部分随机整数矩阵余核的通用性
Universality for cokernels of partially random integral matrices
AI总结：
研究部分随机整数矩阵余核的通用性，放宽元素平衡和独立性条件，证明即使有一定数量元素不满足\(\varepsilon\)-平衡及存在特定依赖关系时，余核分布仍趋近科恩 - 莱斯特拉分布，还证明特定随机带矩阵余核也有此性质，回答了相关问题。
AI中文摘要：
给定任意\(\varepsilon > 0\)，设\(M(n)\)是\(\mathbb{Z}_p\)上的随机\(n\times(n + u)\)矩阵，其所有元素独立且\(\varepsilon\)-平衡（在每个模\(p\)的剩余类中的概率至多为\(1 - \varepsilon\)）。伍德证明当\(n \to \infty\)时，\(\mathrm{cok}(M(n))\)的分布趋近于科恩和莱斯特拉关于类群的猜想分布。给定\(\alpha,\beta > 0\)且\(\alpha + \beta < 1\)，证明即使放宽假设，允许\(M(n)\)每列至多\(\alpha n\)个元素且每行至多\(\beta n\)个元素不\(\varepsilon\)-平衡，\(\mathrm{cok}(M(n))\)的分布仍趋近于科恩 - 莱斯特拉分布。还放宽了独立性条件，允许列元素间有特定依赖关系。此外，证明对于任意\(\delta > 0\)，带宽为\(\log(n)^{1 + \delta}\)且带内元素\(\varepsilon\)-平衡、带外元素任意的随机带矩阵的余核也趋近于科恩 - 莱斯特拉分布，回答了康 - 李 - 于的一个问题。
英文摘要：
Given any $\varepsilon > 0$, let $M(n)$ be a random $n \times (n+u)$ matrix over $\mathbb{Z}_p$, with all entries independent and $\varepsilon$-balanced (lying in each residue class mod $p$ with probability at most $1-\varepsilon$). Wood proved that as $n \to \infty$ the distribution of $\mathrm{cok}(M(n))$ approaches Cohen and Lenstra's conjectured distribution of class groups. Given $α,β>0$ such that $α+ β<1$, we prove that the distribution of $\mathrm{cok}(M(n))$ still approaches the Cohen--Lenstra distribution even if we weaken the hypothesis by allowing up to $αn$ entries per column and up to $βn$ entries per row of $M(n)$ to not be $\varepsilon$-balanced. We also weaken the independence condition by allowing certain types of dependence between the entries of each column. In addition, we prove that, for any $δ> 0$, the cokernels of random band matrices of width $\log(n)^{1+δ}$ with $\varepsilon$-balanced entries in the band and arbitrary entries outside of it will also approach the Cohen--Lenstra distribution, which answers a question of Kang--Lee--Yu.
