75,429 research outputs found
Spectral measure of large random Hankel, Markov and Toeplitz matrices
We study the limiting spectral measure of large symmetric random matrices of
linear algebraic structure. For Hankel and Toeplitz matrices generated by
i.i.d. random variables of unit variance, and for symmetric Markov
matrices generated by i.i.d. random variables of zero mean
and unit variance, scaling the eigenvalues by we prove the almost
sure, weak convergence of the spectral measures to universal, nonrandom,
symmetric distributions , and of unbounded
support. The moments of and are the sum of volumes of
solids related to Eulerian numbers, whereas has a bounded smooth
density given by the free convolution of the semicircle and normal densities.
For symmetric Markov matrices generated by i.i.d. random variables
of mean and finite variance, scaling the eigenvalues by
we prove the almost sure, weak convergence of the spectral measures to
the atomic measure at . If , and the fourth moment is finite, we prove
that the spectral norm of scaled by converges
almost surely to 1.Comment: Published at http://dx.doi.org/10.1214/009117905000000495 in the
Annals of Probability (http://www.imstat.org/aop/) by the Institute of
Mathematical Statistics (http://www.imstat.org
Tridiagonal realization of the anti-symmetric Gaussian -ensemble
The Householder reduction of a member of the anti-symmetric Gaussian unitary
ensemble gives an anti-symmetric tridiagonal matrix with all independent
elements. The random variables permit the introduction of a positive parameter
, and the eigenvalue probability density function of the corresponding
random matrices can be computed explicitly, as can the distribution of
, the first components of the eigenvectors. Three proofs are given.
One involves an inductive construction based on bordering of a family of random
matrices which are shown to have the same distributions as the anti-symmetric
tridiagonal matrices. This proof uses the Dixon-Anderson integral from Selberg
integral theory. A second proof involves the explicit computation of the
Jacobian for the change of variables between real anti-symmetric tridiagonal
matrices, its eigenvalues and . The third proof maps matrices from the
anti-symmetric Gaussian -ensemble to those realizing particular examples
of the Laguerre -ensemble. In addition to these proofs, we note some
simple properties of the shooting eigenvector and associated Pr\"ufer phases of
the random matrices.Comment: 22 pages; replaced with a new version containing orthogonal
transformation proof for both cases (Method III
Random transition-rate matrices for the master equation
Random-matrix theory is applied to transition-rate matrices in the Pauli
master equation. We study the distribution and correlations of eigenvalues,
which govern the dynamics of complex stochastic systems. Both the cases of
identical and of independent rates of forward and backward transitions are
considered. The first case leads to symmetric transition-rate matrices, whereas
the second corresponds to general, asymmetric matrices. The resulting matrix
ensembles are different from the standard ensembles and show different
eigenvalue distributions. For example, the fraction of real eigenvalues scales
anomalously with matrix dimension in the asymmetric case.Comment: 15 pages, 12 figure
Truncations of Random Orthogonal Matrices
Statistical properties of non--symmetric real random matrices of size ,
obtained as truncations of random orthogonal matrices are
investigated. We derive an exact formula for the density of eigenvalues which
consists of two components: finite fraction of eigenvalues are real, while the
remaining part of the spectrum is located inside the unit disk symmetrically
with respect to the real axis. In the case of strong non--orthogonality,
const, the behavior typical to real Ginibre ensemble is found. In the
case with fixed , a universal distribution of resonance widths is
recovered.Comment: 4 pages, final revised version (one reference added, minor changes in
Introduction
2*2 random matrix ensembles with reduced symmetry: From Hermitian to PT-symmetric matrices
A possibly fruitful extension of conventional random matrix ensembles is
proposed by imposing symmetry constraints on conventional Hermitian matrices or
parity-time- (PT-) symmetric matrices. To illustrate the main idea, we first
study 2*2 complex Hermitian matrix ensembles with O(2) invariant constraints,
yielding novel level-spacing statistics such as singular distributions,
half-Gaussian distribution, distributions interpolating between GOE (Gaussian
Orthogonal Ensemble) distribution and half Gaussian distributions, as well as
gapped-GOE distribution. Such a symmetry-reduction strategy is then used to
explore 2*2 PT-symmetric matrix ensembles with real eigenvalues. In particular,
PT-symmetric random matrix ensembles with U(2) invariance can be constructed,
with the conventional complex Hermitian random matrix ensemble being a special
case. In two examples of PT-symmetric random matrix ensembles, the
level-spacing distributions are found to be the standard GUE (Gaussian Unitary
Ensemble) statistics or "truncated-GUE" statistics
Random matrix ensembles for -symmetric systems
Recently much effort has been made towards the introduction of non-Hermitian
random matrix models respecting -symmetry. Here we show that there is a
one-to-one correspondence between complex -symmetric matrices and
split-complex and split-quaternionic versions of Hermitian matrices. We
introduce two new random matrix ensembles of (a) Gaussian split-complex
Hermitian, and (b) Gaussian split-quaternionic Hermitian matrices, of arbitrary
sizes. They are related to the split signature versions of the complex and the
quaternionic numbers, respectively. We conjecture that these ensembles
represent universality classes for -symmetric matrices. For the case of
matrices we derive analytic expressions for the joint probability
distributions of the eigenvalues, the one-level densities and the level
spacings in the case of real eigenvalues.Comment: 9 pages, 3 figures, typos corrected, small changes, accepted for
publication in Journal of Physics
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