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2017 Local circular law for the product of a deterministic matrix with a random matrix
Haokai Xi, Fan Yang, Jun Yin
Electron. J. Probab. 22: 1-77 (2017). DOI: 10.1214/17-EJP76

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 $\epsilon >0$.

Citation

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Haokai Xi. Fan Yang. Jun Yin. "Local circular law for the product of a deterministic matrix with a random matrix." Electron. J. Probab. 22 1 - 77, 2017. https://doi.org/10.1214/17-EJP76

Information

Received: 5 June 2016; Accepted: 15 June 2017; Published: 2017
First available in Project Euclid: 21 July 2017

zbMATH: 1373.15058
MathSciNet: MR3683369
Digital Object Identifier: 10.1214/17-EJP76

Subjects:
Primary: 15B52
Secondary: 60B20, 82B44

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