The range of thresholds for diameter 2 in random Cayley graphs

Abstract

Given a group G, the model G(G,p) denotes the probability space of all Cayley graphs of G where each element of the generating set is chosen independently at random with probability p.

Given a family of groups (G_k) and a $c \in \mathbb{R}_+$ we say that c is the threshold for diameter 2 for (G_k) if for any ε > 0 with high probability $\Gamma \in \mathcal{G}(G_k,p)$ has diameter greater than 2 if $p \leqslant \sqrt{(c - \eps)\frac{\log{n}}{n}}$ and diameter at most 2 if $p \geqslant \sqrt{(c + \eps)\frac{\log{n}}{n}}$. In [5] we proved that if c is a threshold for diameter 2 for a family of groups (G_k) then $c \in [1/4,2]$ and provided two families of groups with thresholds 1/4 and 2 respectively.

In this paper we study the question of whether every $c \in [1/4,2]$ is the threshold for diameter 2 for some family of groups. Rather surprisingly it turns out that the answer to this question is negative. We show that every $c \in [1/4,4/3]$ is a threshold but a $c \in (4/3,2]$ is a threshold if and only if it is of the form 4n/(3n-1) for some positive integer n.