Related papers: Eigenvectors of non normal random matrices

We develop a method to calculate left-right eigenvector correlations of the product of $m$ independent $N\times N$ complex Ginibre matrices. For illustration, we present explicit analytical results for the vector overlap for a couple of…

We consider $n^2\times n^2$ real symmetric and hermitian matrices $M_n$, which are equal to sum of $m_n$ tensor products of vectors $X^\mu=B(Y^\mu\otimes Y^\mu)$, $\mu=1,\dots,m_n$, where $Y^\mu$ are i.i.d. random vectors from $\mathbb R^n…

The eigenvalue PDF for some well known classes of non-Hermitian random matrices --- the complex Ginibre ensemble for example --- can be interpreted as the Boltzmann factor for one-component plasma systems in two-dimensional domains. We…

Suppose that $\Gamma=(V,E)$ is a graph with vertices $V$, edges $E$, a free group action on the vertices $\mathbb{Z}^d \curvearrowright V$ with finitely many orbits, and a linear operator $D$ on the Hilbert space $l^2(V)$ such that $D$…

Pencils of Hankel matrices whose elements have a joint Gaussian distribution with nonzero mean and not identical covariance are considered. An approximation to the distribution of the squared modulus of their determinant is computed which…

Complex eigenvalues of random matrices $J=\text{GUE }+ i\gamma \diag (1, 0, \ldots, 0)$ provide the simplest model for studying resonances in wave scattering from a quantum chaotic system via a single open channel. It is known that in the…

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 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…

Let $X_{m} = G_{1}\ldots G_{m}$ denote the product of $m$ independent random matrices of size $N \times N$, with each matrix in the product consisting of independent standard Gaussian variables. Denoting by $N_{\mathbb{R}}(m)$ the total…

We investigate the statistical properties of eigenvalues of pseudo-Hermitian random matrices whose eigenvalues are real or complex conjugate. It is shown that when the spectrum splits into separated sets of real and complex conjugate…

We consider quadratic forms of deterministic matrices $A$ evaluated at the random eigenvectors of a large $N \times N$ GOE or GUE matrix, or equivalently evaluated at the columns of a Haar-orthogonal or Haar-unitary random matrix. We prove…

Observables in random tensor theory are polynomials in the entries of a tensor of rank $d$ which are invariant under $U(N)^d$. It is notoriously difficult to evaluate the expectations of such polynomials, even in the Gaussian distribution.…

In this paper, we prove a universality result of convergence for a bivariate random process defined by the eigenvectors of a sample covariance matrix. Let $V_n=(v_{ij})_{i \leq n,\, j\leq m}$ be a $n\times m$ random matrix, where $(n/m)\to…

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 consider the eigenvectors of symmetric matrices with independent heavy tailed entries, such as matrices with entries in the domain of attraction of $\alpha$-stable laws, or adjacencymatrices of Erdos-Renyi graphs. We denote by…

We study the phenomenon of "crowding" near the largest eigenvalue $\lambda_{\max}$ of random $N \times N$ matrices belonging to the Gaussian Unitary Ensemble (GUE) of random matrix theory. We focus on two distinct quantities: (i) the…

Let $M_n$ be an $n\times n$ real (resp. complex) Wigner matrix and $U_n\Lambda_n U_n^*$ be its spectral decomposition. Set $(y_1,y_2...,y_n)^T=U_n^*x$, where $x=(x_1,x_2,...,$ $x_n)^T$ is a real (resp. complex) unit vector. Under the…

Let v_1,...,v_{n-1} be n-1 independent vectors in R^n (or C^n). We study x, the unit normal vector of the hyperplane spanned by the v_i. Our main finding is that x resembles a random vector chosen uniformly from the unit sphere, under some…

We discuss regularization by noise of the spectrum of large random non-Normal matrices. Under suitable conditions, we show that the regularization of a sequence of matrices that converges in *-moments to a regular element $a$, by the…

Hu and Ye conjectured that for a $k$-th order and $n$-dimensional tensor $\mathcal{A}$ with an eigenvalue $\lambda$ and the corresponding eigenvariety $\mathcal{V}_\lambda(\mathcal{A})$, $$\mathrm{am}(\lambda) \ge \sum_{i=1}^\kappa…