CLT for non-Hermitian random band matrices with variance profiles

Abstract

We show that the fluctuations of the linear eigenvalue statistics of a non-Hermitian random band matrix of increasing bandwidth $b_n$ with a continuous variance profile $w_{\nu }(x)$ converges to a $N(0,{\sigma }_f^2(\nu ))$, where $\nu ={\lim }_{n\to \infty }(2b_n/n)\in [0,1]$ and $f$ is the test function. When $\nu \in (0,1]$, we obtain an explicit formula for ${\sigma }_f^2(\nu )$, which depends on $f$, and variance profile $w_{\nu }$. When $\nu =1$, the formula is consistent with Rider and Silverstein (2006) \cite{rider2006gaussian}. We also independently compute an explicit formula for ${\sigma }_f^2(0)$ i.e., when the bandwidth $b_n$ grows slower compared to $n$. In addition, we show that ${\sigma }_f^2(\nu )\to {\sigma }_f^2(0)$ as $\nu \downarrow 0$.

Typos corrected; a few more explanations and a couple of pictures have been added

References in corpus (5)

• Central limit theorem for linear eigenvalue statistics of random matrices with independent entries
• The dimension-free structure of nonhomogeneous random matrices
• Non-Hermitian random matrices with a variance profile (I): Deterministic equivalents and limiting ESDs
• Circular Law for Random Block Band Matrices with Genuinely Sublinear Bandwidth
• On Spectral Norm of Large Band Random Matrices

Cited by in corpus (2)

• Spectrum of random centrosymmetric matrices; CLT and Circular law
• Circular Law for Random Block Band Matrices with Genuinely Sublinear Bandwidth