随机指派问题的中心极限定理
A central limit theorem for the random assignment problem
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中文总结 AI 辅助
该研究证明了n×n独立均匀随机矩阵中完美匹配最小代价的中心极限定理，通过变量替换、有向树因子处理及三角阵列极限分析完成推导，方法可推广至其他问题。
中文摘要 AI 辅助
设\n\\(C_n\\)为元素服从独立均匀分布的\\(n\times n\\)矩阵中完美匹配的最小代价。我们证明：\\[\n\sqrt n\{C_n-\zeta(2)\}\n\\ \Longrightarrow\n\mathcal N\bigl(0,4\zeta(2)-4\zeta(3)\bigr). \\]证明始于基于最优对偶势的均匀根最短路径选择的精确变量替换。在对未使用的约化成本积分后，参考律在势场条件下分离各行，而有序势间隙变为独立指数分布。唯一的剩余依赖是有向树因子。对势排序将其0-1支撑转化为Ferrers矩阵，其矩阵树行列式为三角形式。随后，奇异逆度估计与精确归一化得到向参考律的全变差收敛。最后，条件三角阵列中心极限定理解释行噪声，第二个三角阵列解释势场的线性响应。本文所用策略有望应用于其他问题。
英文摘要
Let \(C_n\) be the minimum cost of a perfect matching in an \(n\times n\) matrix of independent uniform random variables. We prove that \[ \sqrt n\{C_n-ζ(2)\} \ \Longrightarrow\ \mathcal N\bigl(0,4ζ(2)-4ζ(3)\bigr). \] The proof begins with an exact change of variables based on a uniformly rooted shortest-path selection of an optimal dual potential. After the unused reduced costs are integrated out, a reference law separates the rows conditionally on the potential field, while the ordered potential gaps become independent exponentials. The only residual dependence is a directed-tree factor. Ordering the potentials turns its zero--one support into a Ferrers matrix, whose matrix-tree determinant is triangular. A singular inverse-degree estimate and exact normalization then yield total-variation convergence to the reference law. Finally, a conditional triangular-array central limit theorem accounts for row noise, and a second triangular array accounts for the linear response of the potential field. The strategy used here is likely to be applicable to other problems.
