Spectral density of generalized Wishart matrices and free multiplicative convolution.

@article{Mlotkowski2015SpectralDO,
  title={Spectral density of generalized Wishart matrices and free multiplicative convolution.},
  author={Wojciech Mlotkowski and Maciej A. Nowak and Karol A. Penson and Karol Życzkowski},
  journal={Physical review. E, Statistical, nonlinear, and soft matter physics},
  year={2015},
  volume={92 1},
  pages={
          012121
        }
}
We investigate the level density for several ensembles of positive random matrices of a Wishart-like structure, W=XX(†), where X stands for a non-Hermitian random matrix. In particular, making use of the Cauchy transform, we study the free multiplicative powers of the Marchenko-Pastur (MP) distribution, MP(⊠s), which for an integer s yield Fuss-Catalan distributions corresponding to a product of s-independent square random matrices, X=X(1)⋯X(s). New formulas for the level densities are derived… 

Figures from this paper

Products of random matrices from polynomial ensembles

  • M. KieburgH. Kosters
  • Mathematics, Computer Science
    Annales de l'Institut Henri Poincaré, Probabilités et Statistiques
  • 2019
TLDR
A transformation formula is derived for the joint densities of a product of two independent bi-unitarily invariant random matrices, the first from a polynomial ensemble and the second from aPolynomialsemble of derivative type.

Equilibrium problems for Raney densities

The Raney numbers are a class of combinatorial numbers generalising the Fuss–Catalan numbers. They are indexed by a pair of positive real numbers (p, r) with p > 1 and 0 < r ⩽ p, and form the moments

Limiting spectral distributions of sums of products of non-Hermitian random matrices

For fixed l≥0 and m≥1, let Xn0, Xn1,..., Xnl be independent random n × n matrices with independent entries, let Fn0 := Xn0, Xn1-1,..., Xnl-1, and let Fn1,..., Fnm be independent random matrices of

Asymptotic distributions of Wishart type products of random matrices

We study asymptotic distributions of large dimensional random matrices of the form $BB^{*}$, where $B$ is a product of $p$ rectangular random matrices, using free probability and combinatorics of

Relating the Bures Measure to the Cauchy Two-Matrix Model

The Bures metric is a natural choice in measuring the distance of density operators representing states in quantum mechanics. In the past few years a random matrix ensemble and the corresponding

Jacobi polynomial moments and products of random matrices

TLDR
An appropriately general class of measures is introduced and characterize them by their moments essentially given by specific Jacobi polynomials with varying parameters, which requires a study of the Riemann surfaces associated to a class of algebraic equations.

Generating random quantum channels

TLDR
Three approaches to the problem of sampling of quantum channels are presented and it is shown under which conditions they become mathematically equivalent, and lead to the uniform, Lebesgue measure on the convex set of quantum operations.

Spectral density of dense random networks and the breakdown of the Wigner semicircle law

Although the spectra of random networks have been studied for a long time, the influence of network topology on the dense limit of network spectra remains poorly understood. By considering the

Complete diagrammatics of the single-ring theorem.

Using diagrammatic techniques, we provide explicit functional relations between the cumulant generating functions for the biunitarily invariant ensembles in the limit of large size of matrices. The

Distinguishing Random and Black Hole Microstates

TLDR
The results are interpreted in the language of quantum hypothesis testing and the subsystem eigenstate thermalization hypothesis (ETH), deriving that chaotic systems obey subsystem ETH for all subsystems less than half the total system size.

References

SHOWING 1-10 OF 51 REFERENCES

Spectral density of products of Wishart dilute random matrices. Part I: the dense case

TLDR
This work derives that the spectral density is a solution of a polynomial equation of degree $M+1$ and obtains exact expressions of it for $M=1, $2$ and $3$ and makes some observations for general $M$, based admittedly on some weak numerical evidence.

Products of rectangular random matrices: singular values and progressive scattering.

TLDR
The so-called ergodic mutual information is considered, which gives an upper bound for the spectral efficiency of a MIMO communication channel with multifold scattering.

Product of Ginibre matrices: Fuss-Catalan and Raney distributions.

TLDR
Using similar techniques, involving the Mellin transform and the Meijer G function, it is found exact expressions for the Raney probability distributions, the moments of which are given by a two-parameter generalization of the Fuss-Catalan numbers.

Raney Distributions and Random Matrix Theory

Recent works have shown that the family of probability distributions with moments given by the Fuss–Catalan numbers permit a simple parameterized form for their density. We extend this result to the

Generalized Bures products from free probability

Inspired by the theory of quantum information, I use two non-Hermitian random matrix models - a weighted sum of circular unitary ensembles and a product of rectangular Ginibre unitary ensembles - as

Singular Values of Products of Ginibre Random Matrices

The squared singular values of the product of M complex Ginibre matrices form a biorthogonal ensemble, and thus their distribution is fully determined by a correlation kernel. The kernel permits a

Singular Values of Products of Ginibre Random Matrices, Multiple Orthogonal Polynomials and Hard Edge Scaling Limits

Akemann, Ipsen and Kieburg recently showed that the squared singular values of products of M rectangular random matrices with independent complex Gaussian entries are distributed according to a

Generating random density matrices

We study various methods to generate ensembles of random density matrices of a fixed size N, obtained by partial trace of pure states on composite systems. Structured ensembles of random pure states,

Multiplication law and S transform for non-Hermitian random matrices.

We derive a multiplication law for free non-Hermitian random matrices allowing for an easy reconstruction of the two-dimensional eigenvalue distribution of the product ensemble from the

“Single ring theorem” and the disk-annulus phase transition

Recently, an analytic method was developed to study in the large N limit non-Hermitian random matrices that are drawn from a large class of circularly symmetric non-Gaussian probability
...