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


Author(s): Nemish, Yuriy
Article Title: Local law for the product of independent non-Hermitian random matrices with independent entries
Affiliation IST Austria
Abstract: We consider products of independent square non-Hermitian 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. 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 (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: 1083-6489
Publisher: Institute of Mathematical Statistics  
Date Published: 2017-01-01
Start Page: 1
End Page: 35
URL:
DOI: 10.1214/17-EJP38
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)
IST Austria Authors
  1. Yuriy Nemish
    1 Nemish