Related papers: Extreme values of CUE characteristic polynomials: …
The zeros of the random Laurent series $1/\mu - \sum_{j=1}^\infty c_j/z^j$, where each $c_j$ is an independent standard complex Gaussian, is known to correspond to the scaled eigenvalues of a particular additive rank 1 perturbation of a…
In the present note we obtain new results on two conjectures by Csordas et al. regarding the interlacing property of zeros of special polynomials. These polynomials came from the Jacobi tau methods for the Sturm-Liouville eigenvalue…
For a natural extension of the circular unitary ensemble of order n, we study as n tends to infinity, the asymptotic behavior of the sequence of orthogonal polynomials with respect to the spectral measure. The last term of this sequence is…
We establish the asymptotics of the joint moments of the characteristic polynomial of a random unitary matrix and its derivative for general real values of the exponents, proving a conjecture made by Hughes in 2001. Moreover, we give a…
Supertropical matrix theory was investigated in [6], whose terminology we follow. In this work we investigate eigenvalues, characteristic polynomials and coefficients of characteristic polynomials of supertropical matrices and their powers,…
Correlation functions involving products and ratios of half-integer powers of characteristic polynomials of random matrices from the Gaussian Orthogonal Ensemble (GOE) frequently arise in applications of Random Matrix Theory (RMT) to…
We quantify the large deviations of Gaussian extreme value statistics on closed convex sets in d-dimensional Euclidean space. The asymptotics imply that the extreme value distribution exhibits a rate function that is a simple quadratic…
The circular and Jacobi ensembles of random matrices have their eigenvalue support on the unit circle of the complex plane and the interval $(0,1)$ of the real line respectively. The averaged value of the modulus of the corresponding…
We explore some interesting features of the characteristic polynomial of the Cartan matrix of a simple Lie algebra. The characteristic polynomial is closely related with the Chebyshev polynomials of first and second kind. In addition, we…
This work identifies a solvable (in the sense that spectral correlation functions can be expressed in terms of orthogonal polynomials), rotationally invariant random matrix ensemble with a logarithmic weakly confining potential. The…
Strongly non-Gaussian ensembles of large random matrices possessing unitary symmetry and logarithmic level repulsion are studied both in presence and absence of hard edge in their energy spectra. Employing a theory of polynomials orthogonal…
We study expectations of powers and correlation functions for characteristic polynomials of $N \times N$ non-Hermitian random matrices. For the $1$-point and $2$-point correlation function, we obtain several characterizations in terms of…
The univariate extreme value theory deals with the convergence in type of powers of elements of sequences of cumulative distribution functions on the real line when the power index gets infinite. In terms of convergence of random variables,…
The zeros of complex Gaussian random polynomials, with coefficients such that the density in the underlying complex space is uniform, are known to have the same statistical properties as the zeros of the coherent state representation of…
In this review we summarise recent results for the complex eigenvalues and singular values of finite products of finite size random matrices, their correlation functions and asymptotic limits. The matrices in the product are taken from…
We develop a new method for studying the asymptotics of symmetric polynomials of representation-theoretic origin as the number of variables tends to infinity. Several applications of our method are presented: We prove a number of theorems…
In this paper, we consider sequences of polynomials that satisfy differential--difference recurrences. Our interest is motivated by the fact that polynomials satisfying such recurrences frequently appear as generating polynomials of integer…
We consider a product of an arbitrary number of independent rectangular Gaussian random matrices. We derive the mean densities of its eigenvalues and singular values in the thermodynamic limit, eventually verified numerically. These…
We consider ensembles of random matrices, known as biorthogonal ensembles, whose eigenvalue probability density function can be written as a product of two determinants. These systems are closely related to multiple orthogonal functions. It…
We consider the $n$-correlation of eigenvalues of random unitary matrices in the alternative form that is not the tidy determinant common in random matrix theory, but rather the expression derived from averages of ratios of characteristic…