Let Δn-1 denote the (n - 1)-dimensional simplex. Let Y be a random d-dimensional subcomplex of Δn-1 obtained by starting with the full (d - 1)-dimensional skeleton of Δn-1 and then adding each d-simplex independently with probability p = c/n. We compute an explicit constant γd, with γ2 ≃ 2. 45, γ3 ≃ 3.5, and γd = Θ (log d) as d → ∞, so that for c < γd such a random simplicial complex either collapses to a (d - 1)-dimensional subcomplex or it contains ∂ Δd+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 γd < cd < d + 1 such that for any c > cd and a fixed field F, asymptotically almost surely Hd(Y;F) ≠ 0.
Bibliographical noteFunding 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).
- Random complexes
- Simplicial homology