Connectivity for square percolation and coarse cubical rigidity in random right-angled Coxeter groups

Abstract

Abstract We consider a random right-angled Coxeter group (RACG), upper W Subscript normal upper Gamma W Γ $W_{\Gamma}$ , whose presentation graph normal upper Gamma Γ $\Gamma$ is taken to be an Erdős–Rényi random graph, i.e. normal upper Gamma tilde script upper G Subscript n comma p Γ ∼ G n , p $\Gamma\sim\mathcal{G}_{n,p}$ . We use techniques from probabilistic combinatorics to establish several new results about the geometry of this random group model. Resolving a conjecture of Susse, we determine the connectivity threshold for square percolation on the random graph normal upper Gamma tilde script upper G Subscript n comma p Γ ∼ G n , p $\Gamma\sim\mathcal{G}_{n,p}$ . We use this result to determine a large range of p for which the random RACG upper W Subscript normal upper Gamma W Γ $W_{\Gamma}$ has a unique cubical coarse median structure. Until Fioravanti et al. (2026), there were no non-hyperbolic examples of groups with cubical coarse rigidity. Perhaps surprisingly, we show that the property of coarse cubical rigidity is in fact typically satisfied by a random RACG for a wide range of the parameter p , including p equals one half p = 1 2 $p=\frac12$ , for which the random RACG is also typically non-hyperbolic.