  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.
