Abstract:
We derive exact analytic expressions for the distributions of eigenvalues and singular values for the product of an arbitrary number of independent rectangular Gaussian random matrices in the limit of large matrix dimensions. We show that they both have power-law behavior at zero and determine the corresponding powers. We also propose a heuristic form of finite size corrections to these expressions which very well approximates the distributions for matrices of finite dimensions.

Abstract:
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 densities near zero. Moreover, we propose a finite-size form of the spectral density of the product close to the border of its eigenvalues' domain. Also, led by the striking similarity between the two main equations, we put forward a conjecture about a simple relationship between the eigenvalues and singular values of any non-Hermitian random matrix whose spectrum exhibits rotational symmetry around zero.

Abstract:
Let $\a$ be a real-valued random variable of mean zero and variance 1. Let $M_n(\a)$ denote the $n \times n$ random matrix whose entries are iid copies of $\a$ and $\sigma_n(M_n(\a))$ denote the least singular value of $M_n(\a)$. ($\sigma_n(M_n(\a))^2$ is also usually interpreted as the least eigenvalue of the Wishart matrix $M_n M_n^{\ast}$.) We show that (under a finite moment assumption) the probability distribution $n \sigma_n(M_n(\a))^2$ is {\it universal} in the sense that it does not depend on the distribution of $\a$. In particular, it converges to the same limiting distribution as in the special case when $a$ is real gaussian. (The limiting distribution was computed explicitly in this case by Edelman.) We also proved a similar result for complex-valued random variables of mean zero, with real and imaginary parts having variance 1/2 and covariance zero. Similar results are also obtained for the joint distribution of the bottom $k$ singular values of $M_n(\a)$ for any fixed $k$ (or even for $k$ growing as a small power of $n$) and for rectangular matrices. Our approach is motivated by the general idea of "property testing" from combinatorics and theoretical computer science. This seems to be a new approach in the study of spectra of random matrices and combines tools from various areas of mathematics.

Abstract:
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 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 endpoints of support for the latter are analysed 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 multi-fold scattering.

Abstract:
We consider the problem of distributed joint source-channel coding of correlated Gaussian sources over a Gaussian Multiple Access Channel (GMAC). There may be side information at the decoder and/or at the encoders. First we specialize a general result (for transmission of correlated sources over a MAC with side information) to obtain sufficient conditions for reliable transmission over a Gaussian MAC. This system does not satisfy the source-channel separation. We study and compare three joint source-channel coding schemes available in literature. We show that each of these schemes is optimal under different scenarios. One of the schemes, Amplify and Forward (AF) which simplifies the design of encoders and the decoder, is optimal at low SNR but not at high SNR. Another scheme is asymptotically optimal at high SNR. The third coding scheme is optimal for orthogonal Gaussian channels. We also show that AF is close to the optimal scheme for orthogonal channels even at high SNR.

Abstract:
The Restricted Isometry Constants (RIC) of a matrix $A$ measures how close to an isometry is the action of $A$ on vectors with few nonzero entries, measured in the $\ell^2$ norm. Specifically, the upper and lower RIC of a matrix $A$ of size $n\times N$ is the maximum and the minimum deviation from unity (one) of the largest and smallest, respectively, square of singular values of all ${N\choose k}$ matrices formed by taking $k$ columns from $A$. Calculation of the RIC is intractable for most matrices due to its combinatorial nature; however, many random matrices typically have bounded RIC in some range of problem sizes $(k,n,N)$. We provide the best known bound on the RIC for Gaussian matrices, which is also the smallest known bound on the RIC for any large rectangular matrix. Improvements over prior bounds are achieved by exploiting similarity of singular values for matrices which share a substantial number of columns.

Abstract:
It has been shown by Akemann, Ipsen and Kieburg that the squared singular values of products of $M$ rectangular random matrices with independent complex Gaussian entries are distributed according to a determinantal point process with a correlation kernel that admits a representation in terms of Meijer G-functions. We prove the universality of the local statistics of the squared singular values, namely, the bulk universality given by the sine kernel and the edge universality given by the Airy kernel. The proof is based on the asymptotic analysis for the double contour integral representation of the correlation kernel. Our strategy can be generalized to deal with other models of products of random matrices introduced recently and to establish similar universal results. Two more examples are investigated, one is the product of $M$ Ginibre matrices and the inverse of $K$ Ginibre matrices studied by Forrester, and the other one is the product of $M-1$ Ginibre matrices with one truncated unitary matrix considered by Kuijlaars and Stivigny.

Abstract:
In this paper, we consider the singular values and singular vectors of finite, low rank perturbations of large rectangular random matrices. Specifically, we prove almost sure convergence of the extreme singular values and appropriate projections of the corresponding singular vectors of the perturbed matrix. As in the prequel, where we considered the eigenvalue aspect of the problem, the non-random limiting value is shown to depend explicitly on the limiting singular value distribution of the unperturbed matrix via an integral transforms that linearizes rectangular additive convolution in free probability theory. The large matrix limit of the extreme singular values of the perturbed matrix differs from that of the original matrix if and only if the singular values of the perturbing matrix are above a certain critical threshold which depends on this same aforementioned integral transform. We examine the consequence of this singular value phase transition on the associated left and right singular eigenvectors and discuss the finite $n$ fluctuations above these non-random limits.

Abstract:
The product of M complex random Gaussian matrices of size N has recently been studied by Akemann, Kieburg and Wei. They showed that, for fixed M and N, the joint probability distribution for the squared singular values of the product matrix forms a determinantal point process with a correlation kernel determined by certain biorthogonal polynomials that can be explicitly constructed. We find that, in the case M=2, the relevant biorthogonal polynomials are actually special cases of multiple orthogonal polynomials associated with Macdonald functions (modified Bessel functions of the second kind) which was first introduced by Van Assche and Yakubovich. With known results on asymptotic zero distribution of these polynomials and general theory on multiple orthogonal polynomial ensembles, it is then easy to obtain an explicit expression for the distribution of squared singular values for the product of two complex random Gaussian matrices in the limit of large matrix dimensions.

Abstract:
We consider products of independent large random rectangular matrices with independent entries. The limit distribution of the expected empirical distribution of singular values of such products is computed. The distribution function is described by its Stieltjes transform, which satisfies some algebraic equation. In the particular case of square matrices we get a well-known distribution which moments are Fuss-Catalan numbers.