Algebraic and combinatorial expansion in random simplicial complexes

TL;DR

This paper investigates the expansion properties and spectral characteristics of high-dimensional random simplicial complexes, revealing concentration phenomena and conductance bounds that extend understanding of their combinatorial and spectral structure.

Contribution

It establishes concentration results for the spectral gap and Cheeger constant in random simplicial complexes above the cohomological connectivity threshold.

Findings

Spectral gap concentrates around the minimum co-degree of (d-1)-faces.

Cheeger constant is concentrated around the same co-degree.

Conductance of a generalized random walk is bounded away from zero with high probability.

Abstract

In this paper we consider the expansion properties and the spectrum of the combinatorial Laplace operator of a $d$-dimensional Linial-Meshulam random simplicial complex, above the cohomological connectivity threshold. We consider the spectral gap of the Laplace operator and the Cheeger constant as this was introduced by Parzanchevski, Rosenthal and Tessler ($Combinatorica$ 36, 2016). We show that with high probability the spectral gap of the random simplicial complex as well as the Cheeger constant are both concentrated around the minimum co-degree of among all $d-1$-faces. Furthermore, we consider a generalisation of a random walk on such a complex and show that the associated conductance is with high probability bounded away from 0.