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

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. 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 $\epsilon >0$.

80 pages, 7 figures

References in corpus (3)

• Local semicircle law and complete delocalization for Wigner random matrices
• The Tracy--Widom limit for the largest eigenvalues of singular complex Wishart matrices
• The smallest singular value of deformed random rectangular matrices

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