Related papers: Lyapunov exponents for products of rectangular rea…
We compute the average characteristic polynomial of the hermitised product of $M$ real or complex Wigner matrices of size $N\times N$ and the average of the characteristic polynomial of a product of $M$ such Wigner matrices times the…
The real Ginibre ensemble consists of random $N \times N$ matrices formed from i.i.d. standard Gaussian entries. By using the method of skew orthogonal polynomials, the general $n$-point correlations for the real eigenvalues, and for the…
We investigate real eigenvalues of real elliptic Ginibre matrices of size $n$, indexed by the parameter of asymmetry $\tau \in [0,1]$. In both the strongly and weakly non-Hermitian regimes, where $\tau \in [0,1)$ is fixed or…
Random matrices formed from i.i.d. standard real Gaussian entries have the feature that the expected number of real eigenvalues is non-zero. This property persists for products of such matrices, independently chosen, and moreover it is…
We consider the product of n complex non-Hermitian, independent random matrices, each of size NxN with independent identically distributed Gaussian entries (Ginibre matrices). The joint probability distribution of the complex eigenvalues of…
Products of $M$ i.i.d. non-Hermitian random matrices of size $N \times N$ relate Gaussian fluctuation of Lyapunov and stability exponents in dynamical systems (finite $N$ and large $M$) to local eigenvalue universality in random matrix…
We consider eigenvalues of a product of n non-Hermitian, independent random matrices. Each matrix in this product is of size N\times N with independent standard complex Gaussian variables. The eigenvalues of such a product form a…
For products $P_N$ of $N$ random matrices of size $d \times d$, there is a natural notion of finite $N$ Lyapunov exponents $\{\mu_i\}_{i=1}^d$. In the case of standard Gaussian random matrices with real, complex or real quaternion elements,…
We show that the limiting eigenvalue density of the product of n identically distributed random matrices from an isotropic unitary ensemble (IUE) is equal to the eigenvalue density of n-th power of a single matrix from this ensemble, in the…
Akemann, Ipsen, and Kieburg showed recently that the squared singular values of a product of M complex Ginibre matrices are distributed according to a determinantal point process. We introduce the notion of a polynomial ensemble and show…
Consider the $n\times n$ matrix $X_n=A_n+H_n$, where $A_n$ is a $n\times n$ matrix (either deterministic or random) and $H_n$ is a $n\times n$ matrix independent from $A_n$ drawn from complex Ginibre ensemble. We study the limiting…
We consider the product of \(k_{n}\) independent \(n\times n\) complex Ginibre matrices and denote its eigenvalues by \(Z_{1},\ldots ,Z_{n}\). Let \(\alpha = \lim_{n\to\infty} n / k_{n}\). Using the determinantal point process method, we…
Polynomial ensembles are a sub-class of probability measures within determinantal point processes. Examples include products of independent random matrices, with applications to Lyapunov exponents, and random matrices with an external…
We establish large deviation type estimates for i.i.d. products of two dimensional random matrices with finitely supported probability distribution. The estimates are stable under perturbations and require no irreducibility assumptions. In…
We study the densities of limiting distributions of squared singular values of high-dimensional matrix products composed of independent complex Gaussian (complex Ginibre) and truncated unitary matrices which are taken from Haar distributed…
We study the properties of the eigenvalues of real random matrices and their products. It is known that when the matrix elements are Gaussian-distributed independent random variables, the fraction of real eigenvalues tends to unity as the…
We construct explicit invariant measures for a family of infinite products of random, independent, identically-distributed elements of SL(2,C). The matrices in the product are such that one entry is gamma-distributed along a ray in the…
We derive a precise asymptotic formula for the density of the small singular values of the real Ginibre matrix ensemble shifted by a complex parameter $z$ as the dimension tends to infinity. For $z$ away from the real axis the formula…
This paper is concerned with the explicit computation of the limiting distribution function of the largest real eigenvalue in the real Ginibre ensemble when each real eigenvalue has been removed independently with constant likelihood. We…
We study the singular values (and Lyapunov exponents) for products of $N$ independent $n\times n$ random matrices with i.i.d. entries. Such matrix products have been extensively analyzed using free probability, which applies when $n\to…