Frontiers of Mathematics in China >

2013 , Vol. 8 >Issue 3: 609 - 641

DOI: https://doi.org/10.1007/s11464-012-0259-5

RESEARCH ARTICLE

Fluctuations of deformed Wigner random matrices

  • Zhonggen SU
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  • Department of Mathematics, Zhejiang University, Hangzhou 310027, China

Received date: 03 Mar 2012

Accepted date: 13 Nov 2012

Published date: 01 Jun 2013

Copyright

2014 Higher Education Press and Springer-Verlag Berlin Heidelberg
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Abstract

Let Xn be a standard real symmetric (complex Hermitian) Wigner matrix, y1, y2, . . . , yn a sequence of independent real random variables independent of Xn. Consider the deformed Wigner matrix Hn,α = n-1/2Xn + n-α/2 diag (y1, . . . , yn), where 0<α<1. It is well known that the average spectral distribution is the classical Wigner semicircle law, i.e., the Stieltjes transform mn,α(z) converges in probability to the corresponding Stieltjes transform m(z). In this paper, we shall give the asymptotic estimate for the expectation Emn,α(z) and varianceVar(mn,α(z)), and establish the central limit theorem for linear statistics with sufficiently regular test function. A basic tool in the study is Stein’s equation and its generalization which naturally leads to a certain recursive equation.

Key words： Asymptotic expansion; deformed Wigner matrice; Gaussian fluctuation; linear statistics; Stein’s equation

Cite this article

Zhonggen SU . Fluctuations of deformed Wigner random matrices[J]. Frontiers of Mathematics in China, 2013 , 8(3) : 609 -641 . DOI: 10.1007/s11464-012-0259-5

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