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

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