Random simplicial complexes in the medial regime

Abstract

We describe topology of random simplicial complexes in the lower and upper models in the medial regime, i.e. under the assumption that the probability parameters pσ approach neither 0 nor 1. The medial regime includes as a special case the simplest and most natural assumption that all probability parameters pσ are equal to each other and are independent of n. We show that nontrivial Betti numbers of typical lower and upper random simplicial complexes in the medial regime lie in a narrow range of dimensions. For instance, an upper random simplicial complex Y on n vertices in the medial regime with high probability has non-vanishing Betti numbers bj(Y) only for k+c<n−j<k+log2⁡k+c′ where k=log2⁡ln⁡n and c,c′ are constants. A lower random simplicial complex on n vertices in the medial regime is, with high probability, (k+a)-connected and its dimension d satisfies d∼k+log2⁡k+a′ where a,a′ are constants. The proofs employ a new technique, based on Alexander duality, which relates the lower and upper models.