We study the asymptotic behaviour of the nodal length of random 2d-spherical harmonics fℓ of high degree ℓ → ∞, i.e. the length of their zero set f-1ℓ (0). It is found that the nodal lengths are asymptotically equivalent, in the L2-sense, to the "gsample trispectrum", i.e., the integral of H4(fℓ(x)), the fourth-order Hermite polynomial of the values of fℓ. A particular by-product of this is a Quantitative Central Limit Theorem (in Wasserstein distance) for the nodal length, in the high energy limit.
Marinucci, D., Rossi, M., Wigman, I. (2020). The asymptotic equivalence of the sample trispectrum and the nodal length for random spherical harmonics. ANNALES DE L'INSTITUT HENRI POINCARE-PROBABILITES ET STATISTIQUES, 56(1), 374-390 [10.1214/19-AIHP964].
The asymptotic equivalence of the sample trispectrum and the nodal length for random spherical harmonics
Rossi, M;
2020
Abstract
We study the asymptotic behaviour of the nodal length of random 2d-spherical harmonics fℓ of high degree ℓ → ∞, i.e. the length of their zero set f-1ℓ (0). It is found that the nodal lengths are asymptotically equivalent, in the L2-sense, to the "gsample trispectrum", i.e., the integral of H4(fℓ(x)), the fourth-order Hermite polynomial of the values of fℓ. A particular by-product of this is a Quantitative Central Limit Theorem (in Wasserstein distance) for the nodal length, in the high energy limit.
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