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Bounds for the Stieltjes transform and the density of states of Wigner matrices. (English) Zbl 1330.60014

Summary: We consider ensembles of Wigner matrices, whose entries are (up to the symmetry constraints) independent and identically distributed random variables. We show the convergence of the Stieltjes transform towards the Stieltjes transform of the semicircle law on optimal scales and with the optimal rate. Our bounds improve previous results, in particular from L. Erdős et al. [Adv. Math. 229, No. 3, 1435–1515 (2012; Zbl 1238.15017); Electron. J. Probab. 18, Paper No. 59, 58 p. (2013; Zbl 1373.15053)], by removing the logarithmic corrections. As applications, we establish the convergence of the eigenvalue counting functions with the rate \((\log N)/N\) and the rigidity of the eigenvalues of Wigner matrices on the same scale. These bounds improve the results of Erdős et al. [loc. cit.] and F. Götze and A. Tikhomirov [Prog. Probab. 66, 139–165 (2013; Zbl 1274.60020)].

MSC:

60B20 Random matrices (probabilistic aspects)
60B12 Limit theorems for vector-valued random variables (infinite-dimensional case)
47B80 Random linear operators
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References:

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