Affiliation:
1. Institute for Advanced Study Kyoto University Kyoto Japan
2. Department of Mathematics The Ohio State University Columbus Ohio USA
3. Global Center for Science and Engineering Waseda University Tokyo Japan
Abstract
AbstractWe study the adjacency matrix of the Linial–Meshulam complex model, which is a higher‐dimensional generalization of the Erdős–Rényi 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 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.
Funder
Japan Society for the Promotion of Science
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