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Small Scale CLTs for the Nodal Length of Monochromatic Waves

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Abstract

We consider the nodal length \(L(\lambda )\) of the restriction to a ball of radius \(r_\lambda \) of a Gaussian pullback monochromatic random wave of parameter \(\lambda >0\) associated with a Riemann surface \((\mathcal M,g)\) without conjugate points. Our main result is that, if \(r_\lambda \) grows slower than \((\log \lambda )^{1/25}\), then (as \(\lambda \rightarrow \infty \)) the length \(L(\lambda )\) verifies a Central Limit Theorem with the same scaling as Berry’s random wave model—as established in Nourdin et al. (Commun Math Phys 369(1):99–151, 2019). Taking advantage of some powerful extensions of an estimate by Bérard (Mathematische Zeitschrift 155:249–276, 1977) due to Keeler (A logarithmic improvement in the two-point Weyl law for manifolds without conjugate points, 2019. arXiv:1905.05136), our techniques are mainly based on a novel intrinsic bound on the coupling of smooth Gaussian fields, that is of independent interest, and moreover allow us to improve some estimates for the nodal length asymptotic variance of pullback random waves in Canzani and Hanin (Commun Math Phys 378:1677–1712, 2020). In order to demonstrate the flexibility of our approach, we also provide an application to phase transitions for the nodal length of arithmetic random waves on shrinking balls of the 2-torus.

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Acknowledgements

We heartily thank an anonymous referee for insightful comments, B. Hanin for discussing with us several aspects of his work [CH20] and B. Keeler for sharing his work [Kee19] before publication. We also thank Y. Canzani, P. Kurlberg, L. Pratelli, P. Rigo and M. Sodin for fruitful discussions. M.R. thanks D. Marinucci for raising an interesting question leading to Remark 1.7 (d). The work of Gauthier Dierickx has been supported by the FNR grant STARS(PUL) at Luxembourg University and in part by the Collaborative Research Center “Statistical modeling of nonlinear dynamic processes” (SFB 823, Teilprojekt C2) of the German Research Foundation (DFG). Ivan Nourdin is supported by the FNR grant APOGee at Luxembourg University. Giovanni Peccati is supported by the FNR grant FoRGES at Luxembourg University. The research of Maurizia Rossi has been supported by the Foundation Science Mathématique de Paris and (partially) by the ANR-17-CE40-0008 project Unirandom.

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Authors and Affiliations

  1. Fakultät für Mathematik, Ruhr-Universität Bochum, Bochum, Germany

    Gauthier Dierickx

  2. Département de Mathématiques, Université du Luxembourg, Esch-sur-Alzette, Luxembourg

    Ivan Nourdin & Giovanni Peccati

  3. Dipartimento di Matematica e Applicazioni, Università di Milano-Bicocca, Milan, Italy

    Maurizia Rossi

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  1. Gauthier Dierickx
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Correspondence to Ivan Nourdin.

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Dierickx, G., Nourdin, I., Peccati, G. et al. Small Scale CLTs for the Nodal Length of Monochromatic Waves. Commun. Math. Phys. 397, 1–36 (2023). https://doi.org/10.1007/s00220-022-04422-w

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