On the Diaconis-Shahshahani Method in Random Matrix Theory
On the Diaconis-Shahshahani Method in Random Matrix Theory
复制标题
随机矩阵理论中的 Diaconis-Shahshahani 方法
DOI：
10.1007/s10801-005-4629-x
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发表时间：
2005
影响因子：
0.8
通讯作者：
M. Stolz
中科院分区：
文献类型：
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作者：
M. Stolz
如果Γ是一个值在紧矩阵群K中的随机变量，则迹Tr（Γj）（j <$N）是真实的值或复值随机变量。作为随机矩阵特征值方法的关键一步，Diaconis和Shahshahani计算了任意固定数量迹的联合矩，如果Γ根据Haar测度分布，并且K是Un，On或Spn之一，其中n足够大。在正交和辛情形下，它们的证明是基于Ram关于Brauer代数特征标的工作，本文给出了这些矩公式的另一种证明。它调用经典不变理论（具体地说，外尔意义上的第一基本定理的张量形式）将矩阵积分的计算减少为计数问题，可以通过初等方法解决。
If Γ is a random variable with values in a compact matrix group K, then the traces Tr(Γj) (j ∊ N) are real or complex valued random variables. As a crucial step in their approach to random matrix eigenvalues, Diaconis and Shahshahani computed the joint moments of any fixed number of these traces if Γ is distributed according to Haar measure and if K is one of Un, On or Spn, where n is large enough. In the orthogonal and symplectic cases, their proof is based on work of Ram on the characters of Brauer algebras.The present paper contains an alternative proof of these moment formulae. It invokes classical invariant theory (specifically, the tensor forms of the First Fundamental Theorems in the sense of Weyl) to reduce the computation of matrix integrals to a counting problem, which can be solved by elementary means.