Geometry of Random Cayley Graphs of Abelian Groups · arXivDesk
2102.02801Feb 4, 2021Accepted 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
Geometry of Random Cayley Graphs of Abelian Groups
Consider the random Cayley graph of a finite Abelian group G with respect to k generators chosen uniformly at random, with 1≪logk≪log∣G∣. Draw a vertex U∼Unif(G)
Nearby in the stack
. We show that the graph distance
dist(id,U)
from the identity to
U
concentrates at a particular value
M
, which is the minimal radius of a ball in
Zk
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≳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)≍k
and
∣G∣
lies in a density-
1
subset of
N
or when
k−2d(G)≍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.