The diameter of random Cayley digraphs of given degree

TL;DR

This paper studies the probability that a random Cayley digraph of a given degree has diameter two, revealing a sharp phase transition at a specific degree threshold as the size of the graph grows.

Contribution

It establishes the asymptotic probability and identifies a phase transition for the diameter of random Cayley digraphs based on the generating set size.

Findings

Probability approaches 1 when degree is linear in n

Sharp phase transition at degree around sqrt(n log n)

Exponential convergence to probability 1 for large degrees

Abstract

We consider random Cayley digraphs of order $n$ with uniformly distributed generating set of size $k$. Specifically, we are interested in the asymptotics of the probability such a Cayley digraph has diameter two as $n\to\infty$ and $k=f(n)$. We find a sharp phase transition from 0 to 1 at around $k = \sqrt{n \log n}$. In particular, if $f(n)$ is asymptotically linear in $n$, the probability converges exponentially fast to 1.