Cohomology groups of non-uniform random simplicial complexes

Abstract

We consider a model of a random simplicial complex generated by takingthe downward-closure of a non-uniform binomial random hypergraph, in whicheach set of k+1 vertices forms an edge with some probability pk independently,where pk depends on k and on the number of vertices n. We consider a notion of connectednesson this model according to the vanishing of cohomology groups over an arbitrary abelian group R. We prove that this notionof connectedness displays a phase transition and determine the threshold.We also prove a hitting time result for a natural process interpretation,in which simplices and their downward-closure are added one by one.