Work

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

Year: 2024

Type: article

Abstract: We 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.

Source: Random Structures and Algorithms

DOI: 10.1002/rsa.21257

Authors: Shu Kanazawa, Khanh Duy Trinh

Institutions: Kyoto University, The Ohio State University, Waseda University

Language: English

Topic: Random Matrices and Applications

Subfield: Statistics and Probability

Field: Mathematics

Domain: Physical Sciences

Open Access status: green

Funder: Japan Society for the Promotion of Science

Award: Randomization of persistent homology and its interaction with Schubert calculus