A threshold for relative hyperbolicity in random right-angled Coxeter groups

We consider the random right-angled Coxeter group $W_{\Gamma}$ whose presentation graph $\Gamma\sim \mathcal{G}_{n,p}$ is an Erd{\H o}s--R\'enyi random graph on $n$ vertices with edge probability $p=p(n)$. We establish that $p=1/\sqrt{n}$ is a threshold for relative hyperbolicity of the random group $W_{\Gamma}$. 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 $\Omega(1/\sqrt{n})$ 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.