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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Nodal area distribution for arithmetic random waves


Author: Valentina Cammarota
Journal: Trans. Amer. Math. Soc. 372 (2019), 3539-3564
MSC (2010): Primary 60G60, 60D05, 35P20; Secondary 60B10, 58J50
DOI: https://doi.org/10.1090/tran/7779
Published electronically: April 23, 2019
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Abstract: We obtain the limiting distribution of the nodal area of random Gaussian Laplace eigenfunctions on $ \mathbb{T}^3= \mathbb{R}^3/ \mathbb{Z}^3$ (three-dimensional ``arithmetic random waves"). We prove that, as the multiplicity of the eigenspace goes to infinity, the nodal area converges to a universal, non-Gaussian distribution. Universality follows from the equidistribution of lattice points on the sphere. Our arguments rely on the Wiener chaos expansion of the nodal area: we show that, analogous to the two-dimensional case addressed by Marinucci et al., [Geom. Funct. Anal. 26 (2016), pp. 926-960] the fluctuations are dominated by the fourth-order chaotic component. The proof builds upon recent results from Benatar and Maffiucci [Int. Math. Res. Not. IMRN (to appear)] that establish an upper bound for the number of nondegenerate correlations of lattice points on the sphere. We finally discuss higher-dimensional extensions of our result.


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Additional Information

Valentina Cammarota
Affiliation: Department of Mathematics, King’s College London, London, England; and Dipartimento di Scienze Statistiche, Università degli Studi di Roma “La Sapienza", Rome, Italy
Email: valentina.cammarota@uniroma1.it

DOI: https://doi.org/10.1090/tran/7779
Received by editor(s): February 24, 2018
Received by editor(s) in revised form: December 10, 2018
Published electronically: April 23, 2019
Additional Notes: The research leading to these results received funding from the European Research Council under the European Union’s Seventh Framework Programme (FP7/2007–2013)/ERC grant agreement no. 335141.
Article copyright: © Copyright 2019 American Mathematical Society