Author(s):

Nemish, Yuriy

Article Title: 
Local law for the product of independent nonHermitian random matrices with independent entries

Affiliation 
IST Austria 
Abstract: 
We consider products of independent square nonHermitian random matrices. More precisely, let X1,…, Xn be independent N × N random matrices with independent entries (real or complex with independent real and imaginary parts) with zero mean and variance 1/N. SoshnikovO’Rourke [19] and GötzeTikhomirov [15] showed that the empirical spectral distribution of the product of n random matrices with iid entries converges to (equation found). We prove that if the entries of the matrices X1,…, Xn are independent (but not necessarily identically distributed) and satisfy uniform subexponential decay condition, then in the bulk the convergence of the ESD of X1,…, Xn to (0.1) holds up to the scale N–1/2+ε.

Keywords: 
Local law; Circular law; Random matrices; Stieltjes transform

Journal Title:

Electronic Journal of Probability

Volume: 
22

ISSN:

10836489

Publisher:

Institute of Mathematical Statistics

Date Published:

20170101

Start Page: 
1

End Page:

35

URL: 

DOI: 
10.1214/17EJP38

Notes: 
I would like to thank my PhD advisors Mireille Capitaine and Michel Ledoux for introducing the problem to me, fruitful discussions and careful reading of the manuscript. I’m also grateful to the anonymous referee for numerous valuable remarks.

Open access: 
yes (OA journal) 