Related papers: Non-Hermitean Wishart random matrices (I)
These expository notes are centered around the circular law theorem, which states that the empirical spectral distribution of a nxn random matrix with i.i.d. entries of variance 1/n tends to the uniform law on the unit disc of the complex…
We briefly review the random matrix theory for large N by N matrices viewed as free random variables in a context of stochastic diffusion. We establish a surprising link between the spectral properties of matrix-valued multiplicative…
The random matrix ensembles (RMT) of quantum statistical Hamiltonian operators, e.g.Gaussian random matrix ensembles (GRME) and Ginibre random matrix ensembles (Ginibre RME), are applied to following quantum statistical systems: nuclear…
The problem of estimating the smallest singular value of random square matrices is important in connection with matrix computations and analysis of the spectral distribution. In this survey, we consider recent developments in the study of…
We discuss the short-time perturbative expansion of the linear entropy for finite-dimensional quantum systems whose dynamics can be effectively described by a non-Hermitian Hamiltonian. We derive a timescale for the degree of mixedness for…
In random matrix theory, Marchenko-Pastur law states that random matrices with independent and identically distributed entries have a universal asymptotic eigenvalue distribution under large dimension limit, regardless of the choice of…
We show in this note that the asymptotic spectral distribution, location and distribution of the largest eigenvalue of a large class of random density matrices coincide with that of Wishart-type random matrices using proper scaling. As an…
We analyze statistical properties of complex eigenvalues of random matrices $\hat{A}$ close to unitary. Such matrices appear naturally when considering quantized chaotic maps within a general theory of open linear stationary systems with…
We consider a new class of non-Hermitian random matrices, namely the ones which have the form of sums of freely independent terms involving unitary matrices. To deal with them, we exploit the recently developed quaternion technique. After…
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 characteristics of the individual ensembles. We define…
We investigate an additive perturbation of a complex Wishart random matrix and prove that a large deviation principle holds for the spectral measures. The rate function is associated to a vector equilibrium problem coming from logarithmic…
This paper studies the asymptotic spectral properties of the sample covariance matrix for high dimensional compositional data, including the limiting spectral distribution, the limit of extreme eigenvalues, and the central limit theorem for…
I present here some results on the statistical behaviour of large random matrices in an ensemble where the probability distribution is not a function of the eigenvalues only. The perturbative expansion can be cast in a closed form and the…
We analyze statistical properties of the complex system with conditions which manifests through specific constraints on the column/row sum of the matrix elements. The presence of additional constraints besides symmetry leads to new…
Random matrix theory has become a cornerstone in modern statistics and data science, providing fundamental tools for understanding high-dimensional covariance structures. Within this framework, the Wishart matrix plays a central role in…
We consider large non-Hermitian real or complex random matrices $X$ with independent, identically distributed centred entries. We prove that their local eigenvalue statistics near the spectral edge, the unit circle, coincide with those of…
Time-independent scattering methods are widely employed to analyze transport in non-Hermitian systems. Their application, however, rests on a critical yet often overlooked assumption: that an incident wave is a pure superposition of…
We consider real symmetric and complex Hermitian random matrices with the additional symmetry $h_{xy}=h_{N-x,N-y}$. The matrix elements are independent (up to the fourfold symmetry) and not necessarily identically distributed. This ensemble…
One of the major themes of random matrix theory is that many asymptotic properties of traditionally studied distributions of random matrices are universal. We probe the edges of universality by studying the spectral properties of random…
We examine the asymptotics of the moments of characteristic polynomials of $N\times N$ matrices drawn from the Hermitian ensembles of Random Matrix Theory, in the limit as $N\to\infty$. We focus in particular on the Gaussian Unitary…