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/
Cooley, Oliver ;
Del Giudice, Nicola ;
Kang, Mihyun ;
Sprüssel, Philipp
Vanishing of Cohomology Groups of Random Simplicial Complexes (Keynote Speakers)
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: |
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Random hypergraphs, random simplicial complexes, sharp threshold, hitting time, connectedness |
Collection: |
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29th International Conference on Probabilistic, Combinatorial and Asymptotic Methods for the Analysis of Algorithms (AofA 2018) |
Issue Date: |
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2018 |
Date of publication: |
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18.06.2018 |