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• W3100145254 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 $Ntimes M$ matrix and $X$ is a random $Mtimes N$ matrix with independent entries having zero mean and variance $(Nwedge M)^{-1}$. 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 $Nsim M$, 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 $(Nwedge M)^{-1/4+epsilon }$ 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 $(Nwedge M)^{-1/2+epsilon }$ for any $epsilon >0$." @default.
• W3100145254 created "2020-11-23" @default.
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• W3100145254 date "2017-01-01" @default.
• W3100145254 modified "2023-09-26" @default.
• W3100145254 title "Local circular law for the product of a deterministic matrix with a random matrix" @default.
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• W3100145254 doi "https://doi.org/10.1214/17-ejp76" @default.
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