Local spectral expansion approach to high dimensional expanders part I: Descent of spectral gaps

TL;DR

This paper introduces local spectral expansion for simplicial complexes, showing it implies spectral gaps in links and global Laplacians, advancing understanding of high-dimensional expanders.

Contribution

It defines local spectral expansion for simplicial complexes and demonstrates its implications for spectral gaps in links and global Laplacians.

Findings

Local spectral expansion leads to spectral gaps in links.

It establishes a connection between local and global spectral properties.

The approach provides a new framework for analyzing high-dimensional expanders.

Abstract

This paper introduces the notion of local spectral expansion of a simplicial complex as a possible analogue of spectral expansion defined for graphs. We then show that the condition of local spectral expansion for a complex yields various spectral gaps in both the links of the complex and the global Laplacians of the complex.