We study the asymptotic behaviour of the nodal length of random 2d-spherical harmonics f(l) of high degree l -> infinity i.e. the length of their zero set f(l)(-1)(0). It is found that the nodal lengths are asymptotically equivalent, in the L-2-sense, to the "sample trispectrum", i.e., the integral of H-4(f(l)(x)), the fourth-order Hermite polynomial of the values of f(l). 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
Marinucci, D
;
2020-01-01
Abstract
We study the asymptotic behaviour of the nodal length of random 2d-spherical harmonics f(l) of high degree l -> infinity i.e. the length of their zero set f(l)(-1)(0). It is found that the nodal lengths are asymptotically equivalent, in the L-2-sense, to the "sample trispectrum", i.e., the integral of H-4(f(l)(x)), the fourth-order Hermite polynomial of the values of f(l). A particular by-product of this is a Quantitative Central Limit Theorem (in Wasserstein distance) for the nodal length, in the high energy limit.| File | Dimensione | Formato | |
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