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

Abstract: We consider products of independent square non-Hermitian random matrices. More precisely, let X(1),...,X(n) be random matrices with independent entries (real or complex with independent real and imaginary parts) with zero mean and variance 1/N. Soshnikov and O'Rourke showed that the empirical spectral distribution of the product X(1)X(2)..X(n) converges to the n-th power of the circular law. We prove that if the entries of the matrices X(1),...,X(n) satisfy uniform subexponential decay condition, then in the bulk the convergence of the ESD holds up to the optimal scale.