Article

Boletim da Sociedade Brasileira de Matemática - Bulletin/Brazilian Mathematical Society

, Volume 29, Issue 1, pp 1-24

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

  • Ya. SinaiAffiliated withMathematics Department, Princeton UniversityLandau Institute of Theoretical Physics
  • , A. SoshnikovAffiliated withInstitute for Advanced Study

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Abstract

We study Wigner ensembles of symmetric random matricesA=(a ij ),i, j=1,...,n with matrix elementsa ij ,ij 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 ξ ij ≤ const and that all higher moments of ξ 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 ξ 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.

Keywords

Random matrices Wigner semi-circle law Central limit theorem Moments