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April 1999 On the Norm and Eigenvalue Distribution of Large Random Matrices
Anne Boutet de Monvel, Alexei Khorunzhy
Ann. Probab. 27(2): 913-944 (April 1999). DOI: 10.1214/aop/1022677390


We study the eigenvalue distribution of $N \times N$ symmetric random matrices $H_N (x, y) = N^-1/2h(x,y),x,y=1,\ldots,N$, where $h(x, y), x \leq y$ are Gaussian weakly dependent random variables. We prove that the normalized eigenvalue counting function of $H_{N}$ converges with probability 1 to a nonrandom function $\mu(\lambda)$ as $N\rightarrow\infty$. We derive an equation for the Stieltjes transform of the measure $d\mu(\lambda)$ and show that the latter has a compact support $\Lambda_\mu$. We find the upper bound for $\lim\sup_{N\rightarrow\infty}\|H_N\|$ and study asymptotically the case when there are no eigenvalues of $H_N$ outside of $\Lambda_\mu$ when $N \rightarrow \infty$.


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Anne Boutet de Monvel. Alexei Khorunzhy. "On the Norm and Eigenvalue Distribution of Large Random Matrices." Ann. Probab. 27 (2) 913 - 944, April 1999.


Published: April 1999
First available in Project Euclid: 29 May 2002

zbMATH: 0941.15021
MathSciNet: MR1698983
Digital Object Identifier: 10.1214/aop/1022677390

Primary: 60F15
Secondary: 15A18‎ , 15A52

Keywords: eigenvalue distribution , limiting theorem , random matrices , spectral norm

Rights: Copyright © 1999 Institute of Mathematical Statistics

Vol.27 • No. 2 • April 1999
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