CLT for non-Hermitian random band matrices with variance profiles

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 $\nu\downarrow 0$.