Global Structural Properties of Random Graphs. (24th December 2016) - Behrstock, Jason; Falgas-Ravry, Victor; Hagen, Mark F; Susse, Tim
Abstract: We study two global structural properties of a graph $Γ$, denoted $\mathcal{AS}$ and $\mathcal{CFS}$, which arise in a natural way from geometric group theory. We study these properties in the Erdős–Rényi random graph model ${\mathcal G}(n, p)$, proving the existence of a sharp threshold for a random graph to have the $\mathcal{AS}$ property asymptotically almost surely, and giving fairly tight bounds for the corresponding threshold for the $\mathcal{CFS}$ property. As an application of our results, we show that for any constant $p$ and any $Γ\in{\mathcal G}(n, p)$, the right-angled Coxeter group $W_Γ$ asymptotically almost surely has quadratic divergence and thickness of order $1$, generalizing and strengthening a result of Behrstock–Hagen–Sisto [8 ]. Indeed, we show that at a large range of densities a random right-angled Coxeter group has quadratic divergence.