Square percolation and the threshold for quadratic divergence in random right-angled Coxeter groups

Given a graph $\Gamma$, its auxiliary \emph{square-graph} $\square(\Gamma)$ is the graph whose vertices are the non-edges of $\Gamma$ and whose edges are the pairs of non-edges which induce a square (i.e., a $4$-cycle) in $\Gamma$. We determine the threshold edge-probability $p=p_c(n)$ at which the Erd{\H o}s--Renyi random graph $\Gamma=\Gamma_{n,p}$ begins to asymptotically almost surely have a square-graph with a connected component whose squares together cover all the vertices of $\Gamma_{n,p}$. We show $p_c(n)=\sqrt{\sqrt{6}-2}/\sqrt{n}$, a polylogarithmic improvement on earlier bounds on $p_c(n)$ due to Hagen and the authors. As a corollary, we determine the threshold $p=p_c(n)$ at which the random right-angled Coxeter group $W_{\Gamma_{n,p}}$ asymptotically almost surely becomes strongly algebraically thick of order $1$ and has quadratic divergence.