## 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 p_{σ} approach neither 0 nor 1. The medial regime includes as a special case the simplest and most natural assumption that all probability parameters p_{σ} 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 b_{j}(Y) only for k+c<n−j<k+log_{2}k+c^{′} where k=log_{2}lnn 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 proofs employ a new technique, based on Alexander duality, which relates the lower and upper models.

Original language | English |
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Article number | 107065 |

Journal | Topology and its Applications |

Volume | 272 |

DOIs | |

State | Published - 1 Mar 2020 |

Externally published | Yes |

## Keywords

- Alexander duality
- Betti numbers
- Random simplicial complex