Integral Homology of Random Simplicial Complexes

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→∞\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n\rightarrow \infty $$\end{document}, the first homology group over Z\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb {Z}$$\end{document} vanishes at the very moment when all the edges are covered by triangular faces.