Local laws for non-Hermitian random matrices and their products

Götze F, Naumov A, Tikhomirov A (2020)
Random Matrices: Theory and Applications 9(4): 2150004.

Zeitschriftenaufsatz | Veröffentlicht | Englisch
 
Download
Es wurden keine Dateien hochgeladen. Nur Publikationsnachweis!
Autor*in
Götze, FriedrichUniBi; Naumov, Alexey; Tikhomirov, Alexander
Abstract / Bemerkung
We consider products of independent n x n non-Hermitian random matrices X-(1), ..., X-(m). Assume that their entries, Xj(jk)((q)), 1 <= j, k <= n, q = 1, ..., m, are independent identically distributed random variables with zero mean, unit variance. Gotze and Tikhomirov [On the asymptotic spectrum of products of independent random matrices, preprint (2010), arXiv:1012.2710] and O'Rourke and Sochnikov [Products of independent non-Hermitian random matrices, Electron. J. Probab. 16 (2011) 2219-2245] proved that under these assumptions the empirical spectral distribution (ESD) of X-(1) ... X-(m) converges to the limiting distribution which coincides with the distribution of the mth power of random variable uniformly distributed in the unit circle. In this paper, we provide a local version of this result. More precisely, assuming additionally that E vertical bar X-11((q))vertical bar(4+delta) < infinity for some delta > 0, we prove that ESD of X-(1) ... X-(m) converges to the limiting distribution on the optimal scale up to n(-1+2a) , 0 < a < 1/2 (up to some logarithmic factor). Our results generalize the recent results of Bourgade et al. [Local circular law for random matrices, Probab. Theory Related Fields 159 (2014) 545-595], Tao and Vu [Random matrices: Universality of local spectral statistics of non-Hermitian matrices, Ann. Probab. 43 (2015) 782-874] and Nemish [Local law for the product of independent non-hermitian random matrices with independent entries, Electron. J. Probab. 22 (2017) 1-35]. We also give further development of Stein's type approach to estimate the Stieltjes transform of ESD.
Stichworte
Random matrices; local circle law; product of non-Hermitian random; matrices; Stieltjes transform; logarithmic potential; Stein's method
Erscheinungsjahr
2020
Zeitschriftentitel
Random Matrices: Theory and Applications
Band
9
Ausgabe
4
Art.-Nr.
2150004
ISSN
2010-3263
eISSN
2010-3271
Page URI
https://pub.uni-bielefeld.de/record/2945600

Zitieren

Götze F, Naumov A, Tikhomirov A. Local laws for non-Hermitian random matrices and their products. Random Matrices: Theory and Applications. 2020;9(4): 2150004.
Götze, F., Naumov, A., & Tikhomirov, A. (2020). Local laws for non-Hermitian random matrices and their products. Random Matrices: Theory and Applications, 9(4), 2150004. doi:10.1142/S2010326321500040
Götze, F., Naumov, A., and Tikhomirov, A. (2020). Local laws for non-Hermitian random matrices and their products. Random Matrices: Theory and Applications 9:2150004.
Götze, F., Naumov, A., & Tikhomirov, A., 2020. Local laws for non-Hermitian random matrices and their products. Random Matrices: Theory and Applications, 9(4): 2150004.
F. Götze, A. Naumov, and A. Tikhomirov, “Local laws for non-Hermitian random matrices and their products”, Random Matrices: Theory and Applications, vol. 9, 2020, : 2150004.
Götze, F., Naumov, A., Tikhomirov, A.: Local laws for non-Hermitian random matrices and their products. Random Matrices: Theory and Applications. 9, : 2150004 (2020).
Götze, Friedrich, Naumov, Alexey, and Tikhomirov, Alexander. “Local laws for non-Hermitian random matrices and their products”. Random Matrices: Theory and Applications 9.4 (2020): 2150004.

Export

Markieren/ Markierung löschen
Markierte Publikationen

Open Data PUB

Web of Science

Dieser Datensatz im Web of Science®

Suchen in

Google Scholar