“Arithmetic random waves” are the Gaussian Laplace eigenfunctions on the two-dimensional torus (Rudnick and Wigman in Annales de l’Insitute Henri Poincaré 9(1):109–130, 2008; Krishnapur et al. in Annals of Mathematics (2) 177(2):699–737, 2013). In this paper we find that their nodal length converges to a non-universal (non-Gaussian) limiting distribution, depending on the angular distribution of lattice points lying on circles. Our argument has two main ingredients. An explicit derivation of the Wiener–Itô chaos expansion for the nodal length shows that it is dominated by its 4th order chaos component (in particular, somewhat surprisingly, the second order chaos component vanishes). The rest of the argument relies on the precise analysis of the fourth order chaotic component.

Marinucci, D., Peccati, G., Rossi, M., Wigman, I. (2016). Non-Universality of Nodal Length Distribution for Arithmetic Random Waves. GEOMETRIC AND FUNCTIONAL ANALYSIS, 26(3), 926-960 [10.1007/s00039-016-0376-5].

### Non-Universality of Nodal Length Distribution for Arithmetic Random Waves

#### Abstract

“Arithmetic random waves” are the Gaussian Laplace eigenfunctions on the two-dimensional torus (Rudnick and Wigman in Annales de l’Insitute Henri Poincaré 9(1):109–130, 2008; Krishnapur et al. in Annals of Mathematics (2) 177(2):699–737, 2013). In this paper we find that their nodal length converges to a non-universal (non-Gaussian) limiting distribution, depending on the angular distribution of lattice points lying on circles. Our argument has two main ingredients. An explicit derivation of the Wiener–Itô chaos expansion for the nodal length shows that it is dominated by its 4th order chaos component (in particular, somewhat surprisingly, the second order chaos component vanishes). The rest of the argument relies on the precise analysis of the fourth order chaotic component.
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Articolo in rivista - Articolo scientifico
Arithmetic random waves; Berry’s cancellation; Nodal lines; Non-central limit theorem;
Arithmetic random waves
English
2016
26
3
926
960
none
Marinucci, D., Peccati, G., Rossi, M., Wigman, I. (2016). Non-Universality of Nodal Length Distribution for Arithmetic Random Waves. GEOMETRIC AND FUNCTIONAL ANALYSIS, 26(3), 926-960 [10.1007/s00039-016-0376-5].
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Utilizza questo identificativo per citare o creare un link a questo documento: `https://hdl.handle.net/10281/253305`
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