Quantitative Limit Theorems for Local Functionals of Arithmetic Random Waves
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We consider Gaussian Laplace eigenfunctions on the two-dimensional flat torus (arithmetic random waves), and provide explicit Berry-Esseen bounds in the 1-Wasserstein distance for the normal and non-normal high-energy approximation of the associated Leray measures and total nodal lengths, respectively. Our results provide substantial extensions (as well as alternative proofs) of findings by Oravecz et al. (Ann Inst Fourier (Grenoble) 58(1):299–335, 2008), Krishnapur et al. (Ann Math 177(2):699–737, 2013), and Marinucci et al. (Geom Funct Anal 26(3):926–960, 2016). Our techniques involve Wiener-Itô chaos expansions, integration by parts, as well as some novel estimates on residual terms arising in the chaotic decomposition of geometric quantities that can implicitly be expressed in terms of the coarea formula.
We thank Ch. Döbler for useful discussions (in particular, for pointing out the relevance of ), as well as two Referees for several useful remarks. The research leading to this work has been supported by the grant F1R-MTH-PUL-15STAR (STARS) at the University of Luxembourg.
- 1.Abramovitz, M., Stegun, I.-A.: Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. National Bureau of Standards Applied Mathematics Series, vol. 55. United States Department of Commerce, National Bureau of Standards (NBS), Washington, DC (1964)Google Scholar
- 5.Dalmao, F., Nourdin, I., Peccati, G., Rossi, M.: Phase singularities in complex arithmetic random waves. Preprint (2016). ArXiv 1608.05631Google Scholar
- 6.Döbler, C.: New developments in Stein’s method with applications. Ph.D. Thesis, Ruhr-Universität Bochum (2012)Google Scholar
- 7.Döbler, C.: The Stein equation beyond the support with applications. In preparation (2018+)Google Scholar
- 12.Kurlberg, P., Wigman, I.: On probability measures arising from lattice points on circles. Math. Ann. (2015, in press). ArXiv: 1501.01995Google Scholar
- 14.Nourdin, I., Peccati, G.: Normal approximations with Malliavin calculus. From Stein’s method to universality. Cambridge Tracts in Mathematics, vol. 192. Cambridge University Press, Cambridge (2012)Google Scholar
- 18.Wigman, I.: On the nodal lines of random and deterministic Laplace eigenfunctions. In: Spectral Geometry. Proceedings of Symposia in Pure Mathematics, vol. 84, pp. 285–297. American Mathematical Society, Providence (2012)Google Scholar
- 19.Yau, S.T.: Survey on partial differential equations in differential geometry. Seminar on Differential Geometry. Annals of Mathematics Studies, vol. 102, pp. 3–71. Princeton University Press, Princeton (1982)Google Scholar