UC Irvine

This thesis studies the asymptotic distribution of sandpile groups of random bipartite graphs in both the directed and undirected settings. Fix a prime p and let the two parts of the bipartition have sizes n and ⌈αn⌉, where 1 p < α ≤ 1. For directed Erdős-Rényi bipartite graphs, this thesis proves that the p-Sylow subgroup of the sandpile group converges to the conjectured Cohen-Lenstra distribution. For undirected Erdős-Rényi bipartite graphs, it proves the corresponding conjecture for odd primes p, where the conjectured limit is the symmetric Cohen-Lenstra distribution. In the undirected case with p = 2, the limiting distribution has been conjectured to be a variant of the symmetric Cohen-Lenstra distribution. Our results provide strong evidence for this conjecture.For a fixed finite abelian p-group G, the surjective moment of a random finite abelian pgroup X is the expected number of surjective homomorphisms from X to G. The surjective moment method seeks to identify a limiting distribution by computing these expectations for every G and then showing that they determine the distribution. A central difficulty in both settings is that the usual surjective moment method does not apply directly: the surjective moments can diverge even though the conjectured limiting distributions have finite surjective moments.The main idea is to isolate a high-probability subset of graphs on which the relevant moments are well behaved, and to show that the complementary exceptional set is rare enough that it does not affect the limiting distribution. In all cases, after restricting to these highprobability subsets, the resulting surjective moments agree with the moments predicted by the conjectured limiting distributions. In the directed case for all primes p, and in the undirected case for odd primes p, Wood’s universality result then yields the conjectured limiting distribution. For p = 2 in the undirected case, the computed moments agree with those of the conjectured distribution, but these moments do not by themselves determine a unique limiting law. These results extend the known picture for random graphs to the bipartite directed and undirected settings.