{"title":"The Rank of the Sandpile Group of Random Directed Bipartite Graphs","authors":"Atal Bhargava, Jack DePascale, Jake Koenig","doi":"10.1007/s00026-023-00637-3","DOIUrl":null,"url":null,"abstract":"<div><p>We identify the asymptotic distribution of <i>p</i>-rank of the sandpile group of random directed bipartite graphs which are not too imbalanced. We show this matches exactly with that of the Erdös–Rényi random directed graph model, suggesting that the Sylow <i>p</i>-subgroups of this model may also be Cohen–Lenstra distributed. Our work builds on the results of Koplewitz who studied <i>p</i>-rank distributions for unbalanced random bipartite graphs, and showed that for sufficiently unbalanced graphs, the distribution of <i>p</i>-rank differs from the Cohen–Lenstra distribution. Koplewitz (sandpile groups of random bipartite graphs, https://arxiv.org/abs/1705.07519, 2017) conjectured that for random balanced bipartite graphs, the expected value of <i>p</i>-rank is <i>O</i>(1) for any <i>p</i>. This work proves his conjecture and gives the exact distribution for the subclass of directed graphs.</p></div>","PeriodicalId":50769,"journal":{"name":"Annals of Combinatorics","volume":"27 4","pages":"979 - 992"},"PeriodicalIF":0.6000,"publicationDate":"2023-04-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s00026-023-00637-3.pdf","citationCount":"1","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Annals of Combinatorics","FirstCategoryId":"100","ListUrlMain":"https://link.springer.com/article/10.1007/s00026-023-00637-3","RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q4","JCRName":"MATHEMATICS, APPLIED","Score":null,"Total":0}
引用次数: 1
Abstract
We identify the asymptotic distribution of p-rank of the sandpile group of random directed bipartite graphs which are not too imbalanced. We show this matches exactly with that of the Erdös–Rényi random directed graph model, suggesting that the Sylow p-subgroups of this model may also be Cohen–Lenstra distributed. Our work builds on the results of Koplewitz who studied p-rank distributions for unbalanced random bipartite graphs, and showed that for sufficiently unbalanced graphs, the distribution of p-rank differs from the Cohen–Lenstra distribution. Koplewitz (sandpile groups of random bipartite graphs, https://arxiv.org/abs/1705.07519, 2017) conjectured that for random balanced bipartite graphs, the expected value of p-rank is O(1) for any p. This work proves his conjecture and gives the exact distribution for the subclass of directed graphs.
期刊介绍：
Annals of Combinatorics publishes outstanding contributions to combinatorics with a particular focus on algebraic and analytic combinatorics, as well as the areas of graph and matroid theory. Special regard will be given to new developments and topics of current interest to the community represented by our editorial board.
The scope of Annals of Combinatorics is covered by the following three tracks:
Algebraic Combinatorics:
Enumerative combinatorics, symmetric functions, Schubert calculus / Combinatorial Hopf algebras, cluster algebras, Lie algebras, root systems, Coxeter groups / Discrete geometry, tropical geometry / Discrete dynamical systems / Posets and lattices
Analytic and Algorithmic Combinatorics:
Asymptotic analysis of counting sequences / Bijective combinatorics / Univariate and multivariable singularity analysis / Combinatorics and differential equations / Resolution of hard combinatorial problems by making essential use of computers / Advanced methods for evaluating counting sequences or combinatorial constants / Complexity and decidability aspects of combinatorial sequences / Combinatorial aspects of the analysis of algorithms
Graphs and Matroids:
Structural graph theory, graph minors, graph sparsity, decompositions and colorings / Planar graphs and topological graph theory, geometric representations of graphs / Directed graphs, posets / Metric graph theory / Spectral and algebraic graph theory / Random graphs, extremal graph theory / Matroids, oriented matroids, matroid minors / Algorithmic approaches
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