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

Abstract

Given a graph Γ, its auxiliary square‐graph □(Γ) is the graph whose vertices are the non‐edges of Γ and whose edges are the pairs of non‐edges which induce a square (i.e., a 4‐cycle) in Γ. We determine the threshold edge‐probability p=pc(n) at which the Erdős–Rényi random graph Γ=Γn,p begins to asymptotically almost surely (a.a.s.) have a square‐graph with a connected component whose squares together cover all the vertices of Γn,p. We show pc(n)=6−2/n, a polylogarithmic improvement on earlier bounds on pc(n) due to Hagen and the authors. As a corollary, we determine the threshold p=pc(n) at which the random right‐angled Coxeter group WΓn,p a.a.s. becomes strongly algebraically thick of order 1 and has quadratic divergence.