Sandpile Groups of Random Bipartite Graphs
Sandpile Groups of Random Bipartite Graphs
复制标题
随机二分图的沙堆组
DOI：
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发表时间：
2017
影响因子：
0.5
通讯作者：
Shaked Koplewitz
中科院分区：
文献类型：
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作者：
Shaked Koplewitz
我们确定随机二分图的沙堆群的 p 秩的渐近分布。我们看到，这取决于每条边上的顶点数之间的比率，当边之间的比率等于 $$frac{1}{p}$$ 1 p 时有一个阈值。我们遵循 Wood 的方法（J Am Math Soc 30(4):915–958, 2017），将随机图视为随机矩阵的特例，并依靠 Maples 给出的最小熵定义的变体（随机矩阵的 Cokernels 满足 Cohen-Lenstra 启发式，2013）来获得有关这些随机矩阵的有用结果。我们的结果表明，与 Erdős-Rényi 随机图的沙堆群不同，随机二分图的沙堆群的分布取决于图的属性，而不是来自一些更一般的随机群模型。
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 $$frac{1}{p}$$ 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.