The range of thresholds for diameter 2 in random Cayley graphs

TL;DR

This paper investigates the thresholds for diameter 2 in random Cayley graphs of groups, revealing that only specific values within a range are thresholds, contrary to initial expectations that all values in that range could be.

Contribution

It characterizes which thresholds in [1/4,2] are attainable for diameter 2 in random Cayley graphs, showing only certain discrete values are possible.

Findings

Thresholds in [1/4,4/3] are all attainable.

Thresholds in (4/3,2] occur only at specific rational points.

Not all values in [1/4,2] are thresholds, contrary to initial conjecture.

Abstract

Given a group G, the model \mathcal{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 \varepsilon > 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.