## Abstract

Let Δ_{n-1} denote the (n - 1)-dimensional simplex. Let Y be a random d-dimensional subcomplex of Δ_{n-1} obtained by starting with the full (d - 1)-dimensional skeleton of Δ_{n-1} and then adding each d-simplex independently with probability p = c/n. We compute an explicit constant γ_{d}, with γ_{2} ≃ 2. 45, γ_{3} ≃ 3.5, and γ_{d} = Θ (log d) as d → ∞, so that for c < γ_{d} such a random simplicial complex either collapses to a (d - 1)-dimensional subcomplex or it contains ∂ Δ_{d+1}, the boundary of a (d + 1)-dimensional simplex. We conjecture this bound to be sharp. In addition, we show that there exists a constant γ_{d} < c_{d} < d + 1 such that for any c > c_{d} and a fixed field F, asymptotically almost surely H_{d}(Y;F) ≠ 0.

Original language | American English |
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Pages (from-to) | 317-334 |

Number of pages | 18 |

Journal | Discrete and Computational Geometry |

Volume | 49 |

Issue number | 2 |

DOIs | |

State | Published - Mar 2013 |

### Bibliographical note

Funding Information:N. Linial was supported by ISF and BSF grants; T. Łuczak was supported by the Foundation for Polish Science; and R. Meshulam was supported by ISF grant with additional partial support from ERC Advanced Research Grant no. 267165 (DISCONV).

## Keywords

- Collapsibility
- Random complexes
- Simplicial homology