Geometry of Random Cayley Graphs of Abelian Groups

TL;DR

The paper analyzes random Cayley graphs of finite Abelian groups with k uniformly random generators in the regime 1 << log k << log |G|, establishing that typical distance from the identity concentrates at a value M determined by minimal lattice-ball radius in Z^k. It proves that the diameter concentrates at the same scale as the typical distance in the dense regime k ≳ log |G|, and shows universal-like behavior (depending only on k and |G|) in the spirit of the Aldous–Diaconis conjecture; it also proves a lower bound trel ≥ c |G|^{2/k} for the spectral gap, with matching upper bounds under stronger hypotheses. For nilpotent groups, the results extend via dominance by the Abelianisation, clarifying how non-Abelian structure affects distance statistics. The work advances the universality program for random Cayley graphs, providing sharp asymptotics and concentration phenomena for Abelian and certain nilpotent settings, with implications for expansion and mixing properties. Overall, the paper develops a unified framework linking lattice-ball geometry, typical-distance concentration, diameter behavior, and spectral-gap bounds in random Cayley graphs, highlighting universality in high-dimensional generator regimes with broad applicability to abelian and nilpotent 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 $U \sim \operatorname{Unif}(G)$. We show that the graph distance $\operatorname{dist}(\mathsf{id},U)$ 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 $k - 2 d(G) \asymp k$. 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.