 98650 Sinai Ya., Soshnikov A.
 A Refinement of Wigner's Semicirle Law in a Neighborhood of the Spectrum
Edge for Random Symmetric Matrices
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Oct 15, 98

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Abstract. This is a continuation of 98647 (``Central Limiy Theorem for Traces of
Large Random Symmetric Matrices With Independent Matrix Elements'').
We study the Wigner ensembles of symmetric random matrices
$A= (a_{ij}) \; i,j = 1, \ldots , n$ with matrix elements
$a_{ij} , \quad i\leq j$ being independent symmetrically distributed
random variables
$$
a_{ij}= \frac{\xi_(ij}}{n^{\frac{1}{2}}}
$$
such that $Var(\xi_{ij})= \frac{1}{4}$
for $i<j$, and all higher moments of $\xi_{ij}$ also exist and grow not
faster than the Gaussian ones.
Under formulated conditions we prove the central limit theorem for the
traces of powers of $A$ growing with $n$ more slowly than
$n^{2/3}$. The limit of $ Var( Trace A^p), \; 1 \ll p \ll n^{2/3}$
does not depend on the fourth and higher moments of $\xi_{ij}$ and the
rate of growth of $p$, and equals to $ \frac{1}{\pi}$.
Developed technoque allows us to show that for the typical (from a measure
viewpoint) matrices in the Wigner ensemble the distance between the maximal
(minimal) eigenvalue and the corresponding endpoints $(+1,1) $ of the support
of a semicircle distribution is $O(N^{1/3})$.
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