Nodal area distribution for arithmetic random waves
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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.References
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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
- MR Author ID: 829478
- Email: valentina.cammarota@uniroma1.it
- 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.
- © Copyright 2019 American Mathematical Society
- 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
- MathSciNet review: 3988618