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

We consider random right-angled Coxeter groups, $W_{\mathrm{\Gamma }}$, whose presentation graph $\mathrm{\Gamma }$ is taken to be an Erdős–Rényi random graph, i.e., $\mathrm{\Gamma }\sim {\mathcal{G}}_{n,p}$. We use techniques from probabilistic combinatorics to establish several new results about the geometry of these random groups. We resolve a conjecture of Susse and determine the connectivity threshold for square percolation on the random graph $\mathrm{\Gamma }\sim {\mathcal{G}}_{n,p}$. We use this result to determine a large range of $p$ for which the random right-angled Coxeter group $W_{\mathrm{\Gamma }}$ has a unique cubical coarse median structure. Until recent work of Fioravanti, Levcovitz and Sageev, there were no non-hyperbolic examples of groups with cubical coarse rigidity; our present results show the property is in fact typically satisfied by a random RACG for a wide range of the parameter $p$, including $p=1\mathrm{/}2$.