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Connectivity for square percolation and coarse cubical rigidity in random right-angled Coxeter groups
- Part of
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- Journal:
- Journal of Applied Probability , First View
- Published online by Cambridge University Press:
- 09 September 2026, pp. 1-25
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We consider a random right-angled Coxeter group (RACG),
$W_{\Gamma}$, whose presentation graph 
$\Gamma$ is taken to be an Erdős–Rényi random graph, i.e. 
$\Gamma\sim\mathcal{G}_{n,p}$. We use techniques from probabilistic combinatorics to establish several new results about the geometry of this random group model. Resolving a conjecture of Susse, we determine the connectivity threshold for square percolation on the random graph 
$\Gamma\sim\mathcal{G}_{n,p}$. We use this result to determine a large range of p for which the random RACG 
$W_{\Gamma}$ has a unique cubical coarse median structure. Until Fioravanti et al. (2026), there were no non-hyperbolic examples of groups with cubical coarse rigidity. Perhaps surprisingly, we show that the property of coarse cubical rigidity is in fact typically satisfied by a random RACG for a wide range of the parameter p, including 
$p=\frac12$, for which the random RACG is also typically non-hyperbolic.
