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

We consider random right-angled Coxeter groups, W_, whose presentation graph is taken to be an Erdos--R\'enyi random graph, i. e. , G₍, ₏. 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 G₍, ₏. We use this result to determine a large range of p for which the random right-angled Coxeter group W_ 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=1/2.