Local Universality for Zeros and Critical Points of Monochromatic Random Waves
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Communications in
Mathematical Physics
Local Universality for Zeros and Critical Points of Monochromatic Random Waves Yaiza Canzani1 , Boris Hanin2,3 1 Department of Mathematics, University of North Carolina, Chapel Hill, USA. E-mail: [email protected] 2 Department of Mathematics, Texas A&M, Texas, USA. E-mail: [email protected] 3 Facebook AI Research, New York, USA.
Received: 17 January 2019 / Accepted: 21 April 2020 Published online: 12 August 2020 – © Springer-Verlag GmbH Germany, part of Springer Nature 2020
Abstract: 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. 1. Introduction This article gives new local and global results about the measures of integration over the zero and critical point sets of a monochromatic random wave φλ of frequency λ. To introduce our results, let (M, g) be a compact, smooth, Riemannian manifold without boundary of dimension n ≥ 2, and write g for the positive definite Laplace-Beltrami 2 operator. Consider an orthonormal basis {ϕ j }∞ j=1 of L (M, g) consisting of real-valued eigenfunctions g ϕ j = λ2j ϕ j with 0 = λ0 < λ1 ≤ λ2 ≤ · · · ∞, normalized so that ϕ j = 1. Monochromatic random waves of frequency λ are Gaussian fields on M 2 defined by 1 φλ := √dim(H ajϕj, (1) ) λ
λ j ∈[λ,λ+1]
where the coefficients a j ∼ N (0, 1) are real valued i.i.d standard Gaussians and Hλ := ker(g − λ2j ). λ j ∈[λ,λ+1]
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Y. Canzani, B. Hanin
We write φλ ∈ RWλ (M, g) for short. The ensembles φλ are Gaussian models for eigenfunctions of the Laplacian with eigenvalue approximately equal to λ2 on a compact Riemannian manifold (M, g). In the setting of a general smooth manifold the ensembles RWλ were first defined by Zelditch in [32]. Zelditch was inspired in part by the influential work of Berry [5], which proposes that random waves on Euclidean space and flat tori are good semiclassical models for high frequency wavefunctions in quantum systems whose classical dynamics are chaotic. Specifically, Berry proposed his Random Wave Conjecture: at a fixed x ∈ M the local behavior of deterministic eigenfunctions ϕ j (x + u/λ) should be well-approximated, as λ → ∞, by the behavior of frequency 1 random waves on Tx M ∼ = Rn , so long as geodesics on (M, g) are chaotic. The idea is that the rescaled eigenfunctions ϕ j (x + u/λ) approximately solve f = f for the “frozen” constant coefficient Laplacian on Tx M ∼ = Rn and hence are almost frequency 1 eigenfunctions of on Rn (see (7)). Moreover,
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