Fluctuations of the nodal length of random spherical harmonics, erratum
Igor Wigman
备注：
This is to correct a sign mistake that has been made in the previous version (that was published in Comm. Math. Phys.). As a result the leading constant in all the theorems was wrong, and the constants are now consistent with the one predicted by Berry. A corrected manuscript plus a detailed erratum with all the corrections that were made relatively to the version published is attached
Using the multiplicities of the Laplace eigenspace on the sphere (the space of spherical harmonics) we endow the space with Gaussian probability measure. This induces a notion of random Gaussian spherical harmonics of degree $n$ having Laplace eigenvalue $E=n(n+1)$. We study the length distribution of the nodal lines of random spherical harmonics. It is known that the expected length is of order $n$. It is natural to conjecture that the variance should be of order $n$, due to the natural scaling. Our principal result is that, due to an unexpected cancelation, the variance of the nodal length of random spherical harmonics is of order $\log{n}$. This behaviour is consistent with the one predicted by Berry for nodal lines on chaotic billiards (Random Wave Model). In addition we find that a similar result is applicable for "generic" linear statistics of the nodal lines.
Moderate Deviation estimates for Nodal Lengths of Random Spherical Harmonics
证明了随机球谐函数节点长度的中偏差估计在全球面上和收缩球面域上都是成立的。后者的中心极限定理最近分别在Marinucci，Rossi and Wigman（2020）和Todino（2020）中建立。我们的证明是基于Schulte和Thäle（2016）对生活在固定Wiener混沌中的随机变量序列的中偏差原理与基于指数等价概念的众所周知的结果的组合。
