The Asymptotic Equivalence of the Sample Trispectrum and the Nodal Length for Random Spherical Harmonics
Marinucci, Domenico ; Rossi, Maurizia ; Wigman, Igor
arXiv, 1705.05747 / Harvested from arXiv
We study the asymptotic behaviour of the nodal length of random $2d$-spherical harmonics $f_{\ell}$ of high degree $\ell \rightarrow\infty$, i.e. the length of their zero set $f_{\ell}^{-1}(0)$. It is found that the nodal lengths are asymptotically equivalent, in the $L^{2}$-sense, to the "sample trispectrum", i.e., the integral of $H_{4}(f_{\ell}(x))$, the fourth-order Hermite polynomial of the values of $f_{\ell}$. A particular by-product of this is a Quantitative Central Limit Theorem (in Wasserstein distance) for the nodal length, in the high energy limit.
Publié le : 2017-05-16
Classification:  Mathematics - Probability,  Mathematical Physics,  60G60, 62M15, 53C65, 42C10, 33C55
@article{1705.05747,
     author = {Marinucci, Domenico and Rossi, Maurizia and Wigman, Igor},
     title = {The Asymptotic Equivalence of the Sample Trispectrum and the Nodal
  Length for Random Spherical Harmonics},
     journal = {arXiv},
     volume = {2017},
     number = {0},
     year = {2017},
     language = {en},
     url = {http://dml.mathdoc.fr/item/1705.05747}
}
Marinucci, Domenico; Rossi, Maurizia; Wigman, Igor. The Asymptotic Equivalence of the Sample Trispectrum and the Nodal
  Length for Random Spherical Harmonics. arXiv, Tome 2017 (2017) no. 0, . http://gdmltest.u-ga.fr/item/1705.05747/