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

TL;DR

This paper establishes a central limit theorem for linear eigenvalue statistics of adjacency matrices in a higher-dimensional random complex model, extending spectral analysis beyond traditional graph models.

Contribution

It proves a CLT for eigenvalue statistics of the adjacency matrix of the Linial-Meshulam complex, a higher-dimensional generalization of Erdős-Rényi graphs.

Findings

Spectral distribution follows Wigner's semicircle law in a diluted regime.

Established CLT for polynomial growth test functions.

Derived explicit variance formula for polynomial test functions.

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.