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. 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 + log 2 k + c ′ where k = log 2 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 + log 2 k + a ′ where a, a ′ are constants. The paper develops a new technique, based on Alexander duality, which relates the lower and upper models.

Ample simplicial complexes

“…Then all vertexes of , other than v, w, lie in the intersection lk (v) ∩ lk (w). Using the above arguments, we see that in this case V ( ) 6, which contradicts our assumption V ( ) 19.…”

“…One of the results of [21] states that the geometric realisation of the Rado complex is homeomorphic to the infinite dimensional simplex and hence it is contractible. A related mathematical object is the medial regime random simplicial complex [19] which, as we show in this paper, is r -ample, with probability tending to one.…”

“…First we recall a model of random simplicial complexes developed in [19,20]; it is a generalisation of the well-known Linial-Meshulam model.…”

“…where p ∈ (0, 1/2] does not depend on n. (Note however that in Remark 5.2 we relax this assumption.) Paper [19] contains information about geometry and topology of medial regime random simplicial complexes; in particular, it is shown that a random simplicial complex in the medial regime is simply connected and has vanishing Betti numbers below dimension ∼ log log n, asymptotically almost surely.…”

“…It was proven in [19] that the medial regime random simplicial complex is simply connected and has vanishing Betti numbers in dimensions log 2 ln n. For these reasons one expects that any r -ample simplicial complex is highly connected, for large r . This question is discussed below in Sect.…”