Translator Disclaimer
2017 Local law for the product of independent non-Hermitian random matrices with independent entries
Yuriy Nemish
Electron. J. Probab. 22: 1-35 (2017). DOI: 10.1214/17-EJP38

Abstract

We consider products of independent square non-Hermitian random matrices. More precisely, let $X_1,\ldots ,X_n$ be independent $N\times N$ random matrices with independent entries (real or complex with independent real and imaginary parts) with zero mean and variance $\frac{1} {N}$. Soshnikov-O’Rourke [19] and Götze-Tikhomirov [15] showed that the empirical spectral distribution of the product of $n$ random matrices with iid entries converges to \[ \frac{1} {n\pi }1_{|z|\leq 1}|z|^{\frac{2} {n}-2}dz d\overline{z} .\tag{0.1} \] We prove that if the entries of the matrices $X_1,\ldots ,X_n$ are independent (but not necessarily identically distributed) and satisfy uniform subexponential decay condition, then in the bulk the convergence of the ESD of $X_1\cdots X_n$ to (0.1) holds up to the scale $N^{-1/2+\varepsilon }$.

Citation

Download Citation

Yuriy Nemish. "Local law for the product of independent non-Hermitian random matrices with independent entries." Electron. J. Probab. 22 1 - 35, 2017. https://doi.org/10.1214/17-EJP38

Information

Received: 1 April 2016; Accepted: 6 February 2017; Published: 2017
First available in Project Euclid: 25 February 2017

zbMATH: 06691469
MathSciNet: MR3622892
Digital Object Identifier: 10.1214/17-EJP38

Subjects:
Primary: 60B20

JOURNAL ARTICLE
35 PAGES


SHARE
Vol.22 • 2017
Back to Top