On Simple Connectivity of Random 2-Complexes

Zur Luria, Yuval Peled*

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

1 Scopus citations

Abstract

The fundamental group of the 2-dimensional Linial–Meshulam random simplicial complex Y2(n, p) was first studied by Babson, Hoffman, and Kahle. They proved that the threshold probability for simple connectivity of Y2(n, p) is about p≈ n- 1 / 2. In this paper, we show that this threshold probability is at most p≤ (γn) - 1 / 2, where γ= 4 4/ 3 3, and conjecture that this threshold is sharp. In fact, we show that p= (γn) - 1 / 2 is a sharp threshold probability for the stronger property that every cycle of length 3 is the boundary of a subcomplex of Y2(n, p) that is homeomorphic to a disk. Our proof uses the Poisson paradigm, and relies on a classical result of Tutte on the enumeration of planar triangulations.

Original languageAmerican English
Pages (from-to)17-32
Number of pages16
JournalDiscrete and Computational Geometry
Volume67
Issue number1
DOIs
StatePublished - Jan 2022

Bibliographical note

Publisher Copyright:
© 2021, The Author(s), under exclusive licence to Springer Science+Business Media, LLC, part of Springer Nature.

Keywords

  • Poisson paradigm
  • Simple connectivity
  • Simplicial complexes
  • Threshold probability
  • Triangulations

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