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI), Volume 160 (1987), pp. 54-71

[11] Krishnapur, M.; Kurlberg, P.; Wigman, I. Nodal length fluctuations for arithmetic random waves, Ann. of Math., Volume 177 (2013) no. 2, pp. 699-737 | DOI

[12] Rudnick, Z.; Wigman, I. Nodal intersections for random eigenfunctions on the torus (to appear in Amer. J. of Math.)

[13] Rudnick, Z.; Wigman, I. On the volume of nodal sets for eigenfunctions of the Laplacian on the torus, Ann. Henri Poincaré, Volume 9 (2008) no. 1, pp. 109-130 | DOI

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