8,134 research outputs found
On the probability that all eigenvalues of Gaussian, Wishart, and double Wishart random matrices lie within an interval
We derive the probability that all eigenvalues of a random matrix lie
within an arbitrary interval ,
, when is a real or complex finite dimensional Wishart,
double Wishart, or Gaussian symmetric/Hermitian matrix. We give efficient
recursive formulas allowing the exact evaluation of for Wishart
matrices, even with large number of variates and degrees of freedom. We also
prove that the probability that all eigenvalues are within the limiting
spectral support (given by the Mar{\v{c}}enko-Pastur or the semicircle laws)
tends for large dimensions to the universal values and for
the real and complex cases, respectively. Applications include improved bounds
for the probability that a Gaussian measurement matrix has a given restricted
isometry constant in compressed sensing.Comment: IEEE Transactions on Information Theory, 201
Orthogonal polynomial ensembles in probability theory
We survey a number of models from physics, statistical mechanics, probability
theory and combinatorics, which are each described in terms of an orthogonal
polynomial ensemble. The most prominent example is apparently the Hermite
ensemble, the eigenvalue distribution of the Gaussian Unitary Ensemble (GUE),
and other well-known ensembles known in random matrix theory like the Laguerre
ensemble for the spectrum of Wishart matrices. In recent years, a number of
further interesting models were found to lead to orthogonal polynomial
ensembles, among which the corner growth model, directed last passage
percolation, the PNG droplet, non-colliding random processes, the length of the
longest increasing subsequence of a random permutation, and others. Much
attention has been paid to universal classes of asymptotic behaviors of these
models in the limit of large particle numbers, in particular the spacings
between the particles and the fluctuation behavior of the largest particle.
Computer simulations suggest that the connections go even farther and also
comprise the zeros of the Riemann zeta function. The existing proofs require a
substantial technical machinery and heavy tools from various parts of
mathematics, in particular complex analysis, combinatorics and variational
analysis. Particularly in the last decade, a number of fine results have been
achieved, but it is obvious that a comprehensive and thorough understanding of
the matter is still lacking. Hence, it seems an appropriate time to provide a
surveying text on this research area.Comment: Published at http://dx.doi.org/10.1214/154957805100000177 in the
Probability Surveys (http://www.i-journals.org/ps/) by the Institute of
Mathematical Statistics (http://www.imstat.org
Eigenvalue Dynamics of a Central Wishart Matrix with Application to MIMO Systems
We investigate the dynamic behavior of the stationary random process defined
by a central complex Wishart (CW) matrix as it varies along a
certain dimension . We characterize the second-order joint cdf of the
largest eigenvalue, and the second-order joint cdf of the smallest eigenvalue
of this matrix. We show that both cdfs can be expressed in exact closed-form in
terms of a finite number of well-known special functions in the context of
communication theory. As a direct application, we investigate the dynamic
behavior of the parallel channels associated with multiple-input
multiple-output (MIMO) systems in the presence of Rayleigh fading. Studying the
complex random matrix that defines the MIMO channel, we characterize the
second-order joint cdf of the signal-to-noise ratio (SNR) for the best and
worst channels. We use these results to study the rate of change of MIMO
parallel channels, using different performance metrics. For a given value of
the MIMO channel correlation coefficient, we observe how the SNR associated
with the best parallel channel changes slower than the SNR of the worst
channel. This different dynamic behavior is much more appreciable when the
number of transmit () and receive () antennas is similar. However, as
is increased while keeping fixed, we see how the best and worst
channels tend to have a similar rate of change.Comment: 15 pages, 9 figures and 1 table. This work has been accepted for
publication at IEEE Trans. Inf. Theory. Copyright (c) 2014 IEEE. Personal use
of this material is permitted. However, permission to use this material for
any other purposes must be obtained from the IEEE by sending a request to
[email protected]
On Finite Rank Deformations of Wigner Matrices
We study the distribution of the outliers in the spectrum of finite rank
deformations of Wigner random matrice under the assumption that the
off-diagonal matrix entries have uniformly bounded fifth moment and the
diagonal entries have uniformly bounded third moment. Using our recent results
on the fluctuation of resolvent entries [31],[28], and ideas from [9], we
extend results by M.Capitaine, C.Donati-Martin, and D.F\'eral [12], [13].Comment: accepted for publication in Annales de l'Institut Henri Poincar\'e
(B) Probabilit\'es et Statistique
A random matrix decimation procedure relating to
Classical random matrix ensembles with orthogonal symmetry have the property
that the joint distribution of every second eigenvalue is equal to that of a
classical random matrix ensemble with symplectic symmetry. These results are
shown to be the case of a family of inter-relations between eigenvalue
probability density functions for generalizations of the classical random
matrix ensembles referred to as -ensembles. The inter-relations give
that the joint distribution of every -st eigenvalue in certain
-ensembles with is equal to that of another
-ensemble with . The proof requires generalizing a
conditional probability density function due to Dixon and Anderson.Comment: 19 pages, 1 figur
On Finite Rank Deformations of Wigner Matrices II: Delocalized Perturbations
We study the distribution of the outliers in the spectrum of finite rank
deformations of Wigner random matrices. We assume that the matrix entries have
finite fourth moment and extend the results by Capitaine, Donati-Martin, and
F\'eral for perturbations whose eigenvectors are delocalized.Comment: We explained some proofs in greater detail, corrected several small
misprints, and updated the bibliograph
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