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

2020/09/30 by Behrstock, Jason, Falgas-Ravry, Victor, Susse, Tim
#Combinatorics (math.CO) #FOS: Mathematics #Geometric Topology (math.GT) #Group Theory (math.GR) #Probability (math.PR)

paper · doi:10.48550/arxiv.2009.14442

Abstract

Given a graph Γ, its auxiliary square-graph \square(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=p_c(n) at which the Erd\H os--Rényi random graph Γ=Γ_n,p begins to asymptotically almost surely have a square-graph with a connected component whose squares together cover all the vertices of Γ_n,p. We show p_c(n)=√(√(6)-2)/√(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_Γ_n,p asymptotically almost surely becomes strongly algebraically thick of order 1 and has quadratic divergence.

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