3,350 research outputs found
Some asymptotic properties of the spectrum of the Jacobi ensemble
For the random eigenvalues with density corresponding to the Jacobi ensemble
a strong uniform approximation by the roots of the Jacobi polynomials is
derived if the parameters depend on and .
Roughly speaking, the eigenvalues can be uniformly approximated by roots of
Jacobi polynomials with parameters , where
the error is of order . These results are used to
investigate the asymptotic properties of the corresponding spectral
distribution if and the parameters and vary with
. We also discuss further applications in the context of multivariate random
-matrices.Comment: 20 pages, 2 figure
Product of random projections, Jacobi ensembles and universality problems arising from free probability
We consider the product of two independent randomly rotated projectors. The
square of its radial part turns out to be distributed as a Jacobi ensemble. We
study its global and local properties in the large dimension scaling relevant
to free probability theory. We establish asymptotics for one point and two
point correlation functions, as well as properties of largest and smallest
eigenvalues.Comment: 28 pages, no figure, pdfLaTe
Biorthogonal ensembles
One object of interest in random matrix theory is a family of point ensembles
(random point configurations) related to various systems of classical
orthogonal polynomials. The paper deals with a one--parametric deformation of
these ensembles, which is defined in terms of the biorthogonal polynomials of
Jacobi, Laguerre and Hermite type.
Our main result is a series of explicit expressions for the correlation
functions in the scaling limit (as the number of points goes to infinity). As
in the classical case, the correlation functions have determinantal form. They
are given by certain new kernels which are described in terms of the Wright's
generalized Bessel function and can be viewed as a generalization of the
well--known sine and Bessel kernels.
In contrast to the conventional kernels, the new kernels are non--symmetric.
However, they possess other, rather surprising, symmetry properties.
Our approach to finding the limit kernel also differs from the conventional
one, because of lack of a simple explicit Christoffel--Darboux formula for the
biorthogonal polynomials.Comment: AMSTeX, 26 page
Limits for circular Jacobi beta-ensembles
Bourgade, Nikeghbali and Rouault recently proposed a matrix model for the
circular Jacobi -ensemble, which is a generalization of the Dyson
circular -ensemble but equipped with an additional parameter , and
further studied its limiting spectral measure. We calculate the scaling limits
for expected products of characteristic polynomials of circular Jacobi
-ensembles. For the fixed constant , the resulting limit near the
spectrum singularity is proven to be a new multivariate function. When , the scaling limits in the bulk and at the soft edge agree with those of
the Hermite (Gaussian), Laguerre (Chiral) and Jacobi -ensembles proved
in the joint work with P Desrosiers "Asymptotics for products of characteristic
polynomials in classical beta-ensembles", Constr. Approx. 39 (2014),
arXiv:1112.1119v3. As corollaries, for even the scaling limits of point
correlation functions for the ensemble are given. Besides, a transition from
the spectrum singularity to the soft edge limit is observed as goes to
infinity. The positivity of two special multivariate hypergeometric functions,
which appear as one factor of the joint eigenvalue densities for spiked
Jacobi/Wishart -ensembles and Gaussian -ensembles with source,
will also be shown.Comment: 26 page
Random Subsets of Structured Deterministic Frames have MANOVA Spectra
We draw a random subset of rows from a frame with rows (vectors) and
columns (dimensions), where and are proportional to . For a
variety of important deterministic equiangular tight frames (ETFs) and tight
non-ETF frames, we consider the distribution of singular values of the
-subset matrix. We observe that for large they can be precisely
described by a known probability distribution -- Wachter's MANOVA spectral
distribution, a phenomenon that was previously known only for two types of
random frames. In terms of convergence to this limit, the -subset matrix
from all these frames is shown to be empirically indistinguishable from the
classical MANOVA (Jacobi) random matrix ensemble. Thus empirically the MANOVA
ensemble offers a universal description of the spectra of randomly selected
-subframes, even those taken from deterministic frames. The same
universality phenomena is shown to hold for notable random frames as well. This
description enables exact calculations of properties of solutions for systems
of linear equations based on a random choice of frame vectors out of
possible vectors, and has a variety of implications for erasure coding,
compressed sensing, and sparse recovery. When the aspect ratio is small,
the MANOVA spectrum tends to the well known Marcenko-Pastur distribution of the
singular values of a Gaussian matrix, in agreement with previous work on highly
redundant frames. Our results are empirical, but they are exhaustive, precise
and fully reproducible
On Eigenvalues of the sum of two random projections
We study the behavior of eigenvalues of matrix P_N + Q_N where P_N and Q_N
are two N -by-N random orthogonal projections. We relate the joint eigenvalue
distribution of this matrix to the Jacobi matrix ensemble and establish the
universal behavior of eigenvalues for large N. The limiting local behavior of
eigenvalues is governed by the sine kernel in the bulk and by either the Bessel
or the Airy kernel at the edge depending on parameters. We also study an
exceptional case when the local behavior of eigenvalues of P_N + Q_N is not
universal in the usual sense.Comment: 14 page
Entropy and the Shannon-McMillan-Breiman theorem for beta random matrix ensembles
We show that beta ensembles in Random Matrix Theory with generic real
analytic potential have the asymptotic equipartition property. In addition, we
prove a Central Limit Theorem for the density of the eigenvalues of these
ensembles.Comment: We made small changes suggested by the referees. In particular, we
updated the bibliograph
Can chaotic quantum energy levels statistics be characterized using information geometry and inference methods?
In this paper, we review our novel information geometrodynamical approach to
chaos (IGAC) on curved statistical manifolds and we emphasize the usefulness of
our information-geometrodynamical entropy (IGE) as an indicator of chaoticity
in a simple application. Furthermore, knowing that integrable and chaotic
quantum antiferromagnetic Ising chains are characterized by asymptotic
logarithmic and linear growths of their operator space entanglement entropies,
respectively, we apply our IGAC to present an alternative characterization of
such systems. Remarkably, we show that in the former case the IGE exhibits
asymptotic logarithmic growth while in the latter case the IGE exhibits
asymptotic linear growth. At this stage of its development, IGAC remains an
ambitious unifying information-geometric theoretical construct for the study of
chaotic dynamics with several unsolved problems. However, based on our recent
findings, we believe it could provide an interesting, innovative and
potentially powerful way to study and understand the very important and
challenging problems of classical and quantum chaos.Comment: 21 page
- …