Fluctuations of the Nodal Length of Random Spherical Harmonics
Fluctuations of the Nodal Length of Random Spherical Harmonics
复制标题
随机球谐函数节点长度的涨落
DOI：
10.1007/s00220-010-1078-8
复制
发表时间：
2009
影响因子：
2.4
通讯作者：
I. Wigman
中科院分区：
文献类型：
--
作者：
I. Wigman
利用球面上的拉普拉斯本征空间（球谐函数空间）的重数，我们赋予该空间高斯概率测度。这就引出了具有拉普拉斯本征值E = n（n + 1）的n次随机高斯球谐函数的概念。本文研究了随机球谐函数节线的长度分布，已知节线的期望长度为n阶。由于自然缩放，很自然地推测方差应该是n阶。我们的主要结果是，由于一个意想不到的取消，随机球谐函数的节点长度的方差是为了log n。这种行为与Berry对混沌台球（随机波模型）上的节线的预测一致。此外，我们发现，类似的结果是适用于“通用”的线性统计的节线。
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.
影响因子：
4.9
作者：
Krishnapur M
通讯作者：
Krishnapur M