13 research outputs found
Singular value distribution of products of random matrices
Recently, the study of products of random matrices gained a lot of interest. Motivated by this, we will be particularly focussing on three different models: (1) the product of r Ginibre random matrices, (2) the product of r truncated unitary matrices, and (3) the Laguerre Muttalib-Borodin ensemble, which was recently shown to have an interpretation in terms of products of random matrices. The squared singular values of model (1) and (2) and the eigenvalues of model (3) all form a determinantal point process on the positive half-line. To be more concrete, the density defines a biorthogonal ensemble with a very specific structure which we will call a polynomial ensemble. In this thesis we analyze the singular value (or eigenvalue) density for model (1), (2) and (3).status: publishe
Asymptotics for characteristic polynomials of Wishart type products of complex Gaussian and truncated unitary random matrices
Based on the multivariate saddle point method we study the asymptotic behavior
of the characteristic polynomials associated to Wishart type random matrices that
are formed as products consisting of independent standard complex Gaussian and a
truncated Haar distributed unitary random matrix. These polynomials form a general
class of hypergeometric functions of type 2Fr
. We describe the oscillatory behavior
on the asymptotic interval of zeros by means of formulae of Plancherel–Rotach type
and subsequently use it to obtain the limiting distribution of the suitably rescaled
zeros. Moreover, we show that the asymptotic zero distribution lies in the class of
Raney distributions and by introducing appropriate coordinates elementary and explicit
characterizations are derived for the densities as well as for the distribution functions.status: publishe
Large gap asymptotics at the hard edge for product random matrices and Muttalib-Borodin ensembles
We study the distribution of the smallest eigenvalue for certain classes of positive-definite Hermitian random matrices, in the limit where the size of the matrices becomes large. Their limit distributions can be expressed as Fredholm determinants of integral operators associated to kernels built out of Meijer G-functions or Wright's generalized Bessel functions. They generalize in a natural way the hard edge Bessel kernel Fredholm determinant. We express the logarithmic derivatives of the Fredholm determinants identically in terms of a 2×2 Riemann-Hilbert problem, and use this representation to obtain the so-called large gap asymptotics
Singular Value Statistics of Matrix Products with Truncated Unitary Matrices
We prove that the squared singular values of a fixed
matrix multiplied with a truncation of a Haar
distributed unitary matrix are distributed by a
polynomial ensemble. This result is applied to a
multiplication of a truncated unitary matrix with a
random matrix.
We show that the structure of polynomial ensembles and
of certain Pfaffian ensembles is preserved. Furthermore
we derive the joint singular value density of a product
of truncated unitary matrices and its corresponding
correlation kernel which can be written as a double
contour integral. This leads to hard edge scaling limits
that also include new finite rank perturbations of the
Meijer G-kernels found for products of complex Ginibre
random matrices.29 pagesstatus: publishe
Large gap asymptotics at the hard edge for product random matrices and Muttalib-Borodin ensembles
We study the distribution of the smallest eigenvalue for certain classes of positive-definite Hermitian random matrices, in the limit where the size of the matrices becomes large. Their limit distributions can be expressed as Fredholm determinants of integral operators associated to kernels built out of Meijer G-functions or Wright's generalized Bessel functions. They generalize in a natural way the hard edge Bessel kernel Fredholm determinant. We express the logarithmic derivatives of the Fredholm determinants identically in terms of a 2×2 Riemann-Hilbert problem, and use this representation to obtain the so-called large gap asymptotics
JACOBI POLYNOMIAL MOMENTS AND PRODUCTS OF RANDOM MATRICES
Motivated by recent results in random matrix theory we will study the distributions arising from products of complex Gaussian random matrices and truncations of Haar distributed unitary matrices. We introduce an appropriately general class of measures and characterize them by their moments essentially given by specific Jacobi polynomials with varying parameters. Solving this moment problem requires a study of the Riemann surfaces associated to a class of algebraic equations. The connection to random matrix theory is then established using methods from free probability.status: publishe