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.

We consider the product of M quadratic random matrices with complex elements and no further symmetry, where all matrix elements of each factor have a Gaussian distribution. This generalises the classical Wishart-Laguerre Gaussian Unitary Ensemble with M = 1. In this paper we first compute the joint probability distribution for the singular values of the product matrix when the matrix size N and the number M are fixed but arbitrary. This leads to a determinantal point process which can be realised in two different ways. First, it can be written as a one-matrix singular value model with a non-standard Jacobian, or second, for M ≥ 2, as a two-matrix singular value model with a set of auxiliary singular values and a weight proportional to the Meijer G-function. For both formulations we determine all singular value correlation functions in terms of the kernels of biorthogonal polynomials which we explicitly construct. They are given in terms of hypergeometric and Meijer G-functions, generalising the Laguerre polynomials for M = 1. Our investigation was motivated from applications in telecommunication of multi-layered scattering MIMO channels. We present the ergodic mutual information for finite-N for such a channel model with M − 1 layers of scatterers as an example.

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 been discussed for infinite matrix size. Moreover we derive the joint probability densities of the eigenvalues. To illustrate our results we apply them to a product of random matrices drawn from Ginibre ensembles and Jacobi ensembles as well as a mixed version thereof. For these weights we show that the product of complex random matrices yield a determinantal point process, while the real and quaternion matrix ensembles correspond to Pfaffian point processes. Our results are visualized by numerical simulations. Furthermore, we present an application to a transport on a closed, disordered chain coupled to a particle bath.

Recently, the supersymmetry method was extended from Gaussian ensembles to arbitrary unitarily invariant matrix ensembles by generalizing the Hubbard-Stratonovich transformation. Here, we complete this extension by including arbitrary orthogonally and unitary-symplectically invariant matrix ensembles. The results are equivalent to, but the approach is different from the superbosonization formula. We express our results in a unifying way. We also give explicit expressions for all one-point functions and discuss features of the higher order correlations.

We prove that the squared singular values of a fixed matrix multiplied with a truncation of a Haar distributed unitary matrix are distributed by a polynomial ensemble. This result is applied to a multiplication of a truncated unitary matrix with a random matrix. We show that the structure of polynomial ensembles and of certain Pfaffian ensembles is preserved. Furthermore we derive the joint singular value density of a product of truncated unitary matrices and its corresponding correlation kernel which can be written as a double contour integral. This leads to hard edge scaling limits that also include new finite rank perturbations of the Meijer G-kernels found for products of complex Ginibre random matrices.

Abstract. There are several methods to treat ensembles of random matrices in symmetric spaces, circular matrices, chiral matrices and others. Orthogonal polynomials and the supersymmetry method are particular powerful techniques.Here, we present a new approach to calculate averages over ratios of characteristic polynomials. At first sight paradoxically, one can coin our approach "supersymmetry without supersymmetry" because we use structures from supersymmetry without actually mapping onto superspaces. We address two kinds of integrals which cover a wide range of applications for random matrix ensembles. For probability densities factorizing in the eigenvalues we find determinantal structures in a unifying way. As a new application we derive an expression for the k-point correlation function of an arbitrary rotation invariant probability density over the Hermitian matrices in the presence of an external field.

We use classical results from harmonic analysis on matrix spaces to investigate the relation between the joint densities of the singular values and the eigenvalues for complex random matrices which are bi-unitarily invariant (also known as isotropic or unitary rotation invariant). We prove that one of these joint densities determines the other one. Moreover we construct an explicit formula relating both joint densities at finite matrix dimension. This relation covers probability densities as well as signed densities. With the help of this relation we derive general analytical relations among the corresponding kernels and biorthogonal functions for a specific class of polynomial ensembles. Furthermore we show how to generalize the relation between the eigenvalue and singular value statistics to certain situations when the ensemble is deformed by a term which breaks the bi-unitary invariance.

Abstract. Starting from exact analytical results on singular values and complex eigenvalues of products of independent Gaussian complex random N × N matrices also called Ginibre ensemble we rederive the Lyapunov exponents for an infinite product. We show that for a large number t of product matrices the distribution of each Lyapunov exponent is normal and compute its t-dependent variance as well as corrections in a large-t expansion. Originally Lyapunov exponents are defined for the singular values of the product matrix that represents a linear time evolution. Surprisingly a similar construction for the moduli of the complex eigenvalues yields the very same exponents and normal distributions to leading order. We discuss a general mechanism for 2 × 2 matrices why the singular values and the radii of complex eigenvalues collapse onto the same value in the large-t limit. Thereby we rederive Newman's triangular law which has a simple interpretation as the radial density of complex eigenvalues in the circular law and study the commutativity of the two limits t → ∞ and N → ∞ on the global and the local scale. As a mathematical byproduct we show that a particular asymptotic expansion of a Meijer G-function with large index leads to a Gaussian.