Connectivity for square percolation and coarse cubical rigidity in random right-angled Coxeter groups

TL;DR

This work analyzes random right-angled Coxeter groups $W_{\Gamma}$ arising from presentation graphs $\Gamma \sim \mathcal{G}_{n,p}$ by examining the square graphs $S(\Gamma)$ and $T_1(\Gamma)$ that encode induced $4$-cycles. Using Chernoff and Janson inequalities and giant-connecting arguments, the authors determine precise connectivity thresholds for these square graphs and prove a unique giant square component emerges in a broad range of $p$. They then translate these combinatorial thresholds into geometric conclusions for $W_{\Gamma}$, showing a wide interval of $p$ yields a unique cubical coarse median structure, i.e., coarse cubical rigidity, including the regime $p=1/2$. The results imply that typical random RACGs are non-hyperbolic and rigid at the cubical level, and they connect square-percolation phenomena to questions about quasi-isometric RAAGs and the structure of their cubulations, while also presenting intriguing extremal constructions and open questions on component size and diameter.

Abstract

We consider random right-angled Coxeter groups, $W_Γ$, whose presentation graph $Γ$ is taken to be an Erdős--Rényi random graph, i.e., $Γ\sim \mathcal{G}_{n,p}$. We use techniques from probabilistic combinatorics to establish several new results about the geometry of these random groups. We resolve a conjecture of Susse and determine the connectivity threshold for square percolation on the random graph $Γ\sim \mathcal{G}_{n,p}$. We use this result to determine a large range of $p$ for which the random right-angled Coxeter group $W_Γ$ has a unique cubical coarse median structure. Until recent work of Fioravanti, Levcovitz and Sageev, there were no non-hyperbolic examples of groups with cubical coarse rigidity; our present results show the property is in fact typically satisfied by a random RACG for a wide range of the parameter $p$, including $p=1/2$.

Figures (3)

• Figure 1: The graph $G'$.
• Figure 2: The graph $G"$: note the bottom indices are shifted to emphasize the similarity with $G'$.
• Figure 3: Two perspectives on the graph $G$. In both are highlighted all the squares which contain the vertices 5 and 16 (in blue) and the vertices 7 and 14 (in red). Both drawings emphasize the bipartite division of the vertices, the drawing on the left highlights the distinction (as seen by the different "slopes") between the blue squares which are in the first component of full support and the red squares which are in the second component of full support. The figure at right is drawn to emphasize the symmetry and to avoid the multiple collinear vertices which hide some of the edges in the graph on the left (for instance those connecting the vertices at the bottom to the ones at the top).

Theorems & Definitions (32)

• Theorem 1.1
• Theorem 1.3: Coarse cubical rigidity
• Theorem 1.4
• Conjecture 1.5
• Definition 2.1: $T_1(\Gamma)$: the square graph
• Remark 2.2: The other square graph: $S(\Gamma)$
• Proposition 2.3
• proof
• Proposition 2.4: Extremal bound for thick graphs behrstock2018globalBehrstock2024thickness
• Proposition 2.5: Chernoff bound