Integral Homology of Random Simplicial Complexes

Tomasz Łuczak; Yuval Peled

doi:10.1007/s00454-017-9938-z

Integral Homology of Random Simplicial Complexes

Tomasz Łuczak^*, Yuval Peled

^*Corresponding author for this work

Research output: Contribution to journal › Article › peer-review

18 Scopus citations

Abstract

The random 2-dimensional simplicial complex process starts with a complete graph on n vertices, and in every step a new 2-dimensional face, chosen uniformly at random, is added. We prove that with probability tending to 1 as n→ ∞, the first homology group over Z vanishes at the very moment when all the edges are covered by triangular faces.

Original language	American English
Pages (from-to)	131-142
Number of pages	12
Journal	Discrete and Computational Geometry
Volume	59
Issue number	1
DOIs	https://doi.org/10.1007/s00454-017-9938-z
State	Published - 1 Jan 2018

Bibliographical note

Funding Information:
Acknowledgements This work was carried out when TŁ visited the Institute for Mathematical Research (FIM) of ETH Zürich. He would like to thank FIM for the hospitality and for creating a stimulating research environment. TŁ partially supported by NCN Grant 2012/06/A/ST1/00261. YP is grateful to the Azrieli foundation for the award of an Azrieli fellowship.

Publisher Copyright:
© 2017, Springer Science+Business Media, LLC.

Keywords

Hitting time
Homology Shadow
Random simplicial complexes

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AU - Peled, Yuval

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N2 - The random 2-dimensional simplicial complex process starts with a complete graph on n vertices, and in every step a new 2-dimensional face, chosen uniformly at random, is added. We prove that with probability tending to 1 as n→ ∞, the first homology group over Z vanishes at the very moment when all the edges are covered by triangular faces.

AB - The random 2-dimensional simplicial complex process starts with a complete graph on n vertices, and in every step a new 2-dimensional face, chosen uniformly at random, is added. We prove that with probability tending to 1 as n→ ∞, the first homology group over Z vanishes at the very moment when all the edges are covered by triangular faces.

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Integral Homology of Random Simplicial Complexes

Abstract

Bibliographical note

Keywords

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