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

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 \ 1 n 1|ₙ| ₁|z|^2 {n-2}dz dz. 0. 1 \ 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+.