Related papers: Characteristic Polynomials of Complex Random Matri…
We consider the asymptotics of the second-order correlation function of the characteristic polynomial of a random matrix. We show that the known result for a random matrix from the Gaussian Unitary Ensemble essentially continues to hold for…
We show that in the large matrix limit, the eigenvalues of the normal matrix model for matrices with spectrum inside a compact domain with a special class of potentials homogeneously fill the interior of a polynomial curve uniquely defined…
The complexity measures of the Cr\'amer-Rao, Fisher-Shannon and LMC (L\'opez-Ruiz, Mancini and Calvet) types of the Rakhmanov probability density $\rho_n(x)=\omega(x) p_n^2(x)$ of the polynomials $p_n(x)$ orthogonal with respect to the…
We consider asymptotics of ratios of random characteristic polynomials associated with orthogonal polynomial ensembles. Under some natural conditions on the measure in the definition of the orthogonal polynomial ensemble we establish a…
This is a review of the Riemann-Hilbert approach to the large $N$ asymptotics in random matrix models and its applications. We discuss the following topics: random matrix models and orthogonal polynomials, the Riemann-Hilbert approach to…
We prove that the expectation of the logarithm of the condition number of each of the zeros of the characteristic polynomial of a complex standard Gaussian matrix is ${\Omega}(n)$. This may provide an explanation for the common wisdom in…
Ensembles of complex symmetric, and complex self dual random matrices are known to exhibit local statistical properties distinct from those of the non-Hermitian Ginibre ensembles. On the other hand, in distinction to the latter, the joint…
We characterize the biorthogonal polynomials that appear in the theory of coupled random matrices via a Riemann-Hilbert problem. Our Riemann-Hilbert problem is different from the ones that were proposed recently by Ercolani and McLaughlin,…
We investigate the second-order correlation function of the characteristic polynomial of a sample covariance matrix. Starting from an explicit formula for the generating function, we re-obtain several well-known kernels from random matrix…
The product of M complex random Gaussian matrices of size N has recently been studied by Akemann, Kieburg and Wei. They showed that, for fixed M and N, the joint probability distribution for the squared singular values of the product matrix…
The paper is concerned with the asymptotic behavior of the correlation functions of the characteristic polynomials of non-Hermitian random matrices with independent entries. It is shown that the correlation functions behave like that for…
Starting from the expression for the superdeterminant of $ (xI-M)$, where $M$ is an arbitrary supermatrix , we propose a definition for the corresponding characteristic polynomial and we prove that each supermatrix satisfies its…
Given a collection $\{\lambda_1, \dots, \lambda_n\} $ of real numbers, there is a canonical probability distribution on the set of real symmetric or complex Hermitian matrices with eigenvalues $\lambda_1,\ldots,\lambda_n$. In this paper, we…
In this paper, we consider the problem of representing a multivariate polynomial as the determinant of a definite (monic) symmetric/Hermitian linear matrix polynomial (LMP). Such a polynomial is known as determinantal polynomial.…
We obtain a novel formula for characteristic polynomials of deformations of the Braid arrangement using the notion of levels of regions. As an application, we recover and strengthen results of Chen et al. on the characteristic polynomial of…
There is a nice combinatorial formula of P. Beelen and M. Datta for the $r$-th generalized Hamming weight of an affine cartesian code. Using this combinatorial formula we give an easy to evaluate formula to compute the $r$-th generalized…
On the one hand, we prove that almost surely, for large dimension, there is no eigenvalue of a Hermitian polynomial in independent Wigner and deterministic matrices, in any interval lying at some distance from the supports of a sequence of…
It is known that computing the permanent of the matrix $1+A$, where $A$ is a finite-rank matrix, requires a number of operations polynomial in the matrix size. Motivated by the boson-sampling proposal of restricted quantum computation, I…
We develop a coordinate-free probabilistic framework for determinantal point processes associated with Bergman kernels on compact complex manifolds. The basic issue is that Bergman kernels are naturally line-bundle-valued:…
Pseudo-hermitian matrices are matrices hermitian with respect to an indefinite metric. They can be thought of as the truncation of pseudo-hermitian operators, defined over some Krein space, together with the associated metric, to a finite…