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Central limit theorem for traces of large random symmetric matrices with independent matrix elements

Authors:

Affiliation:

(1) Mathematics Department, Princeton University, 08544-1000 Princeton, NJ, USA;(2) Landau Institute of Theoretical Physics, Moscow, Russia;(3) Institute for Advanced Study, Olden Lane, 08540 Princeton, NJ, USA

Abstract:

We study Wigner ensembles of symmetric random matricesA=(a_ij),i, j=1,...,n with matrix elementsa_ij,i

j being independent symmetrically distributed random variables

$a_{ij} = a_{ji} = frac{{xi _{ij} }}{{n^{tfrac{1}{2}} }}.$

We assume that Var $xi _{ij} = frac{1}{4}$ , fori<j, Var xgr

_ij

const and that all higher moments of xgr

_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 ofA growing withn more slowly than $$sqrt n$$

. The limit of Var (TraceA^p), $$1 ll p ll sqrt n$$

, does not depend on the fourth and higher moments of xgr

_ij and the rate of growth ofp, and equals to $frac{1}{pi }$ . As a corollary we improve the estimates on the rate of convergence of the maximal eigenvalue to 1 and prove central limit theorem for a general class of linear statistics of the spectra.Dedicated to the memory of R. Mañé

Keywords:

Random matrices Wigner semi-circle law Central limit theorem Moments

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