Local laws for non-Hermitian random matrices
- Authors: Götze F.1, Naumov A.A.2,3, Tikhomirov A.N.4
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Affiliations:
- Bielefeld University
- Skolkovo Institute of Science and Technology
- Institute for Information Transmission Problems
- Komi Scientific Center, Ural Branch
- Issue: Vol 96, No 3 (2017)
- Pages: 558-560
- Section: Mathematics
- URL: https://journals.rcsi.science/1064-5624/article/view/225417
- DOI: https://doi.org/10.1134/S1064562417060072
- ID: 225417
Cite item
Abstract
The product of m ∈ 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 + δ,δ > 0,. Recent results of Bourgade, Yau, and Yin, of Tao and Vu, and of Nemish are generalized.
About the authors
F. Götze
Bielefeld University
Email: a.naumov@skoltech.ru
Germany, Bielefeld, 33501
A. A. Naumov
Skolkovo Institute of Science and Technology; Institute for Information Transmission Problems
Author for correspondence.
Email: a.naumov@skoltech.ru
Russian Federation, Skolkovo, Moscow oblast, 143025; Moscow, 127994
A. N. Tikhomirov
Komi Scientific Center, Ural Branch
Email: a.naumov@skoltech.ru
Russian Federation, Syktyvkar, 167982