@article{11740,
abstract = {We consider a generalised model of a random simplicial complex, which arises from a random hypergraph. Our model is generated by taking the downward-closure of a non-uniform binomial random hypergraph, in which for each k, each set of k+1 vertices forms an edge with some probability pk independently. As a special case, this contains an extensively studied model of a (uniform) random simplicial complex, introduced by Meshulam and Wallach [Random Structures & Algorithms 34 (2009), no. 3, pp. 408–417].
We consider a higher-dimensional notion of connectedness on this new model according to the vanishing of cohomology groups over an arbitrary abelian group R. We prove that this notion of 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. In addition, we determine the asymptotic behaviour of cohomology groups inside the critical window around the time of the phase transition.},
author = {Cooley, Oliver and Del Giudice, Nicola and Kang, Mihyun and Sprüssel, Philipp},
issn = {1077-8926},
journal = {Electronic Journal of Combinatorics},
number = {3},
publisher = {Electronic Journal of Combinatorics},
title = {{Phase transition in cohomology groups of non-uniform random simplicial complexes}},
doi = {10.37236/10607},
volume = {29},
year = {2022},
}