Random simplicial complexes in the medial regime
Michael Färber, Lewis Mead · Topology and its Applications · 2020
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