Electronic Journal of Probability

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

Haokai Xi, Fan Yang, and Jun Yin

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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 $\epsilon >0$.

Article information

Electron. J. Probab. Volume 22 (2017), paper no. 60, 77 pp.

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

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Digital Object Identifier

Primary: 15B52: Random matrices
Secondary: 60B20: Random matrices (probabilistic aspects; for algebraic aspects see 15B52) 82B44: Disordered systems (random Ising models, random Schrödinger operators, etc.)

local circular law anisotropic local law deformation universality

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Xi, Haokai; Yang, Fan; Yin, Jun. Local circular law for the product of a deterministic matrix with a random matrix. Electron. J. Probab. 22 (2017), paper no. 60, 77 pp. doi:10.1214/17-EJP76. https://projecteuclid.org/euclid.ejp/1500602612

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