Large Random Simplicial Complexes, III; The Critical Dimension

TL;DR

This paper investigates the critical dimension of random simplicial complexes across a multi-parameter model, linking it to topological invariants like Betti numbers and fundamental groups, and extends previous models with new geometric insights.

Contribution

It characterizes the critical dimension in terms of geometric and topological properties within a general multi-parameter framework, broadening understanding of random simplicial complexes.

Findings

Characterization of critical dimension via Betti numbers and fundamental group

Analysis of minimal cycles and simplex degrees in relation to critical dimension

Extension of models to include clique complexes and other special cases

Abstract

In this paper we study the notion of critical dimension of random simplicial complexes in the general multi-parameter model described in our previous papers of this series. This model includes as special cases the Linial-Meshulam-Wallach model as well as the clique complexes of random graphs. We characterise the concept of critical dimension in terms of various geometric and topological properties of random simplicial complexes such as their Betti numbers, the fundamental group, the size of minimal cycles and the degrees of simplexes. We mention in the text a few interesting open questions.