Geometric and Functional Analysis

, Volume 26, Issue 3, pp 926–960 | Cite as

Non-Universality of Nodal Length Distribution for Arithmetic Random Waves

  • Domenico Marinucci
  • Giovanni Peccati
  • Maurizia Rossi
  • Igor Wigman


“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.

Keywords and phrases

Arithmetic random waves Nodal lines Non-central limit theorem Berry’s cancellation 

Mathematics Subject Classification

60G60 60D05 60B10 58J50 35P20 


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Copyright information

© Springer International Publishing 2016

Authors and Affiliations

  • Domenico Marinucci
    • 1
  • Giovanni Peccati
    • 2
  • Maurizia Rossi
    • 2
  • Igor Wigman
    • 3
  1. 1.Dipartimento di MatematicaUniversità di Roma Tor VergataRomeItaly
  2. 2.Unité de Recherche en Mathématiques, Faculté des Sciences, de la Technologie et de la CommunicationUniversité du LuxembourgLuxembourg CityLuxembourg
  3. 3.Department of MathematicsKing’s College LondonLondonUK

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