Morse subgroups and boundaries of random right-angled Coxeter groups

We study Morse subgroups and Morse boundaries of random right-angled Coxeter group in the Erdos-Renyi model. We show that at densities below (root 1/2-is an element of) root logn/n random right-angled Coxeter groups almost surely have Morse hyperbolic surface subgroups. This implies their Morse boundaries contain embedded circles. Further, at densities above (root 1/2-is an element of) root logn/n we show that, almost surely, the hyperbolic Morse special subgroups of a random right-angled Coxeter group are virtually free. We also apply these methods to log n show that for a random graph Gamma at densities below (1-is an element of) root log n/n , rectangle(Gamma) almost surely contains an isolated vertex. This shows, in particular, that at densities below (1-is an element of) root log n/n a random right-angled Coxeter group is almost surely not quasi-isometric to a right-angled Artin group