CLT for Non-Hermitian Random Band Matrices with Variance Profiles

Abstract

We study the linear eigenvalue statistics of a non-Hermitian random band matrix with a continuous variance profile \(w_{\nu }(x)\) and increasing bandwidth \(b_{n}\). We show that the fluctuations of the linear eigenvalue statistics converges to \(N(0,\sigma _{f}^{2}(\nu ))\), where \(\nu =\lim _{n\rightarrow \infty }(2b_{n}/n)\in [0,1]\) and f is an analytic test function. We obtain explicit formulae of \(\sigma _{f}^{2}(\nu )\) in two different cases, namely when \(\nu \in (0,1]\) and when \(\nu = 0\). In addition, we show that \(\sigma _{f}^{2}(\nu )\rightarrow \sigma _{f}^{2}(0)\) as \(\nu \downarrow 0\). In particular by setting \(\nu =1\), we obtain the result for full non-Hermitian matrices with a constant variance profile, which was previously found by Rider and Silverstein (Ann Probab 34:2118–2143, 2006).