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 $1\ll log\mathit{k}\ll log|\mathit{G}|$. Draw a vertex $\mathit{U}\sim Unif(\mathit{G})$.

We show that the graph distance $dist(\mathsf{id},\mathit{U})$ from the identity to U concentrates at a particular value M, which is the minimal radius of a ball in ${\mathbb{Z}}^{\mathit{k}}$ of cardinality at least $|\mathit{G}|$, under mild conditions. In other words, the distance from the identity for all but $\mathit{o}(|\mathit{G}|)$ of the elements of G lies in the interval $[\mathit{M}-\mathit{o}(\mathit{M}),\mathit{M}\mathbf{+}\mathit{o}(\mathit{M})]$. In the regime $\mathit{k}\gtrsim log|\mathit{G}|$, we show that the diameter of the graph is also asymptotically M. In the spirit of a conjecture of Aldous and Diaconis (Technical Report 231 (1985)), this M depends only on k and $|\mathit{G}|$, not on the algebraic structure of G.

Write $\mathit{d}(\mathit{G})$ for the minimal size of a generating subset of G. We prove that the order of the spectral gap is $|\mathit{G}{|}^{-2/\mathit{k}}$ when $\mathit{k}-\mathit{d}(\mathit{G})\asymp \mathit{k}$ and $|\mathit{G}|$ lies in a density-1 subset of $\mathbb{N}$ or when $\mathit{k}-2\mathit{d}(\mathit{G})\asymp \mathit{k}$. This extends, for Abelian groups, a celebrated result of Alon and Roichman (Random Structures Algorithms 5 (1994) 271–284).

The aforementioned results all hold with high probability over the random Cayley graph.