Abstract
We determine the asymptotic distribution of the p-rank of the sandpile groups of random bipartite graphs. We see that this depends on the ratio between the number of vertices on each side, with a threshold when the ratio between the sides is equal to \(\frac{1}{p}\). We follow the approach of Wood (J Am Math Soc 30(4):915–958, 2017) and consider random graphs as a special case of random matrices, and rely on a variant the definition of min-entropy given by Maples (Cokernels of random matrices satisfy the Cohen–Lenstra heuristics, 2013) to obtain useful results about these random matrices. Our results show that unlike the sandpile groups of Erdős–Rényi random graphs, the distribution of the sandpile groups of random bipartite graphs depends on the properties of the graph, rather than coming from some more general random group model.
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Acknowledgements
The author is grateful to Sam Payne and Nathan Kaplan for suggesting the problem, as well as their many helpful suggestions along the way. Also to Dan Carmon, for suggesting the proof of Claim 1. This work was partially supported by NSF CAREER DMS-1149054. On behalf of all authors, the corresponding author states that there is no conflict of interest. Data sharing is not applicable to this article as no datasets were generated or analyzed during the current study.
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Communicated by Kolja Knauer.
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Koplewitz, S. Sandpile Groups of Random Bipartite Graphs. Ann. Comb. 27, 1–18 (2023). https://doi.org/10.1007/s00026-022-00616-0
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DOI: https://doi.org/10.1007/s00026-022-00616-0