Eigenvalue confinement and spectral gap for random simplicial complexes

TL;DR

This paper analyzes the spectral properties of the adjacency operator in random simplicial complexes, establishing eigenvalue confinement, spectral gap estimates, and the semicircle law for eigenvalue distribution.

Contribution

It provides the first eigenvalue confinement and spectral gap estimates for the Linial-Meshulam model, extending random matrix theory to dependent entries in simplicial complexes.

Findings

Spectral gap is approximately $np - 2\

Eigenvalues follow the semicircle law asymptotically.

Eigenvalue confinement holds under certain probabilistic conditions.

Abstract

We consider the adjacency operator of the Linial-Meshulam model for random simplicial complexes on $n$ vertices, where each $d$-cell is added independently with probability $p$ to the complete $(d-1)$-skeleton. Under the assumption $np(1-p) \gg \log^4 n$, we prove that the spectral gap between the $\binom{n-1}{d}$ smallest eigenvalues and the remaining $\binom{n-1}{d-1}$ eigenvalues is $np - 2\sqrt{dnp(1-p)} \, (1 + o(1))$ with high probability. This estimate follows from a more general result on eigenvalue confinement. In addition, we prove that the global distribution of the eigenvalues is asymptotically given by the semicircle law. The main ingredient of the proof is a F\"uredi-Koml\'os-type argument for random simplicial complexes, which may be regarded as sparse random matrix models with dependent entries.