Further Results and Discussions on Random Cayley Graphs · arXivDesk
1911.02975Nov 7, 2019This 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. This latest version also updates the second named author's name to his new surname, from "Thomas" to "Olesker-Taylor". arXiv admin note: text overlap with arXiv:1911.02974
Further Results and Discussions on Random Cayley Graphs
Consider the random Cayley graph of a finite group G with respect to k generators chosen uniformly at random, with 1≪k≲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=Zpd
Nearby in the stack
with
p
prime. We then exposit a proof of Diaconis and Saloff-Coste (1994) establishing lack of cutoff when
k≍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
Lq
-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.