Akemann Gernot, Ipsen Jesper R, Kieburg Mario
Department of Physics, Bielefeld University, Postfach 100131, D-33501 Bielefeld, Germany.
Phys Rev E Stat Nonlin Soft Matter Phys. 2013 Nov;88(5):052118. doi: 10.1103/PhysRevE.88.052118. Epub 2013 Nov 11.
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.
我们讨论具有独立高斯元素的(M)个矩形随机矩阵的乘积，其有多种应用，包括无线通信和经济物理学。对于复矩阵，利用哈里什 - 钱德拉 - 伊茨基松 - 祖伯积分公式得到联合概率密度函数的显式表达式。通过双矩阵模型和双正交多项式方法，得到了有限矩阵大小下所有相关函数和矩的显式表达式。这推广了(M = 1)时所谓威沙特 - 拉盖尔高斯酉系综（或手征酉系综）的经典结果以及方阵乘积的先前结果。相关函数由行列式点过程给出，其中核可以用梅杰尔(G)函数表示。我们将结果与数值模拟以及大矩阵极限下宏观水平密度的已知结果进行比较。针对一般的(M)，详细分析了后者支撑端点的位置。最后，我们考虑所谓的遍历互信息，它给出了具有多重散射的多输入多输出通信信道频谱效率的上界。
