173 research outputs found
Deviations from the Circular Law
Consider Ginibre's ensemble of non-Hermitian random matrices in
which all entries are independent complex Gaussians of mean zero and variance
. As the normalized counting measure of the
eigenvalues converges to the uniform measure on the unit disk in the complex
plane. In this note we describe fluctuations about this {\em Circular Law}.
First we obtain finite formulas for the covariance of certain linear
statistics of the eigenvalues. Asymptotics of these objects coupled with a
theorem of Costin and Lebowitz then result in central limit theorems for a
variety of these statistics
Small deviations for beta ensembles
We establish various small deviation inequalities for the extremal (soft
edge) eigenvalues in the beta-Hermite and beta-Laguerre ensembles. In both
settings, upper bounds on the variance of the largest eigenvalue of the
anticipated order follow immediately
Beta ensembles, stochastic Airy spectrum, and a diffusion
We prove that the largest eigenvalues of the beta ensembles of random matrix
theory converge in distribution to the low-lying eigenvalues of the random
Schroedinger operator -d^2/dx^2 + x + (2/beta^{1/2}) b_x' restricted to the
positive half-line, where b_x' is white noise. In doing so we extend the
definition of the Tracy-Widom(beta) distributions to all beta>0, and also
analyze their tails. Last, in a parallel development, we provide a second
characterization of these laws in terms of a one-dimensional diffusion. The
proofs rely on the associated tridiagonal matrix models and a universality
result showing that the spectrum of such models converge to that of their
continuum operator limit. In particular, we show how Tracy-Widom laws arise
from a functional central limit theorem.Comment: Revised content, new results. In particular, Theorems 1.3 and 5.1 are
ne
Quantization Bounds on Grassmann Manifolds and Applications to MIMO Communications
This paper considers the quantization problem on the Grassmann manifold
\mathcal{G}_{n,p}, the set of all p-dimensional planes (through the origin) in
the n-dimensional Euclidean space. The chief result is a closed-form formula
for the volume of a metric ball in the Grassmann manifold when the radius is
sufficiently small. This volume formula holds for Grassmann manifolds with
arbitrary dimension n and p, while previous results pertained only to p=1, or a
fixed p with asymptotically large n. Based on this result, several quantization
bounds are derived for sphere packing and rate distortion tradeoff. We
establish asymptotically equivalent lower and upper bounds for the rate
distortion tradeoff. Since the upper bound is derived by constructing random
codes, this result implies that the random codes are asymptotically optimal.
The above results are also extended to the more general case, in which
\mathcal{G}_{n,q} is quantized through a code in \mathcal{G}_{n,p}, where p and
q are not necessarily the same. Finally, we discuss some applications of the
derived results to multi-antenna communication systems.Comment: 26 pages, 7 figures, submitted to IEEE Transactions on Information
Theory in Aug, 200
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