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

Abstract

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