Eigenvalues and Singular Values of Products of Rectangular Gaussian Random Matrices

Weak Commutation Relations and Eigenvalue Statistics for Products of Rectangular Random Matrices

We study the joint probability density of the eigenvalues of a product of rectangular real, complex or quaternion random matrices in a unified way. The random matrices are distributed according to arbitrary probability densities, whose only restriction is the invariance under left and right multiplication by orthogonal, unitary or unitary symplectic matrices, respectively. We show that a product of rectangular matrices is statistically equivalent to a product of square matrices. Hereby we prove a weak commutation relation of the random matrices at finite matrix sizes, which previously have bee…

Eigenvalues and Singular Values of Products of Rectangular Gaussian Random Matrices (The Extended Version)

We consider a product of an arbitrary number of independent rectangular Gaussian random matrices. We derive the mean densities of its eigenvalues and singular values in the thermodynamic limit, eventually verified numerically. These densities are encoded in the form of the so called M-transforms, for which polynomial equations are found. We exploit the methods of planar diagrammatics, enhanced to the non-Hermitian case, and free random variables, respectively; both are described in the appendices. As particular results of these two main equations, we find the singular behavior of the spectral …

Products of Rectangular Random Matrices: Singular Values and Progressive Scattering

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 bi-orthogonal polynomials. This generalises the classical result for the so-called Wishart--Laguerre Gaussian unita…

二维活性布朗粒子穿越抛物线障碍：有限矩形域中具有吸收边界条件

Two-Dimensional Active Brownian Particles Crossing a Parabolic Barrier: Finite Rectangular Domain with Absorbing Boundary Conditions

研究二维活性布朗粒子在有限矩形域中穿越抛物线障碍的动力学行为，通过扰动理论求解福克-普肯杰方程，分析活动性对生存概率和首次通过时间分布的影响。

矩形随机矩阵之和的奇异向量与高秩信号的最优估计器：扩展尖峰模型

Singular Vectors of Sums of Rectangular Random Matrices and Optimal Estimators of High-Rank Signals: The Extensive Spike Model

本文针对有限秩尖峰模型无法准确描述中等秩信号的问题，提出扩展尖峰模型并解析刻画高秩信号奇异结构在加性噪声下的变形规律，进而推导出最优旋转不变去噪器和协方差估计器。