Integral homology of random simplicial complexes

TL;DR

This paper investigates the behavior of the first homology group in a random 2-dimensional simplicial complex process, showing it vanishes precisely when all edges are covered by faces as the number of vertices grows large.

Contribution

It establishes a precise probabilistic threshold for the vanishing of the first homology in a random simplicial complex process.

Findings

First homology over  vanishes with high probability at the edge coverage point.

Homology vanishing occurs exactly when all edges are included in faces.

Results hold asymptotically as the number of vertices tends to infinity.

Abstract

The random $2$-dimensional simplicial complex process starts with a complete graph on $n$ vertices, and in every step a new $2$-dimensional face, chosen uniformly at random, is added. We prove that with probability tending to $1$ as $n\to\infty$, the first homology group over $\mathbb Z$ vanishes at the very moment when all the edges are covered by triangular faces.