On the distribution of the nodal sets of random spherical harmonics
On the distribution of the nodal sets of random spherical harmonics
复制标题
关于随机球谐函数节点集的分布
DOI：
10.1063/1.3056589
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发表时间：
2008
影响因子：
1.3
通讯作者：
I. Wigman
中科院分区：
文献类型：
--
作者：
I. Wigman
研究了m维球面上拉普拉斯特征函数节点集的体积。众所周知，En=n（n+m−1）对应的特征空间是n维n次球谐波的空间En。我们利用特征值的多重性赋予En高斯概率测度，研究随机选取的函数的节点集的m维体积分布。期望体积与En成正比。我们的主要结果之一是将体积的方差限定为0 （En/N）。除了节点集的体积外，我们还研究了节点集的Leray测度。我们发现它的期望值与n无关。我们可以确定方差的渐近形式是(const)/N。
We study the volume of the nodal set of eigenfunctions of the Laplacian on the m-dimensional sphere. It is well known that the eigenspaces corresponding to En=n(n+m−1) are the spaces En of spherical harmonics of degree n of dimension N. We use the multiplicity of the eigenvalues to endow En with the Gaussian probability measure and study the distribution of the m-dimensional volume of the nodal sets of a randomly chosen function. The expected volume is proportional to En. One of our main results is bounding the variance of the volume to be O(En/N). In addition to the volume of the nodal set, we study its Leray measure. We find that its expected value is n independent. We are able to determine that the asymptotic form of the variance is (const)/N.