Random simplicial complexes, duality and the critical dimension

Michael Farber, Lewis Mead, Tahl Nowik

Research output: Contribution to journalArticlepeer-review

3 Scopus citations


In this paper, we discuss two general models of random simplicial complexes which we call the lower and the upper models. We show that these models are dual to each other with respect to combinatorial Alexander duality. The behavior of the Betti numbers in the lower model is characterized by the notion of critical dimension, which was introduced by Costa and Farber in [Large random simplicial complexes III: The critical dimension, J. Knot Theory Ramifications 26 (2017) 1740010]: random simplicial complexes in the lower model are homologically approximated by a wedge of spheres of dimension equal the critical dimension. In this paper, we study the Betti numbers in the upper model and introduce new notions of critical dimension and spread. We prove that (under certain conditions) an upper random simplicial complex is homologically approximated by a wedge of spheres of the critical dimension.

Original languageEnglish
Pages (from-to)1-31
Number of pages31
JournalJournal of Topology and Analysis
Issue number1
StatePublished - 1 Mar 2022

Bibliographical note

Funding Information:
Michael Farber was partially supported by the EPSRC, by the IIAS and by the Marie Curie Actions, FP7, in the frame of the EURIAS Fellowship Programme. Lewis Mead was supported by an EPSRC PhD fellowship.

Publisher Copyright:
© 2022 The Author(s).


  • Alexander duality
  • Random simplicial complex
  • critical dimension


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