TY - JOUR

T1 - Local Universality for Zeros and Critical Points of Monochromatic Random Waves

AU - Canzani, Yaiza

AU - Hanin, Boris

N1 - Funding Information:
We are grateful to an anonymous referee for finding a gap in a previous version of this article and for significantly improving some aspects of the exposition. That version concerned a wide class of integral statistics (not just zeros and critical points) of monochromatic random waves. However, there was an error in the previous incarnation of what are now Propositions 4 and 5. The new propositions fix the mistake for the special cases of zeros and critical points. We leave the extension of the results in this paper to more general integral statistics for future work. The second author would also like to thank Damien Gayet and Thomas Letendre for several useful discussions pertaining to the arguments in Proposition 6.
Publisher Copyright:
© 2020, Springer-Verlag GmbH Germany, part of Springer Nature.

PY - 2020/9/1

Y1 - 2020/9/1

N2 - This paper concerns the asymptotic behavior of zeros and critical points for monochromatic random waves ϕλ of frequency λ on a compact, smooth, Riemannian manifold (M, g) as λ→ ∞. We prove global variance estimates for the measures of integration over the zeros and critical points of ϕλ. These global estimates hold for a wide class of manifolds—for example when (M, g) has no conjugate points—and rely on new local variance estimates on zeros and critical points of ϕλ in balls of radius ≈ λ- 1 around a fixed point. Our local results hold under conditions about the structure of geodesics that are generic in the space of all metrics on M.

AB - This paper concerns the asymptotic behavior of zeros and critical points for monochromatic random waves ϕλ of frequency λ on a compact, smooth, Riemannian manifold (M, g) as λ→ ∞. We prove global variance estimates for the measures of integration over the zeros and critical points of ϕλ. These global estimates hold for a wide class of manifolds—for example when (M, g) has no conjugate points—and rely on new local variance estimates on zeros and critical points of ϕλ in balls of radius ≈ λ- 1 around a fixed point. Our local results hold under conditions about the structure of geodesics that are generic in the space of all metrics on M.

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U2 - 10.1007/s00220-020-03826-w

DO - 10.1007/s00220-020-03826-w

M3 - Article

AN - SCOPUS:85089356158

SN - 0010-3616

VL - 378

SP - 1677

EP - 1712

JO - Communications in Mathematical Physics

JF - Communications in Mathematical Physics

IS - 3

ER -