On the local semicircular law for Wigner ensembles

Götze F, Naumov A, Tikhomirov A, Timushev D (2018)
BERNOULLI 24(3): 2358-2400.

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Zeitschriftenaufsatz | Veröffentlicht | Englisch
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Abstract / Bemerkung
We consider a random symmetric matrix X = [X-jk] (n)(j, k=1) with upper triangular entries being i. i. d. random variables with mean zero and unit variance. We additionally suppose that E vertical bar X-11 vertical bar(4+delta) =: mu(4+delta) < infinity for some delta > 0. The aim of this paper is to significantly extend a recent result of the authors Gotze, Naumov and Tikhomirov (2015) and show that with high probability the typical distance between the Stieltjes transform of the empirical spectral distribution (ESD) of the matrix n(-1/2) X and Wigner's semicircle law is of order (nv)(-1) log n, where v denotes the distance to the real line in the complex plane. We apply this result to the rate of convergence of the ESD to the distribution function of the semicircle law as well as to rigidity of eigenvalues and eigenvector delocalization significantly extending a recent result by Gotze, Naumov and Tikhomirov (2015). The result on delocalization is optimal by comparison with GOE ensembles. Furthermore the techniques of this paper provide a new shorter proof for the optimal O(n(-1)) rate of convergence of the expected ESD to the semicircle law.
Erscheinungsjahr
Zeitschriftentitel
BERNOULLI
Band
24
Ausgabe
3
Seite(n)
2358-2400
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Götze F, Naumov A, Tikhomirov A, Timushev D. On the local semicircular law for Wigner ensembles. BERNOULLI. 2018;24(3):2358-2400.
Götze, F., Naumov, A., Tikhomirov, A., & Timushev, D. (2018). On the local semicircular law for Wigner ensembles. BERNOULLI, 24(3), 2358-2400. doi:10.3150/17-BEJ931
Götze, F., Naumov, A., Tikhomirov, A., and Timushev, D. (2018). On the local semicircular law for Wigner ensembles. BERNOULLI 24, 2358-2400.
Götze, F., et al., 2018. On the local semicircular law for Wigner ensembles. BERNOULLI, 24(3), p 2358-2400.
F. Götze, et al., “On the local semicircular law for Wigner ensembles”, BERNOULLI, vol. 24, 2018, pp. 2358-2400.
Götze, F., Naumov, A., Tikhomirov, A., Timushev, D.: On the local semicircular law for Wigner ensembles. BERNOULLI. 24, 2358-2400 (2018).
Götze, Friedrich, Naumov, Alexey, Tikhomirov, Alexander, and Timushev, Dmitry. “On the local semicircular law for Wigner ensembles”. BERNOULLI 24.3 (2018): 2358-2400.