Phase transition in cohomology groups of non-uniform random simplicial complexes

Cooley O, Del Giudice N, Kang M, Sprüssel P. 2022. Phase transition in cohomology groups of non-uniform random simplicial complexes. Electronic Journal of Combinatorics. 29(3), P3.27.

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Journal Article | Published | English

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Author
Cooley, OliverISTA; Del Giudice, Nicola; Kang, Mihyun; Sprüssel, Philipp
Department
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.
Publishing Year
Date Published
2022-07-29
Journal Title
Electronic Journal of Combinatorics
Acknowledgement
Supported by Austrian Science Fund (FWF): I3747, W1230.
Volume
29
Issue
3
Article Number
P3.27
eISSN
IST-REx-ID

Cite this

Cooley O, Del Giudice N, Kang M, Sprüssel P. Phase transition in cohomology groups of non-uniform random simplicial complexes. Electronic Journal of Combinatorics. 2022;29(3). doi:10.37236/10607
Cooley, O., Del Giudice, N., Kang, M., & Sprüssel, P. (2022). Phase transition in cohomology groups of non-uniform random simplicial complexes. Electronic Journal of Combinatorics. Electronic Journal of Combinatorics. https://doi.org/10.37236/10607
Cooley, Oliver, Nicola Del Giudice, Mihyun Kang, and Philipp Sprüssel. “Phase Transition in Cohomology Groups of Non-Uniform Random Simplicial Complexes.” Electronic Journal of Combinatorics. Electronic Journal of Combinatorics, 2022. https://doi.org/10.37236/10607.
O. Cooley, N. Del Giudice, M. Kang, and P. Sprüssel, “Phase transition in cohomology groups of non-uniform random simplicial complexes,” Electronic Journal of Combinatorics, vol. 29, no. 3. Electronic Journal of Combinatorics, 2022.
Cooley O, Del Giudice N, Kang M, Sprüssel P. 2022. Phase transition in cohomology groups of non-uniform random simplicial complexes. Electronic Journal of Combinatorics. 29(3), P3.27.
Cooley, Oliver, et al. “Phase Transition in Cohomology Groups of Non-Uniform Random Simplicial Complexes.” Electronic Journal of Combinatorics, vol. 29, no. 3, P3.27, Electronic Journal of Combinatorics, 2022, doi:10.37236/10607.
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2022-08-08
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