Related papers: Characteristic polynomials in real Ginibre ensembl…
Motivated by the connection between the eigenvalues of the complex Ginibre matrix model and the magnetic Laplacian in the complex plane, we derive analogues of the complex Hermite polynomials for the elliptic Ginibre model. To this end, we…
We define a family of symmetric and a family of non-symmetric polynomials in terms of vanishing conditions. These families depend on two paramters, q and t. Their main feature is that they consist of non-homogeneous polynomials. The…
We present an iterative technique to obtain skew-orthogonal polynomials with quartic weight, arising in the study of symplectic ensembles of random matrices.
Let $\sigma_n(\cdot)$ denote the least singular value of a $n \times n$ matrix. It is well-known that $\mathbb{P}[\sigma_n(A) \le \varepsilon] \le \varepsilon n$ if $A$ is drawn from the real Ginibre ensemble of $n \times n$ matrices and…
Form an $n \times n$ matrix by drawing entries independently from $\{\pm1\}$ (or another fixed nontrivial finitely supported distribution in $\mathbf{Z}$) and let $\phi$ be the characteristic polynomial. Conditionally on the extended…
We compute the joint eigenvalue distribution for the rank one Hermitian and non-Hermitian perturbations of chiral Gaussian $\beta$-ensembles ($\beta>0$) of random matrices.
We study, count and locate the exceptional points where eigenvalues collide for certain families of matrices $$R(s,t) = \cos(s \pi / 2)C + \sin(s \pi / 2)U(t), \quad s,t \in [0,1]$$ where $C$ is a realization of a Ginibre random matrix, or…
In this paper, we compute the probability that an $N \times N$ matrix from the generalised Gaussian Unitary Ensemble (gGUE) is positive definite, extending a previous result of Dean and Majumdar \cite{DM}. For this purpose, we work out the…
It has been shown by Akemann, Ipsen and Kieburg that the squared singular values of products of $M$ rectangular random matrices with independent complex Gaussian entries are distributed according to a determinantal point process with a…
We consider the elliptic Ginibre ensembles in the real, complex and symplectic symmetry classes. As the matrix size tends to infinity, we derive the asymptotic behaviour of the upper tail large deviation probabilities for both the spectral…
We study the characteristic polynomials of both the Gaussian Orthogonal and Symplectic Ensembles. We show that for both ensembles, powers of the absolute value of the characteristic polynomials converge in law to Gaussian multiplicative…
Random matrix theory has proven very successful in the understanding of the spectra of chaotic systems. Depending on symmetry with respect to time reversal and the presence or absence of a spin 1/2 there are three ensembles, the Gaussian…
We consider four nontrivial ensembles involving Gaussian Wigner and Wishart matrices. These are relevant to problems ranging from multiantenna communication to random supergravity. We derive the matrix probability density, as well as the…
Skew orthogonal polynomials arise in the calculation of the $n$-point distribution function for the eigenvalues of ensembles of random matrices with orthogonal or symplectic symmetry. In particular, the distribution functions are completely…
Graph polynomials are deemed useful if they give rise to algebraic characterizations of various graph properties, and their evaluations encode many other graph invariants. Algebraic: The complete graphs $K_n$ and the complete bipartite…
One object of interest in random matrix theory is a family of point ensembles (random point configurations) related to various systems of classical orthogonal polynomials. The paper deals with a one--parametric deformation of these…
We develop a simple algorithm to generate random variables described by densities equaling squared Hermite functions. As an application, we show how to generate a randomly chosen eigenvalue of a matrix from the Gaussian Unitary Ensemble…
We consider asymptotic behavior of the correlation functions of the characteristic polynomials of the hermitian sample covariance matrices $H_n=n^{-1}A_{m,n}^*A_{m,n}$, where $A_{m,n}$ is a $m\times n$ complex matrix with independent and…
Kernel mean embedding is a useful tool to represent and compare probability measures. Despite its usefulness, kernel mean embedding considers infinite-dimensional features, which are challenging to handle in the context of differentially…
We demonstrate the convergence of the characteristic polynomial of several random matrix ensembles to a limiting universal function, at the microscopic scale. The random matrix ensembles we treat are classical compact groups and the…