AI 中文总结
该论文研究线性超图中均匀随机匹配的平均空置率，通过采样删除边并应用Molloy-Reed定理，给出了匹配计数下界及空置率的上界，并改进正则情形下的系数。
AI 中文摘要
设 $M$ 从有限线性 $k$-均匀超图 $H$ 的所有匹配中均匀选取，并设 $\overline{q}(H)$ 为顶点未被覆盖的平均概率。若 $H$ 的最大度为 $D$，归一化平均度为 $\beta=k|E(H)|/(|V(H)|D)$，则对每个固定的 $k\geq 2$ 且一致地关于阶数，有 $\overline{q}(H) \leq 1-\beta+(\beta k+o_D(1))\log\log D/\log D$。其基本估计是阶数一致的计数 $\log Z(H)\geq (|E(H)|/D)(\log D-(k+o_D(1))\log\log D)$，并且该计数也适用于大小为 $(1-o_D(1))|E(H)|/D$ 的匹配。我们通过采样边、删除那些具有异常大采样度的边，并在应用 Molloy-Reed 列表边着色定理之前仅控制总删除质量来证明这一点。对于 $d$-正则的 $H$，早期 Asratian-Kuzjurin 采样到计数的定量版本，使用 Gould 和 Kelly 的近完美匹配定理，改进了系数：对每个固定的 $k\geq 3$，$\sup_H \overline{q}(H) \leq (\max\{3,k-1\}+o_d(1))\log\log d/\log d$。Kahn 更强的逐点预测被 Lee 对 $k\geq 3$ 反驳。定性的平均结论已由 Kahn 和 Kim 记录，归功于 Anders Johansson，并且也由 Grable-Asratian-Kuzjurin 枚举得出；但两者均未给出速率。
英文摘要
Let $M$ be chosen uniformly from all matchings of a finite linear $k$-uniform hypergraph $H$, and let $\overline{q}(H)$ be the average probability that a vertex is left uncovered. If $H$ has maximum degree $D$ and normalized average degree $β=k|E(H)|/(|V(H)|D)$, then, for every fixed $k\geq 2$ and uniformly in the order, $\overline{q}(H) \leq 1-β+(βk+o_D(1))\log\log D/\log D$. The underlying estimate is the order-uniform count $\log Z(H)\geq (|E(H)|/D)(\log D-(k+o_D(1))\log\log D)$, and it also counts matchings of size $(1-o_D(1))|E(H)|/D$. We prove this by sampling edges, deleting those incident with unusually large sampled degrees, and controlling only the total deleted mass before applying the Molloy-Reed list edge-colouring theorem. For $d$-regular $H$, a quantitative version of the earlier Asratian-Kuzjurin sampling-to-counting route, using the near-perfect-matching theorem of Gould and Kelly, sharpens the coefficient: for every fixed $k\geq 3$, $\sup_H \overline{q}(H) \leq (\max\{3,k-1\}+o_d(1))\log\log d/\log d$. Kahn's stronger pointwise prediction was disproved by Lee for $k\geq 3$. The qualitative averaged conclusion was already recorded by Kahn and Kim, crediting Anders Johansson, and also follows from the Grable-Asratian-Kuzjurin enumeration; neither source states a rate.
Comments16 pages. Ancillary files include exact verification programs. Companion repository: https://github.com/agupta/average-vacancy-linear-hypergraphs
