We consider the random right-angled Coxeter group WΓ whose presentation graph Γ∼Gn,p is an Erdős–Rényi random graph on n vertices with edge probability p=p(n). We establish that p=1/n is a threshold for relative hyperbolicity of the random group WΓ. As a key step in the proof, we determine the minimal number of pairs of generators that must commute in a right-angled Coxeter group which is not relatively hyperbolic, a result which is of independent interest.
We also show that there is an interval of edge probabilities of width Ω(1/n) in which the random right-angled Coxeter group has precisely cubic divergence. This interval is between the thresholds for relative hyperbolicity (whence exponential divergence) and quadratic divergence. Moreover, a simple random walk on any Cayley graph of the random right-angled Coxeter group for p in this interval satisfies a central limit theorem.