论文标题
两个Wishart矩阵差的光谱统计
Spectral statistics for the difference of two Wishart matrices
论文作者
论文摘要
在这项工作中，我们考虑了两个独立的复杂WishArt矩阵的加权差异，并在有限的差异场景中使用两种不同的方法得出了相应特征值的关节概率密度函数。第一个派生涉及使用统一组积分，而第二个派生依赖于应用衍生原理。后者将从单位不变的合奏中绘制的基质的特征值的关节概率密度与其对角线元件的关节概率密度有关。还获得了任意阶相关函数的精确闭合形式表达式，并将光谱密度与蒙特卡洛模拟结果形成鲜明对比。还得出了量矩的分析结果以及量化频谱阳性方面的概率。此外，我们使用代数随机矩阵的stieltjes变换方法为光谱密度提供了大差异结果。最后，我们指出了这些结果与两个随机密度矩阵差的相应结果的关系，并获得了光谱密度和绝对平均值的一些显式和封闭形式的表达式。
In this work, we consider the weighted difference of two independent complex Wishart matrices and derive the joint probability density function of the corresponding eigenvalues in a finite-dimension scenario using two distinct approaches. The first derivation involves the use of unitary group integral, while the second one relies on applying the derivative principle. The latter relates the joint probability density of eigenvalues of a matrix drawn from a unitarily invariant ensemble to the joint probability density of its diagonal elements. Exact closed form expressions for an arbitrary order correlation function are also obtained and spectral densities are contrasted with Monte Carlo simulation results. Analytical results for moments as well as probabilities quantifying positivity aspects of the spectrum are also derived. Additionally, we provide a large-dimension asymptotic result for the spectral density using the Stieltjes transform approach for algebraic random matrices. Finally, we point out the relationship of these results with the corresponding results for difference of two random density matrices and obtain some explicit and closed form expressions for the spectral density and absolute mean.
