Local Spectral Expansion Approach to High Dimensional Expanders Part II: Mixing and Geometrical overlapping

TL;DR

This paper advances the understanding of high-dimensional expanders by establishing that local spectral expansion in simplicial complexes leads to global mixing and geometric overlapping, with improved bounds on error terms.

Contribution

It proves that local spectral expansion in links implies global mixing and geometric overlapping, providing tighter bounds than previous results.

Findings

Local spectral expansion implies global mixing.

Local spectral expansion implies geometric overlapping.

Tighter bounds on error terms in mixing results.

Abstract

In this paper, we further explore the local-to-global approach for expansion of simplicial complexes that we call local spectral expansion. Specifically, we prove that local expansion in the links imply the global expansion phenomena of mixing and geometric overlapping. Our mixing results also give tighter bounds on the error terms compared to previously known results.