A central limit theorem for integrals of random waves
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and
Abstract
We derive a central limit theorem for the mean-square of random waves in the high-frequency limit over shrinking sets. Our proof applies to any compact Riemannian manifold of dimension 3 or higher, thanks to the universality of the local Weyl law. The key technical step is an estimate capturing some cancellation in a triple integral of Bessel functions, which we achieve using Gegenbauer’s addition formula.
Funding statement: Matthew de Courcy-Ireland thanks the Natural Sciences and Engineering Research Council of Canada for a PGS D grant [PGSD2-471570-2015]. Marius Lemm thanks the Institute for Advanced Study for its hospitality during the 2017–2018 academic year.
Acknowledgements
We thank Yaiza Canzani for helpful discussions about Weyl’s law. Matthew de Courcy-Ireland thanks Peter Sarnak for his advice, encouragement, and support over the course of this work. Marius Lemm thanks Rupert L. Frank for multiple stimulating discussions on the topic. We thank the anonymous referee for a careful reading of the paper and many constructive comments.
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© 2022 Walter de Gruyter GmbH, Berlin/Boston
Articles in the same Issue
- Frontmatter
- A central limit theorem for integrals of random waves
- Connectedness of Kisin varieties associated to absolutely irreducible Galois representations
- The Hilbert–Schinzel specialization property
- A mixed elliptic-parabolic boundary value problem coupling a harmonic-like map with a nonlinear spinor
- Isomorphisms among quantum Grothendieck rings and propagation of positivity
- The Neukirch–Uchida theorem with restricted ramification
- Perverse 𝔽p-sheaves on the affine Grassmannian
- Corrigendum to Intrinsic flat stability of the positive mass theorem for graphical hypersurfaces of Euclidean space (J. reine angew. Math. 727 (2017), 269–299)
Articles in the same Issue
- Frontmatter
- A central limit theorem for integrals of random waves
- Connectedness of Kisin varieties associated to absolutely irreducible Galois representations
- The Hilbert–Schinzel specialization property
- A mixed elliptic-parabolic boundary value problem coupling a harmonic-like map with a nonlinear spinor
- Isomorphisms among quantum Grothendieck rings and propagation of positivity
- The Neukirch–Uchida theorem with restricted ramification
- Perverse 𝔽p-sheaves on the affine Grassmannian
- Corrigendum to Intrinsic flat stability of the positive mass theorem for graphical hypersurfaces of Euclidean space (J. reine angew. Math. 727 (2017), 269–299)
