Random integral matrices and the Cohen-Lenstra heuristics
Random integral matrices and the Cohen-Lenstra heuristics
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DOI：
10.1353/ajm.2019.0008
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发表时间：
2015-04
影响因子：
1.7
通讯作者：
M. Wood
中科院分区：
文献类型：
--
作者：
M. Wood
翻译后摘要：我们证明，给定任何$\n>0$，随机积分$n\times n$矩阵的独立条目，位于任何剩余类模一个素数的概率最多为1-\n $有cokernels渐近（作为$n\rightarrow\infty$）分布在有限阿贝尔群的分布，科恩和Lenstra猜想是分布的类群的虚二次域。这表明Cohen-Lenstra分布对于由生成元和随机关系给出的有限阿贝尔群是普适的--这是关于${\Bbb Z}/ p{\Bbb Z}$中具有独立元素的随机矩阵的秩分布的一个结果的改进。这是有趣的，特别是在光的事实，这些类组是自然的cokernels的方阵。我们还证明了类似的$n\次（n+u）$矩阵。
Abstract:We prove that given any $\epsilon>0$, random integral $n\times n$ matrices with independent entries that lie in any residue class modulo a prime with probability at most $1-\epsilon$ have cokernels asymptotically (as $n\rightarrow\infty$) distributed as in the distribution on finite abelian groups that Cohen and Lenstra conjecture to be the distribution for class groups of imaginary quadratic fields. This shows the Cohen-Lenstra distribution is universal for finite abelian groups given by generators and random relations---that the distribution of quotients does not depend on the way in which we choose (sufficiently nice) relations. This is a refinement of a result on the distribution of ranks of random matrices with independent entries in ${\Bbb Z}/ p{\Bbb Z}$. This is interesting especially in light of the fact that these class groups are naturally cokernels of square matrices. We also prove the analogue for $n\times (n+u)$ matrices.