Geometry of Random Cayley Graphs of Abelian Groups

Abstract

Consider the random Cayley graph of a finite Abelian group $G$ with respect to $k$ generators chosen uniformly at random, with $1\ll \log k\ll \log |G|$. Draw a vertex $U\sim \mathrm{Unif}(G)$. We show that the graph distance $\mathrm{dist}(\mathsf{id},U)$ from the identity to $U$ concentrates at a particular value $M$, which is the minimal radius of a ball in ${\mathbb{Z}}^k$ of cardinality at least $|G|$, under mild conditions. In other words, the distance from the identity for all but $o(|G|)$ of the elements of $G$ lies in the interval $[M-o(M),M+o(M)]$. In the regime $k\gtrsim \log |G|$, we show that the diameter of the graph is also asymptotically $M$. In the spirit of a conjecture of Aldous and Diaconis (1985), this $M$ depends only on $k$ and $|G|$, not on the algebraic structure of $G$. Write $d(G)$ for the minimal size of a generating subset of $G$. We prove that the order of the spectral gap is $|G{|}^{-2/k}$ when $k-d(G)\asymp k$ and $|G|$ lies in a density-$1$ subset of $\mathbb{N}$ or when $k-2d(G)\asymp k$. This extends, for Abelian groups, a celebrated result of Alon and Roichman (1994). The aforementioned results all hold with high probability over the random Cayley graph.

Accepted at Annals of Applied Probability (AAP), Sept '22. This is part of a multi-paper project investigating properties of Cayley graphs with divergently many generators chosen uniformly at random. There is some textual overlap between the introductions of the different papers

References in corpus (6)

• Mixing and cut-off in cycle walks
• Cutoff for Almost All Random Walks on Abelian Groups
• Cutoff for Random Walks on Upper Triangular Matrices
• Supplementary Material for Random Cayley Graphs Project
• Geometry of Random Cayley Graphs of Abelian Groups
• Further Results and Discussions on Random Cayley Graphs

Cited by in corpus (2)

• Cutoff for Almost All Random Walks on Abelian Groups
• Geometry of Random Cayley Graphs of Abelian Groups