Related papers: A limiting random analytic function related to the…
Representation theory and the theory of symmetric functions have played a central role in Random Matrix Theory in the computation of quantities such as joint moments of traces and joint moments of characteristic polynomials of matrices…
Consider a square random matrix with independent and identically distributed entries of mean zero and unit variance. We show that as the dimension tends to infinity, the spectral radius is equivalent to the square root of the dimension in…
We have discussed earlier the correlation functions of the random variables $\det(\la-X)$ in which $X$ is a random matrix. In particular the moments of the distribution of these random variables are universal functions, when measured in the…
We investigate the characteristic polynomials of the Gaussian $\beta$-ensemble for general $\beta>0$ through its transfer matrix recurrence. We show that the rescaled characteristic polynomial converges to a random entire function in a…
We evaluate averages involving characteristic polynomials, inverse characteristic polynomials and ratios of characteristic polynomials for a $N\times N$ random matrix taken from a $L$-deformed Chiral Gaussian Unitary Ensemble with an…
We investigate the asymptotic behaviour of the second-order correlation function of the characteristic polynomial of a Hermitian Wigner matrix at the edge of the spectrum. We show that the suitably rescaled second-order correlation function…
We study moments of characteristic polynomials of truncated Haar distributed matrices from the three classical compact groups O(N), U(N) and Sp(2N). For finite matrix size we calculate the moments in terms of hypergeometric functions of…
Let $p_n$ be the characteristic polynomial of an $n \times n$ random matrix drawn from one of the compact classical matrix groups. We show that the critical points of $p_n$ converge to the uniform distribution on the unit circle as $n$…
We study the characteristic polynomial of random permutation matrices following some measures which are invariant by conjugation, including Ewens' measures which are one-parameter deformations of the uniform distribution on the permutation…
In this paper, we investigate the extremal values of (the logarithm of) the characteristic polynomial of a random unitary matrix whose spectrum is distributed according the Circular Beta Ensemble (C$\beta$E). More precisely, if $X_n$ is…
The ensemble $\CUE^{(q)}$ of truncated random unitary matrices is a deformation of the usual Circular Unitary Ensemble depending on a discrete non-negative parameter $q.$ $\CUE^{(q)}$ is an exactly solved model of random contraction…
We consider the sample covariance matrices of large data matrices which have i.i.d. complex matrix entries and which are non-square in the sense that the difference between the number of rows and the number of columns tends to infinity. We…
We investigate determinants of random unitary pencils (with scalar or matrix coefficients), which generalize the characteristic polynomial of a single unitary matrix. In particular we examine moments of such determinants, obtained by…
We calculate the autocorrelation functions (or shifted moments) of the characteristic polynomials of matrices drawn uniformly with respect to Haar measure from the groups U(N), O(2N) and USp(2N). In each case the result can be expressed in…
Analogously to the space of virtual permutations, we define projective limits of isometries: these sequences of unitary operators are natural in the sense that they minimize the rank norm between successive matrices of increasing sizes. The…
Approximating kernel functions with random features (RFs)has been a successful application of random projections for nonparametric estimation. However, performing random projections presents computational challenges for large-scale…
We compute averages of products and ratios of characteristic polynomials associated with Orthogonal, Unitary, and Symplectic Ensembles of Random Matrix Theory. The pfaffian/determinantal formulas for these averages are obtained, and the…
Polynomial ensembles are determinantal point processes associated with (non necessarily orthogonal) projections onto polynomial subspaces. The aim of this survey article is to put forward the use of recurrence coefficients to obtain the…
According to Dyson's three fold way, from the viewpoint of global time reversal symmetry there are three circular ensembles of unitary random matrices relevant to the study of chaotic spectra in quantum mechanics. These are the circular…
The paper is concerned with the correlation functions of the characteristic polynomials of random matrices with independent complex entries. We investigate how the asymptotic behavior of the correlation functions depends on the second…