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.
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.
Accepted:
Published online:
Pär Kurlberg 1; Igor Wigman 2
@article{CRMATH_2015__353_2_101_0, author = {P\"ar Kurlberg and Igor Wigman}, 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}, }
Pär Kurlberg; Igor Wigman. Non-universality of the Nazarov–Sodin constant. Comptes Rendus. Mathématique, Volume 353 (2015) no. 2, pp. 101-104. doi : 10.1016/j.crma.2014.09.026. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2014.09.026/
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