On a surprising relation between the Marchenko-Pastur law, rectangular and square free convolutions

On a Surprising Relation between Rectangular and Square Free Convolutions

Debbah and Ryan have recently [DR07] proved a result about the limit empirical singular distribution of the sum of two rectangular random matrices whose dimensions tend to infinity. In this paper, we reformulate it in terms of the rectangular free convolution introduced in [BG07b] and then we give a new, shorter, proof of this result under weaker hypothesis: we do not suppose the probability me...

A total positivity property of the Marchenko-Pastur Law

On the Rate of Convergence to the Marchenko–Pastur Distribution

LetX = (Xjk) denote n×p random matrix with entriesXjk, which are independent for 1 ≤ j ≤ n, 1 ≤ k ≤ p. We consider the rate of convergence of empirical spectral distribution function of matrix W = 1 p XX ∗ to the Marchenko–Pastur law. We assume that EXjk = 0, EX 2 jk = 1 and that the distributions of the matrix elements Xjk have a uniformly sub exponential decay in the sense that there exists a...

Lyapunov Exponents of Free Operators

Lyapunov exponents of a dynamical system are a useful tool to gauge the stability and complexity of the system. This paper offers a definition of Lyapunov exponents for a sequence of free linear operators. The definition is based on the concept of the extended Fuglede-Kadison determinant. We establish the existence of Lyapunov exponents, derive formulas for their calculation, and show that Lyap...

Multivariate Fuss-Narayana Polynomials and their Application to Random Matrices

It has been shown recently that the limit moments of W (n) = B(n)B∗(n), where B(n) is a product of p independent rectangular random matrices, are certain homogeneous polynomials Pk(d0, d1, . . . , dp) in the asymptotic dimensions of these matrices. Using the combinatorics of noncrossing partitions, we explicitly determine these polynomials and show that they are closely related to polynomials w...