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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

The threshold function for vanishing of the top homology group of random $ d$-complexes


Author: Dmitry N. Kozlov
Journal: Proc. Amer. Math. Soc. 138 (2010), 4517-4527
MSC (2010): Primary 55U10; Secondary 60B99
Published electronically: July 28, 2010
MathSciNet review: 2680076
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Abstract: For positive integers $ n$ and $ d$, and the probability function $ 0\leq p(n)\leq 1$, we let $ Y_{n,p,d}$ denote the probability space of all at most $ d$-dimensional simplicial complexes on $ n$ vertices, which contain the full $ (d-1)$-dimensional skeleton, and whose $ d$-simplices appear with probability $ p(n)$. In this paper we determine the threshold function for vanishing of the top homology group in $ Y_{n,p,d}$, for all $ d\geq 1$.


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Additional Information

Dmitry N. Kozlov
Affiliation: Department of Mathematics, University of Bremen, 28334 Bremen, Federal Republic of Germany
Email: dfk@math.uni-bremen.de

DOI: http://dx.doi.org/10.1090/S0002-9939-2010-10596-8
PII: S 0002-9939(2010)10596-8
Keywords: Random simplicial complexes, threshold function, second moment method, homology.
Received by editor(s): October 20, 2009
Published electronically: July 28, 2010
Additional Notes: This research was supported by the University of Bremen as part of AG CALTOP
Communicated by: Jim Haglund
Article copyright: © Copyright 2010 American Mathematical Society