# MathSciDoc: An Archive for Mathematician ∫

#### Probabilitymathscidoc:1608.28001

2016
It is well known that the spectral measure of eigenvalues of a rescaled square non-Hermitian random matrix with independent entries satis es the circular law. We consider the product $T X$, where $T$ is a deterministic $N\times M$ matrix and $X$ is a random $M\times  N$ matrix with independent entries having zero mean and variance $(N \wedge M) ^{-1}$. We prove a general local circular law for the empirical spectral distribution (ESD) of $TX$ at any point $z$ away from the unit circle under the assumptions that$N \sim M$, and the matrix entries $X_{ij} have suciently high moments. More precisely, if$z$satisfes$| |z| - 1| \ge \tau$for arbitrarily small$\tau > 0, the ESD of $TX$ converges to $\tilde{X}_ D (z) d A(z)$, where $\tilde{X}_D$ is a rotation-invariant function determined by the singular valuau of $T$ and $dA$ denotes the Lebesgue measure on $\mathbb{C}$. The local circular law is valid around $z$ up to scale $(N \wedge M) ^{-1/4+\epsilon}$ for any  $\epsilon >0$. Moreover, if $|z| >1$ or the matrix entries of $X$ have vanishing third moments, the local circular law is valid around $z$ up to scale $(N \wedge M) ^{-1/2+\epsilon}$ for any  $\epsilon >0$.
Local circular law, deterministic matrix
@inproceedings{haokai2016local,