Large random simplicial complexes, III the critical dimension

Abstract

In this paper, we study the notion of critical dimension of random simplicial complexes in the general multi-parameter model described in [Random simplicial complexes, preprint (2014), arXiv:1412.5805 ; Large random simplicial complexes, I, preprint (2015), arXiv:1503.06285 ; Large random simplical complexes, II, preprint (2015), arXiv:1509.04837 ]. This model includes as special cases the Linial–Meshulam–Wallach model [Homological connectivity of random 2-complexes, Combinatorica 26 (2006) 475–487; Homological connectivity of random [Formula: see text]-complexes, Random Struct. Alogrithms 34 (2009) 408–417.] as well as the clique complexes of random graphs. We characterize 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.