Related papers: Mean left-right eigenvector self-overlap in the re…

The real Ginibre spherical ensemble consists of random matrices of the form $A B^{-1}$, where $A,B$ are independent standard real Gaussian $N \times N$ matrices. The expected number of real eigenvalues is known to be of order $\sqrt{N}$. We…

We study statistical properties of the eigenvectors of non-Hermitian random matrices, concentrating on Ginibre's complex Gaussian ensemble, in which the real and imaginary parts of each element of an N x N matrix, J, are independent random…

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 pairs of GOE (Gaussian Orthogonal Ensemble) matrices which are correlated with each others, and subject to additive and multiplicative rank-one perturbations. We focus on the regime of parameters in which the finite-rank…

Let $\sqrt{N}+\lambda_{max}$ be the largest real eigenvalue of a random $N\times N$ matrix with independent $N(0,1)$ entries (the `real Ginibre matrix'). We study the large deviations behaviour of the limiting $N\rightarrow \infty$…

In this paper we study the limiting distribution of the $k$ smallest gaps between eigenvalues of three kinds of random matrices -- the Ginibre ensemble, the Wishart ensemble and the universal unitary ensemble. All of them follow a…

We study the Ginibre ensemble of $N \times N$ complex random matrices and compute exactly, for any finite $N$, the full distribution as well as all the cumulants of the number $N_r$ of eigenvalues within a disk of radius $r$ centered at the…

We study the angles between the eigenvectors of a random $n\times n$ complex matrix $M$ with density $\propto \mathrm{e}^{-n\operatorname{Tr}V(M^*M)}$ and $x\mapsto V(x^2)$ convex. We prove that for unit eigenvectors…

We consider the singular vectors of any $m \times n$ submatrix of a rectangular $M \times N$ Gaussian matrix and study their asymptotic overlaps with those of the full matrix, in the macroscopic regime where $N \,/\, M\,$, $m \,/\, M$ as…

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…

The real Ginibre ensemble refers to the family of $n\times n$ matrices in which each entry is an independent Gaussian random variable of mean zero and variance one. Our main result is that the appropriately scaled spectral radius converges…

Random matrices from the elliptic Ginibre orthogonal ensemble (GinOE) are a certain linear combination of a real symmetric, and real anti-symmetric, real Gaussian random matrices and controlled by a parameter $\tau$. Our interest is in the…

Statistical properties of eigenvectors in non-Hermitian random matrix ensembles are discussed, with an emphasis on correlations between left and right eigenvectors. Two approaches are described. One is an exact calculation for Ginibre's…

Non-Hermitian random matrices provide a useful framework for understanding universal characteristics of dissipative quantum chaotic systems with loss or gain. We consider a model of two such system represented by two independent $N\times N$…

We consider the standard overlap $\mathcal{O}_{ij}: =\langle \mathbf{r}_j, \mathbf{r}_i\rangle\langle \mathbf{l}_j, \mathbf{l}_i\rangle$ of any bi-orthogonal family of left and right eigenvectors of a large random matrix $X$ with centred…

We study the images of the complex Ginibre eigenvalues under the power maps $\pi_M: z \mapsto z^M$, for any integer $M$. We establish the following equality in distribution, $$ {\rm{Gin}}(N)^M \stackrel{d}{=} \bigcup_{k=1}^M {\rm{Gin}}…

The real Ginibre ensemble consists of $n\times n$ real matrices ${\bf X}$ whose entries are i.i.d. standard normal random variables. In sharp contrast to the complex and quaternion Ginibre ensemble, real eigenvalues in the real Ginibre…

We consider the eigenvectors of the principal minor of dimension $n< N$ of the Dyson Brownian motion in $\mathbb{R}^{N}$ and investigate their asymptotic overlaps with the eigenvectors of the full matrix in the limit of large dimension. We…

This is part II of a review relating to the three classes of random non-Hermitian Gaussian matrices introduced by Ginibre in 1965. While part I restricted attention to the GinUE (Ginibre unitary ensemble) case of complex elements, in this…

Let $G_n$ be an $n \times n$ matrix with real i.i.d. $N(0,1/n)$ entries, let $A$ be a real $n \times n$ matrix with $\Vert A \Vert \le 1$, and let $\gamma \in (0,1)$. We show that with probability $0.99$, $A + \gamma G_n$ has all of its…