Central limit theorem for linear eigenvalue statistics of the adjacency matrices of random simplicial complexes

Abstract: We study the adjacency matrix of the Linial-Meshulam complex model, which is a higher-dimensional generalization of the Erd\H{o}s-R\'enyi graph model. Recently, Knowles and Rosenthal proved that the empirical spectral distribution of the adjacency matrix is asymptotically given by Wigner's semicircle law in a diluted regime. In this article, we prove a central limit theorem for the linear eigenvalue statistics for test functions of polynomial growth that is of class $C^{2}$ on a closed interval. The proof is based on higher-dimensional combinatorial enumerations and concentration properties of random symmetric matrices. Furthermore, when the test function is a polynomial function, we obtain the explicit formula for the variance of the limiting Gaussian distribution.