CLT for LES of correlated Non-Hermitian Random Matrices

We consider two $n\times n$ non-Hermitian random matrices such that the $ij$th entry of one matrix is correlated with the $ij$th entry of the other matrix. However, the entries of any particular matrix are i.i.d. random variables. We study the asymptotic behavior of the combined spectrum, and the limit of the linear eigenvalue statistic defined on the combined spectrum. We show that if the random variables are centered with variance $1/n$ and having finite moments, then the centered \textit{Linear Eigenvalue Statistics} (LESs) converge jointly to a bivariate Gaussian distribution. We assumed that the test function used in the LES belongs to Sobolev $H^{2+\delta}$ space. The variance of the limiting Gaussian distribution depends on correlation structure of the matrix entries and the fourth order mixed cumulants of the matrix entries. This generalizes the previous results by Rider, Silverstein (2006), Cipolloni, Erd\H{o}s, Schr\"oder (2023). In particular, we obtain the limiting LES of random centrosymmetric matrices.