### LOCAL SEMICIRCLE LAW UNDER MOMENT CONDITIONS: THE STIELTJES TRANSFORM, RIGIDITY, AND DELOCALIZATION

Götze F, Naumov AA, Tikhomirov AN (2018)
THEORY OF PROBABILITY AND ITS APPLICATIONS 62(1): 58-83.

Zeitschriftenaufsatz | Veröffentlicht | Englisch

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Autor*in
Götze, FriedrichUniBi; Naumov, A. A.; Tikhomirov, A. N.
Einrichtung
Abstract / Bemerkung
We consider a random symmetric matrix X = [X-jk](j,k=1)(n) where the upper triangular entries are independent identically distributed random variables with zero mean and unit variance. We additionally suppose that E vertical bar X-11 vertical bar(4+delta) =: mu(4+delta) < infinity for some delta > 0. Under these conditions we show that 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), where v is the distance in the complex plane to the real line. Furthermore, we outline applications such as the rate of convergence of the ESD to the distribution function of the semicircle law, rigidity of the eigenvalues, and eigenvector delocalization.
Stichworte
random matrices; local semicircle law; Stieltjes transform
Erscheinungsjahr
2018
Zeitschriftentitel
THEORY OF PROBABILITY AND ITS APPLICATIONS
Band
62
Ausgabe
1
Seite(n)
58-83
Konferenz
International Conference on Probability and Statistics
Konferenzort
Moscow, RUSSIA
ISSN
0040-585X
eISSN
1095-7219
Page URI
https://pub.uni-bielefeld.de/record/2920353

### Zitieren

Götze F, Naumov AA, Tikhomirov AN. LOCAL SEMICIRCLE LAW UNDER MOMENT CONDITIONS: THE STIELTJES TRANSFORM, RIGIDITY, AND DELOCALIZATION. THEORY OF PROBABILITY AND ITS APPLICATIONS. 2018;62(1):58-83.
Götze, F., Naumov, A. A., & Tikhomirov, A. N. (2018). LOCAL SEMICIRCLE LAW UNDER MOMENT CONDITIONS: THE STIELTJES TRANSFORM, RIGIDITY, AND DELOCALIZATION. THEORY OF PROBABILITY AND ITS APPLICATIONS, 62(1), 58-83. doi:10.1137/S0040585X97T988496
Götze, F., Naumov, A. A., and Tikhomirov, A. N. (2018). LOCAL SEMICIRCLE LAW UNDER MOMENT CONDITIONS: THE STIELTJES TRANSFORM, RIGIDITY, AND DELOCALIZATION. THEORY OF PROBABILITY AND ITS APPLICATIONS 62, 58-83.
Götze, F., Naumov, A.A., & Tikhomirov, A.N., 2018. LOCAL SEMICIRCLE LAW UNDER MOMENT CONDITIONS: THE STIELTJES TRANSFORM, RIGIDITY, AND DELOCALIZATION. THEORY OF PROBABILITY AND ITS APPLICATIONS, 62(1), p 58-83.
F. Götze, A.A. Naumov, and A.N. Tikhomirov, “LOCAL SEMICIRCLE LAW UNDER MOMENT CONDITIONS: THE STIELTJES TRANSFORM, RIGIDITY, AND DELOCALIZATION”, THEORY OF PROBABILITY AND ITS APPLICATIONS, vol. 62, 2018, pp. 58-83.
Götze, F., Naumov, A.A., Tikhomirov, A.N.: LOCAL SEMICIRCLE LAW UNDER MOMENT CONDITIONS: THE STIELTJES TRANSFORM, RIGIDITY, AND DELOCALIZATION. THEORY OF PROBABILITY AND ITS APPLICATIONS. 62, 58-83 (2018).
Götze, Friedrich, Naumov, A. A., and Tikhomirov, A. N. “LOCAL SEMICIRCLE LAW UNDER MOMENT CONDITIONS: THE STIELTJES TRANSFORM, RIGIDITY, AND DELOCALIZATION”. THEORY OF PROBABILITY AND ITS APPLICATIONS 62.1 (2018): 58-83.

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