probabilistic topology

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We exhibit a sharp threshold for vanishing of rational cohomology in random flag complexes, providing a generalization of the Erdős–Rényi theorem. As a corollary, almost all $d$-dimensional flag complexes have nontrivial (rational, reduced) homology only in middle degree $\lfloor d/2 \rfloor$.

We describe a natural topological generalization of edge expansion for graphs to regular CW complexes and prove that this property holds with high probability for certain random complexes.

There have been several recent articles studying homology of various types of random simplicial complexes. Some theorems have concerned thresholds for vanishing of homology, and in others estimates for the expectations of the Betti numbers. However little seems known so far about limiting distributions. In this article we establish Poisson and normal approximation theorems for Betti numbers of different kinds of random simplicial complex: Erdos-Renyi random clique complexes, random Vietoris-Rips complexes, and random Cech complexes. These results may be of practical interest in topological data analysis.