Products of rectangular random matrices: Singular values and progressive scattering
Products of rectangular random matrices: Singular values and progressive scattering
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DOI：
10.1103/physreve.88.052118
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发表时间：
2013-11-11
影响因子：
2.4
通讯作者：
Kieburg, Mario
中科院分区：
文献类型：
--
作者：
Akemann, Gernot;Ipsen, Jesper R.;Kieburg, Mario
本文讨论了具有独立高斯项的M个矩形随机矩阵的乘积，它在无线通信和经济物理等领域有着广泛的应用。对于复矩阵，利用Harish-Chandra-Itzykson-Zuber积分公式，得到了联合概率密度函数的显式表达式。利用双矩阵模型和双正交多项式的方法，得到了有限矩阵大小的所有相关函数和矩的显式表达式。这推广了所谓的Wishart-Laguerre Gaus型系综(或手征酉系综)在M=1时的经典结果和前人关于方阵乘积的结果。相关函数由一个行列式点过程给出，其中核可以用Meijer G-函数来表示。在大矩阵极限下，我们将计算结果与数值模拟和已知的宏观能级密度结果进行了比较。最后，我们考虑了所谓的遍历互信息，它给出了具有多重散射的MIMO通信信道频谱效率的一个上界。
We discuss the product of M rectangular random matrices with independent Gaussian entries, which have several applications, including wireless telecommunication and econophysics. For complex matrices an explicit expression for the joint probability density function is obtained using the Harish-Chandra-Itzykson-Zuber integration formula. Explicit expressions for all correlation functions and moments for finite matrix sizes are obtained using a two-matrix model and the method of biorthogonal polynomials. This generalizes the classical result for the so-called Wishart-Laguerre Gaussian unitary ensemble (or chiral unitary ensemble) at M = 1, and previous results for the product of square matrices. The correlation functions are given by a determinantal point process, where the kernel can be expressed in terms of Meijer G-functions. We compare the results with numerical simulations and known results for the macroscopic level density in the limit of large matrices. The location of the end points of support for the latter are analyzed in detail for general M. Finally, we consider the so-called ergodic mutual information, which gives an upper bound for the spectral efficiency of a MIMO communication channel with multifold scattering.