Cohomology groups of non-uniform random simplicial complexes

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Oliver Cooley Nicola Del Giudice Mihyun Kang Philipp Sprüssel


We consider a model of a random simplicial complex generated by takingthe downward-closure of a non-uniform binomial random hypergraph, in whicheach set of k+1 vertices forms an edge with some probability pk independently,where pk depends on k and on the number of vertices n. We consider a notion of connectednesson this model according to the vanishing of cohomology groups over an arbitrary abelian group R. We prove that this notionof connectedness displays a phase transition and determine the threshold.We also prove a hitting time result for a natural process interpretation,in which simplices and their downward-closure are added one by one.

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Cooley, O., Del Giudice, N., Kang, M., & Sprüssel, P. (2019). Cohomology groups of non-uniform random simplicial complexes. Acta Mathematica Universitatis Comenianae, 88(3), 553-560. Retrieved from