Global Structural Properties of Random Graphs

Jason Behrstock, Victor Falgas-Ravry, Mark F. Hagen, Timothy Susse

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We study two global structural properties of a graph $\Gamma$, denoted AS and CFS, which arise in a natural way from geometric group theory. We study these properties in the Erd\"os--R\'enyi random graph model G(n,p), proving a sharp threshold for a random graph to have the AS property asymptotically almost surely, and giving fairly tight bounds for the corresponding threshold for CFS. As an application of our results, we show that for any constant p and any $\Gamma\in G(n,p)$, the right-angled Coxeter group $W_\Gamma$ asymptotically almost surely has quadratic divergence and thickness of order 1, generalizing and strengthening a result of Behrstock--Hagen--Sisto.
Original languageEnglish
Article numberrnw287
Pages (from-to)1411–1441
Number of pages41
JournalInternational Mathematics Research Notices
Issue number5
Early online date24 Dec 2016
Publication statusPublished - Mar 2018

Bibliographical note

21 pages, 5 figures


  • math.PR
  • math.CO
  • math.GR
  • math.GT


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