License: Creative Commons Attribution 3.0 Unported license (CC BY 3.0)
When quoting this document, please refer to the following
DOI: 10.4230/LIPIcs.AofA.2018.7
URN: urn:nbn:de:0030-drops-89006
URL: http://dagstuhl.sunsite.rwth-aachen.de/volltexte/2018/8900/
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Cooley, Oliver ; Del Giudice, Nicola ; Kang, Mihyun ; Sprüssel, Philipp

Vanishing of Cohomology Groups of Random Simplicial Complexes (Keynote Speakers)

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Abstract

We consider k-dimensional random simplicial complexes that are generated from the binomial random (k+1)-uniform hypergraph by taking the downward-closure, where k >= 2. For each 1 <= j <= k-1, we determine when all cohomology groups with coefficients in F_2 from dimension one up to j vanish and the zero-th cohomology group is isomorphic to F_2. This property is not monotone, but nevertheless we show that it has a single sharp threshold. Moreover, we prove a hitting time result, relating the vanishing of these cohomology groups to the disappearance of the last minimal obstruction. Furthermore, we study the asymptotic distribution of the dimension of the j-th cohomology group inside the critical window. As a corollary, we deduce a hitting time result for a different model of random simplicial complexes introduced in [Linial and Meshulam, Combinatorica, 2006], a result which has only been known for dimension two [Kahle and Pittel, Random Structures Algorithms, 2016].

BibTeX - Entry

@InProceedings{cooley_et_al:LIPIcs:2018:8900,
  author =	{Oliver Cooley and Nicola Del Giudice and Mihyun Kang and Philipp Spr{\"u}ssel},
  title =	{{Vanishing of Cohomology Groups of Random Simplicial Complexes (Keynote Speakers)}},
  booktitle =	{29th International Conference on Probabilistic,  Combinatorial and Asymptotic Methods for the Analysis of Algorithms  (AofA 2018)},
  pages =	{7:1--7:14},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-078-1},
  ISSN =	{1868-8969},
  year =	{2018},
  volume =	{110},
  editor =	{James Allen Fill and Mark Daniel Ward},
  publisher =	{Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{http://drops.dagstuhl.de/opus/volltexte/2018/8900},
  URN =		{urn:nbn:de:0030-drops-89006},
  doi =		{10.4230/LIPIcs.AofA.2018.7},
  annote =	{Keywords: Random hypergraphs, random simplicial complexes, sharp threshold, hitting time, connectedness}
}

Keywords: Random hypergraphs, random simplicial complexes, sharp threshold, hitting time, connectedness
Collection: 29th International Conference on Probabilistic, Combinatorial and Asymptotic Methods for the Analysis of Algorithms (AofA 2018)
Issue Date: 2018
Date of publication: 18.06.2018

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