随机二分图的沙堆群分布
Distribution of Sandpile groups of random bipartite graphs
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中文总结 AI 辅助
研究随机厄多斯 - 雷尼二分图\(G_{\alpha}(n,u)\)的沙堆群\(p\)-西罗子群极限分布，通过舍弃特定图解决满射矩发散问题，证明了奇素数\(p\)时的猜想，\(p = 2\)时计算矩与猜想匹配但不唯一确定分布。
中文摘要 AI 辅助
固定一个素数\(p\)和一个常数\(\frac{1}{p}<\alpha\leq1\)。考虑具有二分划\((V_1,V_2)\)（\(|V_1| = n\)，\(|V_2|=\lceil\alpha n\rceil\)）且边概率\(0 < u < 1\)的随机厄多斯 - 雷尼二分图\(G_{\alpha}(n,u)\)。文献[1]和[8]的作者猜想了\(G_{\alpha}(n,u)\)的沙堆群的\(p\)-西罗子群当\(n\to\infty\)时的极限分布。我们证明了对于奇素数\(p\)的这个猜想。之前通过计算从随机阿贝尔\(p\)-群到每个有限阿贝尔\(p\)-群\(H\)的满射的期望数量来证明类似结果，但在我们的设定中，这些满射矩常常发散到无穷，尽管猜想的极限分布有有限矩。我们通过舍弃太多顶点度数可被\(p\)整除的图来解决这个问题。去除这组罕见图的贡献后，满射矩收敛到期望值。当\(p\)为奇数时，应用伍德普遍性定理得到期望的分布收敛。对于\(p = 2\)，我们计算的矩（排除罕见图后）与猜想分布的矩匹配，但这些矩不能唯一确定一个分布。
英文摘要
Fix a prime $p$ and a constant $\frac{1}{p}<α\leq 1$. Consider the random Erdős--Rényi bipartite graph $G_α(n,u)$ with bipartition $(V_1,V_2)$ of sizes $|V_1|=n$ and $|V_2|=\lceilαn\rceil$, and edge probability $0<u<1$. The authors of [1] and [8] conjectured a limiting distribution for the $p$-Sylow subgroup of the sandpile group of $G_α(n,u)$ as $n\to\infty$. We prove this conjecture for odd primes $p$. Similar results have previously been proved by computing the expected number of surjections from the random abelian $p$-group to $H$, for each finite abelian $p$-group $H$. However, in our setting, these surjective moments often diverge to infinity, despite the conjectured limiting distribution having finite moments. We resolve this issue by discarding the graphs for which too many vertices have degrees divisible by $p$. Once we remove the contribution of this rare set of graphs, then the surjective moments converge to the expected values. When $p$ is odd, applying Wood's universality theorem yields the desired convergence in distribution. For $p=2$, our computed moments (after excluding the rare set of graphs) match those of the conjectured distribution. However, these moments do not uniquely determine a distribution.
