Integral Homology of Random Simplicial Complexes

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 $$n\rightarrow \infty $$ , the first homology group over $$\mathbb {Z}$$ vanishes at the very moment when all the edges are covered by triangular faces.