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
- …