On Permanental Polynomials of Certain Random Matrices

Determinants and permanents of Hessenberg matrices and generalized Lucas polynomials

In this paper, we give some determinantal and permanental representations of generalized Lucas polynomials, which are a general form of generalized bivariate Lucas p-polynomials, ordinary Lucas and Perrin sequences etc., by using various Hessenberg matrices. In addition, we show that determinant and permanent of these Hessenberg matrices can be obtained by using combinations. Then we show, the ...

Generalized Bivariate Lucas p-Polynomials and Hessenberg Matrices

In this paper, we give some determinantal and permanental representations of generalized bivariate Lucas p-polynomials by using various Hessenberg matrices. The results that we obtained are important since generalized bivariate Lucas p-polynomials are general forms of, for example, bivariate Jacobsthal-Lucas, bivariate Pell-Lucas ppolynomials, Chebyshev polynomials of the first kind, Jacobsthal...

Determinantal Processes and Independence

Abstract: We give a probabilistic introduction to determinantal and permanental point processes. Determinantal processes arise in physics (fermions, eigenvalues of random matrices) and in combinatorics (nonintersecting paths, random spanning trees). They have the striking property that the number of points in a region D is a sum of independent Bernoulli random variables, with parameters which a...

On the average number of sharp crossings of certain Gaussian random polynomials

The Expected Characteristic and Permanental Polynomials of the Random Gram Matrix

A t × n random matrix A can be formed by sampling n independent random column vectors, each containing t components. The random Gram matrix of size n, Gn = A A, contains the dot products between all pairs of column vectors in the randomly generated matrix A, and has characteristic roots coinciding with the singular values of A. Furthermore, the sequences det (Gi) and perm(Gi) (for i = 0, 1, . ....