We investigate the asymptotic behavior of the nodal lines for random spherical harmonics restricted to shrinking domains, in the 2-dimensional case: for example, the length of the zero set Zℓ,rℓ:= ZBrℓ (Tℓ) = len({x ∈ S2 ∩ Brℓ: Tℓ(x) = 0}), where Brℓ is the spherical cap of radius rℓ. We show that the variance of the nodal length is logarithmic in the high energy limit; moreover, it is asymptotically fully equivalent, in the L2-sense, to the "local sample trispectrum", namely, the integral on the ball of the fourth-order Hermite polynomial. This result extends and generalizes some recent findings for the full spherical case. As a consequence a Central Limit Theorem is established.

Todino, A. (2020). Nodal lengths in shrinking domains for random eigenfunctions on S2. BERNOULLI, 26(4), 3081-3110 [10.3150/20-BEJ1216].

Nodal lengths in shrinking domains for random eigenfunctions on S2

Todino, AP
2020

Abstract

We investigate the asymptotic behavior of the nodal lines for random spherical harmonics restricted to shrinking domains, in the 2-dimensional case: for example, the length of the zero set Zℓ,rℓ:= ZBrℓ (Tℓ) = len({x ∈ S2 ∩ Brℓ: Tℓ(x) = 0}), where Brℓ is the spherical cap of radius rℓ. We show that the variance of the nodal length is logarithmic in the high energy limit; moreover, it is asymptotically fully equivalent, in the L2-sense, to the "local sample trispectrum", namely, the integral on the ball of the fourth-order Hermite polynomial. This result extends and generalizes some recent findings for the full spherical case. As a consequence a Central Limit Theorem is established.
Articolo in rivista - Articolo scientifico
Scientifica
Berry's cancellation; Limit theorem; Random eigenfunctions; Sample trispectrum;
English
Todino, A. (2020). Nodal lengths in shrinking domains for random eigenfunctions on S2. BERNOULLI, 26(4), 3081-3110 [10.3150/20-BEJ1216].
Todino, A
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Utilizza questo identificativo per citare o creare un link a questo documento: http://hdl.handle.net/10281/374829
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