On the norm and eigenvalue distribution of large random matrices

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Abstract

We study the eigenvalue distribution of N × N symmetric random matrices HN(x, y) = N -1/2h(x, y), x, y = 1,. . . , N, where h(x, y), x ≤ y are Gaussian weakly dependent random variables. We prove that the normalized eigenvalue counting function of HN converges with probability 1 to a nonrandom function μ(λ) as N → ∞. We derive an equation for the Stieltjes transform of the measure d μ(λ) and show that the latter has a compact support Λμ. We find the upper bound for lim sup N → ∞ ∥ HN ∥ and study asymptotically the case when there are no eigenvalues of HN outside of Λμ when N → ∞.

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Boutet De Monvel, A., & Khorunzhy, A. (1999). On the norm and eigenvalue distribution of large random matrices. Annals of Probability, 27(2), 913–944. https://doi.org/10.1214/aop/1022677390

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