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 Zh. Mat. Fiz. Anal. Geom., 2014, Volume 10, Number 4, Pages 451–484 (Mi jmag605)

On the Fluctuations of Entries of Matrices whose Randomness is due to Classical Groups

V. Vasilchuk

B. Verkin Institute for Low Temperature Physics and Engineering, National Academy of Sciences of Ukraine, 47 Lenin Ave., Kharkiv, 61103, Ukraine

Abstract: We consider first the $n\times n$ random matrices $H_{n}=A_{n}+U_{n}^{* }B_{n}U_{n}$, where $A_{n}$ and $B_{n}$ are Hermitian, having the limiting normalized counting measure (NCM) of eigenvalues as $n\rightarrow \infty$, and $U_{n}$ is unitary uniformly distributed over $U(n)$. We find the leading term of asymptotic expansion for the covariance of elements of resolvent of $H_{n}$ and establish the Central Limit Theorem for the elements of sufficiently smooth test functions of the corresponding linear statistics. We consider then analogous problems for the matrices $W_{n}=S_{n}U_{n}^{* }T_{n}U_{n}$, where $U_n$ is as above and $S_n$ and $T_n$ are non-random unitary matrices having limiting NCM's as $n\rightarrow \infty$.

Key words and phrases: Random matrices, Central Limit Theorem, Limit Laws.

DOI: https://doi.org/10.15407/mag10.04.451

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MSC: Primary 60F05, 15B52; Secondary 15A18
Revised: 09.09.2014
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Citation: V. Vasilchuk, “On the Fluctuations of Entries of Matrices whose Randomness is due to Classical Groups”, Zh. Mat. Fiz. Anal. Geom., 10:4 (2014), 451–484

Citation in format AMSBIB
\Bibitem{Vas14} \by V.~Vasilchuk \paper On the Fluctuations of Entries of Matrices whose Randomness is due to Classical Groups \jour Zh. Mat. Fiz. Anal. Geom. \yr 2014 \vol 10 \issue 4 \pages 451--484 \mathnet{http://mi.mathnet.ru/jmag605} \crossref{https://doi.org/10.15407/mag10.04.451} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=3309798} \isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=000346136000005} 

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Citing articles on Google Scholar: Russian citations, English citations
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This publication is cited in the following articles:
1. Girko V.L., “30 years of General Statistical Analysis and canonical equation K60 for Hermitian matrices (A+ BUC)(A+ BUC)*, where U is a random unitary matrix”, Random Operators Stoch. Equ., 23:4 (2015), 235–260
2. V. L. Girko, “The Canonical Equations $K_{66}$, $K_{67}$, $K_{68}$ and $K_{69}$”, Random Operators Stoch. Equ., 24:3 (2016), 173–197
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