Local Laws for Non-Hermitian Random Matrices

Götze F, Naumov AA, Tikhomirov AN (2017)
DOKLADY MATHEMATICS 96(3): 558-560.

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Zeitschriftenaufsatz | Veröffentlicht | Englisch
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Abstract / Bemerkung
The product of m is an element of N independent random square matrices whose elements are independent identically distributed random variables with zero mean and unit variance is considered. It is known that, as the size of the matrices increases to infinity, the empirical spectral measure of the normalized eigenvalues of the product converges with probability 1 to the distribution of the mth power of the random variable uniformly distributed on the unit disk of the complex plane. In particular, in the case of m = 1, the circular law holds. The purpose of this paper is to prove the validity of the local circular law (as well as its generalization to the case of any fixed m) in the case where the distribution of the matrix elements has finite absolute moment of order 4 + delta, delta > 0. Recent results of Bourgade, Yau, and Yin, of Tao and Vu, and of Nemish are generalized.
Erscheinungsjahr
Zeitschriftentitel
DOKLADY MATHEMATICS
Band
96
Ausgabe
3
Seite(n)
558-560
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Götze F, Naumov AA, Tikhomirov AN. Local Laws for Non-Hermitian Random Matrices. DOKLADY MATHEMATICS. 2017;96(3):558-560.
Götze, F., Naumov, A. A., & Tikhomirov, A. N. (2017). Local Laws for Non-Hermitian Random Matrices. DOKLADY MATHEMATICS, 96(3), 558-560. doi:10.1134/S1064562417060072
Götze, F., Naumov, A. A., and Tikhomirov, A. N. (2017). Local Laws for Non-Hermitian Random Matrices. DOKLADY MATHEMATICS 96, 558-560.
Götze, F., Naumov, A.A., & Tikhomirov, A.N., 2017. Local Laws for Non-Hermitian Random Matrices. DOKLADY MATHEMATICS, 96(3), p 558-560.
F. Götze, A.A. Naumov, and A.N. Tikhomirov, “Local Laws for Non-Hermitian Random Matrices”, DOKLADY MATHEMATICS, vol. 96, 2017, pp. 558-560.
Götze, F., Naumov, A.A., Tikhomirov, A.N.: Local Laws for Non-Hermitian Random Matrices. DOKLADY MATHEMATICS. 96, 558-560 (2017).
Götze, Friedrich, Naumov, A. A., and Tikhomirov, A. N. “Local Laws for Non-Hermitian Random Matrices”. DOKLADY MATHEMATICS 96.3 (2017): 558-560.