Electronic Journal of Probability

Local law for the product of independent non-Hermitian random matrices with independent entries

Yuriy Nemish

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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 $N^{-1/2+\varepsilon }$.

Article information

Source
Electron. J. Probab., Volume 22 (2017), paper no. 22, 35 pp.

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

Permanent link to this document
https://projecteuclid.org/euclid.ejp/1487991681

Digital Object Identifier
doi:10.1214/17-EJP38

Mathematical Reviews number (MathSciNet)
MR3622892

Zentralblatt MATH identifier
06691469

Subjects
Primary: 60B20: Random matrices (probabilistic aspects; for algebraic aspects see 15B52)

Keywords
random matrices circular law Stieltjes transform local law

Rights
Creative Commons Attribution 4.0 International License.

Citation

Nemish, Yuriy. Local law for the product of independent non-Hermitian random matrices with independent entries. Electron. J. Probab. 22 (2017), paper no. 22, 35 pp. doi:10.1214/17-EJP38. https://projecteuclid.org/euclid.ejp/1487991681


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