Local circular law for the product of a deterministic matrix with a random matrix

Haokai Xi University of Wisconsin-Madison Fan Yang University of Wisconsin-Madison Jun Yin University of Wisconsin-Madison

Probability mathscidoc: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 suciently 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
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@inproceedings{haokai2016local,
  title={Local circular law for the product of a deterministic matrix with a random matrix},
  author={Haokai Xi, Fan Yang, and Jun Yin},
  url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20160823000319891117390},
  year={2016},
}
Haokai Xi, Fan Yang, and Jun Yin. Local circular law for the product of a deterministic matrix with a random matrix. 2016. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20160823000319891117390.
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