Mathematical analysis
Non-universality of the Nazarov–Sodin constant
[Non-universalité de la constante de Nazarov–Sodin]

Présenté par : the Editorial Board

Kurlberg, Pär1 ; Wigman, Igor2
1 Department of Mathematics, KTH Royal Institute of Technology, SE-100 44 Stockholm, Sweden
2 Department of Mathematics, King's College London, UK
Comptes Rendus. Mathématique, Tome 353 (2015) no. 2, pp. 101-104.

On démontre que la constante de Nazarov–Sodin, qui, à un changement d'échelle près, donne le terme principal de l'ordre de croissance du nombre de composantes nodales d'un champ aléatoire gaussien, dépend effectivement du champ. On en déduit que le résultat reste vrai pour les « ondes aléatoires arithmétiques », c'est-à-dire pour les fonctions propres du laplacien aléatoire sur un tore.

We prove that the Nazarov–Sodin constant, which up to a natural scaling gives the leading order growth for the expected number of nodal components of a random Gaussian field, genuinely depends on the field. We then infer the same for “arithmetic random waves”, i.e. random toral Laplace eigenfunctions.

Reçu le :
Accepté le :
Publié le :
DOI : 10.1016/j.crma.2014.09.026
Kurlberg, Pär 1 ; Wigman, Igor 2

1 Department of Mathematics, KTH Royal Institute of Technology, SE-100 44 Stockholm, Sweden
2 Department of Mathematics, King's College London, UK 
@article{CRMATH_2015__353_2_101_0,
     author = {Kurlberg, P\"ar and Wigman, Igor},
     title = {Non-universality of the {Nazarov{\textendash}Sodin} constant},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {101--104},
     publisher = {Elsevier},
     volume = {353},
     number = {2},
     year = {2015},
     doi = {10.1016/j.crma.2014.09.026},
     language = {en},
     url = {http://archive.numdam.org/articles/10.1016/j.crma.2014.09.026/}
}
TY  - JOUR
AU  - Kurlberg, Pär
AU  - Wigman, Igor
TI  - Non-universality of the Nazarov–Sodin constant
JO  - Comptes Rendus. Mathématique
PY  - 2015
SP  - 101
EP  - 104
VL  - 353
IS  - 2
PB  - Elsevier
UR  - http://archive.numdam.org/articles/10.1016/j.crma.2014.09.026/
DO  - 10.1016/j.crma.2014.09.026
LA  - en
ID  - CRMATH_2015__353_2_101_0
ER  - 
%0 Journal Article
%A Kurlberg, Pär
%A Wigman, Igor
%T Non-universality of the Nazarov–Sodin constant
%J Comptes Rendus. Mathématique
%D 2015
%P 101-104
%V 353
%N 2
%I Elsevier
%U http://archive.numdam.org/articles/10.1016/j.crma.2014.09.026/
%R 10.1016/j.crma.2014.09.026
%G en
%F CRMATH_2015__353_2_101_0
Kurlberg, Pär; Wigman, Igor. Non-universality of the Nazarov–Sodin constant. Comptes Rendus. Mathématique, Tome 353 (2015) no. 2, pp. 101-104. doi : 10.1016/j.crma.2014.09.026. http://archive.numdam.org/articles/10.1016/j.crma.2014.09.026/

[1] Bérard, P.; Helffer, B. Dirichlet eigenfunctions of the square membrane: Courant's property, and A. Stern's and A. Pleijel's analyses (Available online) | arXiv

[2] Berry, M.V. Regular and irregular semiclassical wavefunctions, J. Phys. A, Volume 10 (1977) no. 12, pp. 2083-2091

[3] Bogomolny, E.; Schmit, C. Percolation model for nodal domains of chaotic wave functions, Phys. Rev. Lett., Volume 88 (2002), p. 114102

[4] Cilleruelo, J. The distribution of the lattice points on circles, J. Number Theory, Volume 43 (1993) no. 2, pp. 198-202

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

[6] P. Kurlberg, I. Wigman, On probability measures arising from lattice points on circles, in preparation.

[7] Nazarov, F.; Sodin, M. On the number of nodal domains of random spherical harmonics, Amer. J. Math., Volume 131 (2009) no. 5, pp. 1337-1357

[8] Sarnak, P.; Wigman, I. Topologies of nodal sets of random band limited functions (Available online) | arXiv

[9] Sodin, M. Lectures on random nodal portraits, June 2012 http://www.math.tau.ac.il/~sodin/SPB-Lecture-Notes.pdf (Preprint. Lecture notes for a mini-course given at the St. Petersburg Summer School in Probability and Statistical Physics, Available online)

Cité par Sources :