正态分布随机矩阵的矩.双射显式求值
Moments of normally distributed random matrices - Bijective explicit evaluation
摘要：
本文研究了$XUYU^t$的特征值分布，其中$X$和$Y$是给定的对称矩阵，$U$是标准正态分布的随机实值方阵.更具体地说，我们看看它的时刻，即数学期望的轨迹$(XUYU^t)^n$为任意整数$n$。Hanlon，Stanley，Stembridge（1992）证明了这个量可以用超八面体群的双陪集的连接系数的生成级数来表示，其中$X$和$Y$的本征值是不定的。我们提供了一个明确的评价，这些系列的单项对称函数。我们的发展依赖于局部定向超映射和一个新的双射之间的分区局部定向超映射和一些装饰森林的连接系数的解释。作为一个推论，我们提供了一个简单的明确的评价时，$U$是复值和$X$和$Y$的时刻$XUYU^*$厄米矩阵。
英文摘要：
This paper is devoted to the distribution of the eigenvalues of $XUYU^t$ where $X$ and $Y$ are given symmetric matrices and $U$ is a random real valued square matrix of standard normal distribution. More specifically we look at its moments, i.e. the mathematical expectation of the trace of $(XUYU^t)^n$ for arbitrary integer $n$. Hanlon, Stanley, Stembridge (1992) showed that this quantity can be expressed in terms of some generating series for the connection coefficients of the double cosets of the hyperoctahedral group with the eigenvalues of $X$ and $Y$ as indeterminate. We provide an explicit evaluation of these series in terms of monomial symmetric functions. Our development relies on an interpretation of the connection coefficients in terms of locally orientable hypermaps and a new bijective construction between partitioned locally orientable hypermaps and some decorated forests. As a corollary we provide a simple explicit evaluation of the moments of $XUYU^*$ when $U$ is complex valued and $X$ and $Y$ are given hermitian matrices.
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