Spectral gap bounds for the simplicial Laplacian and an application to random complexes

TL;DR

This paper establishes new spectral gap bounds for the simplicial Laplacian by comparing complexes, extending previous results, and applies these bounds to analyze cohomology thresholds in Erdős-Rényi random graph complexes.

Contribution

It introduces two generalized spectral gap bounds for simplicial complexes and applies them to improve bounds on cohomology vanishing thresholds in random complexes.

Findings

Improved upper bounds for cohomology thresholds with a logarithmic factor.

Enhanced lower bounds for cohomology thresholds using probabilistic estimates.

Generalization of spectral gap bounds beyond clique complexes.

Abstract

In this article, we derive two spectral gap bounds for the reduced Laplacian of a general simplicial complex. Our two bounds are proven by comparing a simplicial complex in two different ways with a larger complex and with the corresponding clique complex respectively. Both of these bounds generalize the result of Aharoni et al. (2005) \cite{ABM} which is valid only for clique complexes. As an application, we investigate the thresholds for vanishing of cohomology of the neighborhood complex of the Erd\"{o}s-R\'enyi random graph. We improve the upper bound derived in Kahle (2007) \cite{kahle} by a logarithmic factor using our spectral gap bounds and we also improve the lower bound via finer probabilistic estimates than those in Kahle (2007) \cite{kahle}.