Related papers: From random matrices to random analytic functions
The integrable structure of Ginibre's Orthogonal Ensemble of random matrices is looked at through the prism of the probability "p_{n,k}" to find exactly "k" real eigenvalues in the spectrum of an "n" by "n" real asymmetric Gaussian random…
This article studies the Gram random matrix model $G=\frac1T\Sigma^{\rm T}\Sigma$, $\Sigma=\sigma(WX)$, classically found in the analysis of random feature maps and random neural networks, where $X=[x_1,\ldots,x_T]\in{\mathbb R}^{p\times…
It is known that determinantal point processes have an intimate relation to operator algebras. In the paper, we extend this relationship to encompass dynamical aspects. Especially, we delve into two types of determinantal point processes.…
We consider the eigenvalues of a large dimensional real or complex Ginibre matrix in the region of the complex plane where their real parts reach their maximum value. This maximum follows the Gumbel distribution and that these extreme…
Gaussian processes offers a convenient way to perform nonparametric reconstructions of observational data assuming only a kernel which describes the covariance between neighbouring points in a data set. We approach the ambiguity in the…
A Gaussian process (GP) is a powerful and widely used regression technique. The main building block of a GP regression is the covariance kernel, which characterizes the relationship between pairs in the random field. The optimization to…
Products and sums of random matrices have seen a rapid development in the past decade due to various analytical techniques available. Two of these are the harmonic analysis approach and the concept of polynomial ensembles. Very recently, it…
The real Ginibre ensemble consists of random $N \times N$ matrices formed from i.i.d. standard Gaussian entries. By using the method of skew orthogonal polynomials, the general $n$-point correlations for the real eigenvalues, and for the…
In the present paper, we consider the integral operator, which acts in Hilbert space and has sine kernel. This operator generates two operator identities and two corresponding canonical differential systems. We find the asymptotics of the…
1. A standard Gaussian random matrix has full rank with probability 1 and is well-conditioned with a probability quite close to 1 and converging to 1 fast as the matrix deviates from square shape and becomes more rectangular. 2. If we…
We discuss how generalized multiresolution analyses (GMRAs), both classical and those defined on abstract Hilbert spaces, can be classified by their multiplicity functions $m$ and matrix-valued filter functions $H$. Given a natural number…
Very recently we have shown that the spherical transform is a convenient tool for studying the relation between the joint density of the singular values and that of the eigenvalues for bi-unitarily invariant random matrices. In the present…
We show that quaternionic Gaussian random variables satisfy a generalization of the Wick formula for computing the expected value of products in terms of a family of graphical enumeration problems. When applied to the quaternionic Wigner…
The random matrix ensembles (RMT) of quantum statistical Hamiltonian operators, e.g.Gaussian random matrix ensembles (GRME) and Ginibre random matrix ensembles (Ginibre RME), are applied to following quantum statistical systems: nuclear…
Random matrices formed from i.i.d. standard real Gaussian entries have the feature that the expected number of real eigenvalues is non-zero. This property persists for products of such matrices, independently chosen, and moreover it is…
A random matrix is likely to be well conditioned, and motivated by this well known property we employ random matrix multipliers to advance some fundamental matrix computations. This includes numerical stabilization of Gaussian elimination…
We introduce a family of discrete analytic functions, called expandable discrete analytic functions, which includes discrete analytic polynomials, and define two products in this family. The first one is defined in a way similar to the…
We investigate the complex Gaussian as well as non-Gaussian distributed random analytical and entire functions (complex entire random field) and calculate their domain of definiteness (radius of convergence) as well as some important…
Let $A$ be a normal operator in a Hilbert space $\mathcal{H}$, and let $\mathcal{G} \subset \mathcal{H}$ be a countable set of vectors. We investigate the relations between $A$, $\mathcal{G}$ , and $L$ that makes the system of iterations…
We consider the roots of uniformly chosen complex and real reciprocal polynomials of degree $N$ whose Mahler measure is bounded by a constant. After a change of variables this reduces to a generalization of Ginibre's complex and real…