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 [8], [9], [10]. This model includes as special cases the Linial-MeshulamWallach model [19], [20] as well as the clique complexes of random graphs. We characterise 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.

Random simplicial complexes, duality and the critical dimension

“…It was shown that the Betti numbers have a very specific pattern which can be described using the notion of a critical dimension k * . Roughly, it was established in [5] that homologicaly a random simplicial complex in the lower model can be well approximated by a wedge of spheres of dimension k * .…”

“…In the present paper we study two very general probabilistic models generating random simplicial complexes of arbitrary dimension which we call the lower and upper models. random simplicial complexes was studied in a series of papers [2], [3], [4], [5] under the name of multiparameter random simplicial complexes; the name reflects the fact that the geometric and topological properties of simplicial complexes in this model depend on the set of probability parameters p 0 , p 1 , . .…”

“…In a recent paper [5] the authors established an interesting pattern of behaviour of the Betti numbers of random simplicial complexes in the lower model. It was shown that there exists a specific dimension k (called the critical dimension) such that the Betti number b k (Y ) is large, the Betti numbers b j (Y ) vanish for 0 < j < k and are significantly smaller than b k (Y ) for j > k. In other words, a random simplicial complex in the lower model can be approximated (homologicaly) by a wedge of spheres of the critical dimension.…”

Random simplicial complexes in the medial regime

“…Note that most random complexes which appear in literature are homogeneous. For example the multi-parameter random simplicial complexes of [6], [7], [8], [9] are homogeneous lower random simplicial complexes.…”

“…Consider the dual system of probability parameters p ′ σ = 1 − pσ (see (10)) which satisfies 0 < e −A ′ = 1 − P ≤ p ′ σ ≤ 1 − p = e −a ′ < 1, where a ′ and A ′ are defined in (13). Next, we use the isomorphism c of Theorem 2.2 and the duality for the Betti numbers (9). The complex c(Y ) is a random simplicial complex in the lower model with respect to the system of probability parameters p To prove the third statement we observe that the reduced Betti numbers of c(Y ) vanish in all dimensions except possibly log 2 ln n − log 2 A ′ − 1 − δ 0 < j ≤ log 2 ln n + log 2 log 2 ln n − log 2 a ′ − 1 + ǫ 0 , Since b j (Y ) = b n−2−j (c(Y )) (cf.…”