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

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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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
2. Pastur L., Vasilchuk V., “On the Law of Addition of Random Matrices”, Commun. Math. Phys., 214:2 (2000), 249–286
3. Vladimir Vasilchuk, “On the Gaussian Random Matrix Ensembles with Additional Symmetry Conditions”, SIGMA, 2 (2006), 007, 12 pp.
4. Kuczala A., Sharpee T.O., “Eigenvalue Spectra of Large Correlated Random Matrices”, Phys. Rev. E, 94:5 (2016), 050101
5. Peligrad C., Peligrad M., “The Limiting Spectral Distribution in Terms of Spectral Density”, Random Matrices-Theor. Appl., 5:1 (2016), 1650003
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