随机稳定室友问题可解概率的尖锐渐近性

中文摘要 AI 辅助

对于偶数 $n$，设 $P_n$ 为 $n$ 个参与者上独立均匀严格偏好列表允许稳定完美匹配的概率。我们证明 \\[ P_n\sim\frac{e\\,2^{1/4}\Gamma(3/4)}{\sqrt\pi}\\,n^{-1/4}. \\] 这确立了 Mertens 推测的衰减指数，其领先常数不同于他最初的数值预测。证明从 Mertens 关于稳定排列的精确交错和出发。为保持其相消性，我们为所有具有相同长于二的环中顶点数的环结构构造一个公共近似。由对称性，积分后的一阶修正对每个这样的环结构都相同，其余误差可按绝对值求和。枚举随后将概率化为一维正项和。

英文摘要

For even $n$, let $P_n$ be the probability that independent uniform strict preference lists on $n$ participants admit a stable perfect matching. We prove \[ P_n\sim\frac{e\,2^{1/4}Γ(3/4)}{\sqrtπ}\,n^{-1/4}. \] This establishes Mertens's conjectured exponent of decay, with a leading constant different from his original numerical prediction. The proof starts from Mertens's exact alternating sum over stable permutations. To preserve its cancellation, we construct a common approximation for all cycle structures with the same number of vertices in cycles longer than two. By symmetry, the integrated first-order correction is the same for every such cycle structure, and the remaining errors can be summed in absolute value. The enumeration then reduces the probability to a one-dimensional sum with positive terms.

发表机构

• University of Tulsa（塔尔萨大学）

机构由 AI 辅助整理，请以论文原文为准。