The Distribution of Sandpile Groups of Random Graphs

Monday, September 29, 2014 - 2:00pm - 2:50pm
Keller 3-180
Melanie Wood (University of Wisconsin, Madison)
We determine the distribution of the sandpile group (a.k.a. Jacobian)
of the Erdős–Rényi random graph G(n,q) as n goes to infinity. Since
any particular abelian group appears with asymptotic probability 0 (as
we show), it is natural ask for the asymptotic distribution of Sylow
p-subgroups of sandpile groups. We prove the distributions of Sylow
p-subgroups converge to specific distributions conjectured by Clancy,
Leake, and Payne. These distributions are related to, but different
from, the Cohen-Lenstra distribution. Our proof involves first finding
the expected number of surjections from the sandpile group to any
finite abelian group (the moments of a random variable valued in
finite abelian groups). To achieve this, we show a universality result
for the moments of cokernels of random symmetric integral matrices
that is strong enough to handle dependence in the diagonal entries. We
then show these moments determine a unique distribution despite their
p^{k^2}-size growth.
MSC Code: