The threshold sequence for polynomial divergence in random right-angled Coxeter groups

Let $\mathcal{G}_{n,p}$ denote the Erdős–Rényi random graph, and let $\Gamma\sim\mathcal{G}_{n,p}$. For a group or graph, “thick of order at most $k$” is used in the sense of the paper. Threshold-sequence conjecture. There exists an infinite strictly decreasing sequence of strictly positive real numbers $(\lambda_k)_{k\in\mathbb{N}}$ with

$$\lambda_1=\sqrt{\sqrt{6}-2}>\lambda_2>\lambda_3>\cdots>\lambda_k>\cdots,$$

such that for every $k\in\mathbb{N}$ and every fixed $\varepsilon>0$: (i) if $p=p(n)\leq\frac{\lambda_k-\varepsilon}{\sqrt n}$, then $a.a.s.$ $\Gamma$ is not thick of order at most $k$; (ii) if $p=p(n)$ satisfies

$$\frac{\lambda_k+\varepsilon}{\sqrt n}\leq p(n)\leq 1-\Omega\left(\frac{\log n}{n}\right),$$

then $a.a.s.$ $\Gamma$ is thick of order at most $k$. This conjecture proposes sharp thresholds for each finite thickness order and, consequently, for the associated polynomial divergence behavior.