98-650 Sinai Ya., Soshnikov A.

A Refinement of Wigner's Semicirle Law in a Neighborhood of the Spectrum Edge for Random Symmetric Matrices (399K, PostScript) Oct 15, 98

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Abstract. This is a continuation of 98-647 (``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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