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Generalized Random Simplicial Complexes
Christopher F. Fowler · 2015-03-06 · via math.PR updates on arXiv.org

We consider a multi-parameter model for randomly constructing simplicial complexes. This model interpolates between random clique complexes and Linial-Meshulam random $k$-dimensional complexes, two models that have been extensively studied. While these models asymptotically exhibit nontrivial cohomology in only one or two dimensions, we show that in this generalized setting nontrivial cohomology can occur in several dimensions simultaneously. We establish upper and lower thresholds for the appearance of nontrivial cohomology in a particular dimension, and in some instances characterize the behavior at criticality.

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