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

We consider random right-angled Coxeter groups, WΓ, whose presentation graph Γ is taken to be an Erd{o}s--R\'enyi random graph, i.e., Γ~ Gn,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 Γ ~ Gn,p. We use this result to determine a large range of p for which the random right-angled Coxeter group WΓ 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 $p=1/2$.