Abstract
It is well known that the spectral measure of eigenvalues of a rescaled square non-Hermitian random matrix with independent entries satisfies the circular law. In this paper, we consider the product $TX$, 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 sufficiently high moments. More precisely, if $z$ satisfies $||z|-1|\ge \tau $ for arbitrarily small $\tau >0$, the ESD of $TX$ converges to $\tilde \chi _{\mathbb D}(z) dA(z)$, where $\tilde \chi _{\mathbb D}$ is a rotation-invariant function determined by the singular values 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 $\epsilon >0$.
Highlights
If |z| ą 1 or the matrix entries of X have vanishing third moments, the local circular law is valid around z up to scale pN ^ M q1{2` for any ą 0
Bai [1, 2] analyzed the empirical spectral distribution (ESD) of pXzq:pXzq through its Stieltjes transform and handled the logarithmic singularity by assuming bounded density and bounded high moments for the entries of X
The final result was presented in [38], where the circular law is proved under the optimal L2 assumption. These papers studied the circular law in the global regime, i.e. the convergence of ESD on subsets containing ηN eigenvalues for some small constant η ą 0
Summary
For the Gaussian random matrix with real entries, the joint distribution of the eigenvalues is more complicated but still integrable, which leads to a proof of the circular law as well [6, 10, 18, 35]. As remarked around (1.3), (B) implies the anisotropic local law for a Gaussian X and a general T Based on this fact we perform Step (C) using a self-consistent comparison argument in [24]. |z| ă 1), we need the optimal averaged local law up to the scale η " pN ^ M q1, which can be obtained under the extra assumption that the entries of X have vanishing third moments. If AN is a matrix, we use the notations AN “ OpBN q and AN “ opBN q to mean }AN } “ OpBN q and }AN } “ opBN q, respectively
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