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?(x) and increasing bandwidth bn. We show that the fluctuations of the linear eigenvalue statistics converges to N(0,?f2(?)), where ?= lim n??(2 bn/ n) ? [0 , 1] and f is an analytic test function. We obtain explicit formulae of ?f2(?) in two different cases, namely when ?? (0 , 1] and when ?= 0. In addition, we show that ?f2(?)??f2(0) as ?? 0. In particular by setting ?= 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). � 2022, The Author(s), under exclusive licence to Springer Science+Business Media, LLC, part of Springer Nature.