In this talk, we discuss about a CLT for the linear eigenvalue statistics of non-Hermitian random band matrices. Let is a random matrix with iid entries, and be it's eigenvalue. Define the linear eigenvalue statistics . Rider, and Silverstein '06 had shown that the unscaled and centered with analytic converges to a Gaussian distribution. The variance of the limiting distribution is completely determined by the test function only. We extend that result for non-Hermitian random band matrices. In case of band matrices, the linear eigenvalue statistics needs to be scaled properly. In addition, the variance of the limiting Gaussian distribution depends on the test function, as well as the growth rate of the bandwidth.
Linear eigenvalue statistics of non Hermitian random band matrices