Limiting Laws of Linear Eigenvalue Statistics
for Unitary Invariant Matrix Models

L. Pastur

Department of Mathematics, University of Wales Swansea
Institute for Low Temperatures, Kharkiv, Ukraine



We study the variance and the Laplace transform of the
probability law of linear eigenvalue statistics of hermitian
$n \times n$ random matrices with unitary invariant
probability law. Assuming that the test function of
statistics is smooth enough and using the asymptotic
formulas by Deift et al for orthogonal polynomials
with varying weights, we show first that if the support
of the Density of States of the model consists of
several intervals, then the variance of statistics
is a quasiperiodic function of $n$ as $n\rightarrow \infty $
generically in the potential, determining the model.
We show next that the exponent of the Laplace
transform of the probability law of these linear statistics
is not in general $1/2\ \times $ variance, as it should
be if the Central Limit Theorem would be valid,
and we find the asymptotic form of the Laplace transform of
the probability law in certain cases.