Geodesic
Phase transition in cohomology groups of non-uniform random simplicial complexes
Oliver Cooley1 ; Nicola Del Giudice1 ; Mihyun Kang1 ; Philipp Sprüssel
1Graz University of Technology
The electronic journal of combinatorics, Tome 29 (2022) no. 3
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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 $p_k$ 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.
DOI : 10.37236/10607
Classification : 05E45, 05C65, 05C80
Mots-clés : random hypergraph, connectedness

Oliver Cooley  1   ; Nicola Del Giudice  1   ; Mihyun Kang  1   ; Philipp Sprüssel 

1 Graz University of Technology 
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Oliver Cooley; Nicola Del Giudice; Mihyun Kang; Philipp Sprüssel. Phase transition in cohomology groups of non-uniform random simplicial complexes. The electronic journal of combinatorics, Tome 29 (2022) no. 3. doi: 10.37236/10607

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