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Mat. Fiz. Anal. Geom., 1996, Volume 3, Number 1/2, Pages 80–101 (Mi jmag484)  

This article is cited in 6 scientific papers (total in 6 papers)

Eigenvalue distribution of large random matrices with correlated entries

A. Khorunzhii

Mathematical Division, B. Verkin Institute for Low Temperature Physics and Engineering, National Academy of Sciences of Ukraine, 47, Lenin Ave., 310164, Kharkov, Ukraine

Abstract: We study the normalized eigenvalue counting function $N_n(\lambda)$ of an ensemble of $n\times n$ symmetric random matrices with statistically dependent arbitrary distributed entries $u_n(x,y)$, $x,y=1,…,n$. We prove that if the correlation function $S$ of the entries is the same for each $n$ and the correlation coefficient of random fields $\{u_n(x,y)\}$ decays fast enough, then in the limit $n\to\infty$ the measure $N_n(d\lambda)$ weakly converges in probability to a nonrandom measure $N(d\lambda)$. We derive an equation for the Stieltjes transform of limiting $N_n(d\lambda)$ and show that the latter depends only on the limiting matrix of averages of $u_n(x,y)$ and the correlation function $S$.

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Received: 05.10.1994
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Citation: A. Khorunzhii, “Eigenvalue distribution of large random matrices with correlated entries”, Mat. Fiz. Anal. Geom., 3:1/2 (1996), 80–101

Citation in format AMSBIB
\Bibitem{Kho96}
\by A.~Khorunzhii
\paper Eigenvalue distribution of large random matrices with correlated entries
\jour Mat. Fiz. Anal. Geom.
\yr 1996
\vol 3
\issue 1/2
\pages 80--101
\mathnet{http://mi.mathnet.ru/jmag484}


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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. De Monvel A., Khorunzhy A., “On the Norm and Eigenvalue Distribution of Large Random Matrices”, Ann. Probab., 27:2 (1999), 913–944  crossref  mathscinet  zmath  isi
    2. Pastur L., Vasilchuk V., “On the Law of Addition of Random Matrices”, Commun. Math. Phys., 214:2 (2000), 249–286  crossref  mathscinet  zmath  adsnasa  isi
    3. Vladimir Vasilchuk, “On the Gaussian Random Matrix Ensembles with Additional Symmetry Conditions”, SIGMA, 2 (2006), 007, 12 pp.  mathnet  crossref  mathscinet  zmath
    4. Kuczala A., Sharpee T.O., “Eigenvalue Spectra of Large Correlated Random Matrices”, Phys. Rev. E, 94:5 (2016), 050101  crossref  isi  scopus
    5. Peligrad C., Peligrad M., “The Limiting Spectral Distribution in Terms of Spectral Density”, Random Matrices-Theor. Appl., 5:1 (2016), 1650003  crossref  zmath  isi
    6. Stolz M., “Fluctuations of Wigner-Type Random Matrices Associated With Symmetric Spaces of Class Diii and Ci”, J. Phys. A-Math. Theor., 51:7 (2018), 075203  crossref  zmath  isi  scopus
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