Further Results and Discussions on Random Cayley Graphs

TL;DR

This paper advances the understanding of random Cayley graphs by analyzing cutoff phenomena, mixing properties, connectivity conditions, and distance metrics, with results applicable to various group structures and inspired by universality conjectures.

Contribution

It provides new insights into the cutoff profile, mixing times, and connectivity criteria of random Cayley graphs, extending previous results and exploring diverse group cases.

Findings

Identifies cutoff phenomena for random walks on Cayley graphs.

Establishes conditions for graph connectivity based on group generators.

Analyzes distance metrics in Abelian groups and mixing properties in specific cases.

Abstract

Consider the random Cayley graph of a finite group $G$ with respect to $k$ generators chosen uniformly at random, with $1 \ll k \lesssim \log |G|$. The results of this article supplement those in the three main papers on random Cayley graphs. The majority of the results are inspired by a `universality' conjecture of Aldous and Diaconis (1985). To start, we study the limit profile of cutoff for the simple random walk on this random graph, as well as a detailed investigation into mixing properties when $G = \mathbb Z_p^d$ with $p$ prime. We then exposit a proof of Diaconis and Saloff-Coste (1994) establishing lack of cutoff when $k \asymp 1$. We move onto discussing material from our companion paper on matrix groups. We then study distance of a typical element of $G$ from the identity in an $L_q$-type graph distance in the Abelian set-up. Finally, we give necessary and sufficient conditions for $k$ independent uniform elements of $G$ to generate $G$, ie for the random Cayley graph to be connected, based on work of Pomerance (2001). The aforementioned results all hold with high probability over the random Cayley graph.