The Defect of Random Hyperspherical Harmonics

随机超球谐函数的非线性泛函的全变差收敛

Convergence in Total Variation for nonlinear functionals of random hyperspherical harmonics

本文研究随机超球谐函数非线性统计量在高能极限下的全变差收敛，结合中心极限定理与Malliavin-Sobolev范数新结果，建立二阶高斯波动，并证明Gegenbauer多项式乘积矩的新估计。

The defect variance of random spherical harmonics

The defect of a function $f:M\rightarrow \mathbb{R}$ is defined as the difference between the measure of the positive and negative regions. In this paper, we begin the analysis of the distribution of defect of random Gaussian spherical harmonics. By an easy argument, the defect is non-trivial only for even degree and the expected value always vanishes. Our principal result is obtaining the asymptotic shape of the defect variance, in the high frequency limit. As other geometric functionals of random eigenfunctions, the defect may be used as a tool to probe the statistical properties of spherica…

Fluctuations of Polyspectra in Spherical and Euclidean Random Wave Models

We consider polynomial transforms (polyspectra) of Berry's model -- the Euclidean Random Wave model -- and of Random Hyperspherical Harmonics. We determine the asymptotic behavior of variance for polyspectra of any order in the high-frequency limit. In particular, we are able to treat polyspectra of any odd order $q\geq 5$, whose asymptotic behavior was left as a conjecture in the case of Random Hyperspherical Harmonics by Marinucci and Wigman (\emph{Comm. Math. Phys.} 2014). To this end, we exploit a relation between the variance of polyspectra and the distribution of uniform random walks on …