Diameter Thresholds of Random Cayley Graphs

Given a group $G$, the model $\mathcal{G}(G,p)$ denotes the probability space of all Cayley graphs of $G$ where each element of $G$ is included in the generating set independently at random with probability $p$. In this article, we investigate the threshold probabilities for the diameter of random graphs in this model. Specifically, let $d_N = (1-\gamma)\sqrt{\frac{\log{N}}{2\log{\log{N}}}}$, where $\gamma \in (0,1)$ is any fixed real number. We show that for any $\varepsilon > 0$, any family of groups $G_k$ of order $N_k$ for which $N_k \to \infty$, and any integer $2 \leqslant d\leqslant d_{N_k}$, a graph $\Gamma_k \in \mathcal{G}(G_k,p)$ with high probability has diameter at most $d$ if $p \geqslant \sqrt[d]{(1+\varepsilon) d! \frac{\log{N_k}}{N_k^{d-1}}}$, and diameter greater than $d$ if $p \leqslant \sqrt[d]{\frac{1-\varepsilon}{2^d} \frac{\log{N_k}}{N_k^{d-1}}}$. Up to a constant factor, these thresholds are similar to those for the usual Erd\H{o}s-R\'enyi random graphs. However, the precise thresholds in our model depend on the underlying family of groups. We provide specific examples of group families demonstrating that both of our bounds are best possible.