Phase transition in cohomology groups of non-uniform random simplicial complexes
Cooley, Oliver
Del Giudice, Nicola
Kang, Mihyun
Sprüssel, Philipp
ddc:510
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.
Electronic Journal of Combinatorics
2022
info:eu-repo/semantics/article
doc-type:article
text
http://purl.org/coar/resource_type/c_6501
https://research-explorer.ista.ac.at/record/11740
https://research-explorer.ista.ac.at/download/11740/11742
Cooley O, Del Giudice N, Kang M, Sprüssel P. Phase transition in cohomology groups of non-uniform random simplicial complexes. <i>Electronic Journal of Combinatorics</i>. 2022;29(3). doi:<a href="https://doi.org/10.37236/10607">10.37236/10607</a>
eng
info:eu-repo/semantics/altIdentifier/doi/10.37236/10607
info:eu-repo/semantics/altIdentifier/issn/1077-8926
info:eu-repo/semantics/altIdentifier/wos/000836200300001
info:eu-repo/semantics/altIdentifier/arxiv/2005.07103
info:eu-repo/semantics/openAccess