COHOMOLOGY GROUPS OF NON-UNIFORM RANDOM SIMPLICIAL COMPLEXES

We consider a model of a random simplicial complex generated by taking the downward-closure of a non-uniform binomial random hypergraph, in which each set of k + 1 vertices forms an edge with some probability p(k) independently, where pk depends on k and on the number of vertices n. We consider a notion of connectedness on this model according to the vanishing of cohomology groups over an arbitrary abelian group R. We prove that this notion of 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.