随机最小生成树中的成对边相关性：通用界与完全图负相关性
Pairwise edge correlations in random minimum spanning trees: a universal bound and complete-graph negative correlation
浏览论文内容
中文总结 AI 辅助
本文针对随机最小生成树的成对边相关性，证明正相关性受通用界控制、$n\boldsymbol{\u2265}3$时完全图边对负相关，还给出边分布非相同时无通用常数的反例。
中文摘要 AI 辅助
设$G$为有限连通多重图，其边从某个无原子分布中独立获取权重，记$\text{MST}(G)$为对应的随机最小生成树。该树的分布不满足成对负相关性：Lyons、Peres与Schramm给出了两条正相关边的例子，本文在简单图上也给出了此类例子。本文证明正相关性仍受一致控制：$\boldsymbol{P}(e,f \text{ 属于 } T)\boldsymbol{\u2264}8\boldsymbol{P}(e \text{ 属于 } T)\boldsymbol{P}(f \text{ 属于 } T)$，回答了Tang与Zhang记录的R. Lyons提出的问题。在对所有其他权重取条件后，Harris不等式给出条件负相关性；两个瓶颈距离与精确二阶矩估计控制剩余环境协方差。对$K_n$，本文证明当$n \u2265 3$时每条边对均满足负相关性。关键有限恒等式为$\boldsymbol{E}[\text{deg}(x)^2]=10(n-1)/n-4\boldsymbol{E}[L_n]$，其中$L_n$是率为1的指数权重下最小生成树的总权重。$\boldsymbol{E}[L_n]$的已知展开式给出收敛到$10-4\boldsymbol{\u03b6}(3)$的速率及成对相关比的极限。最后，一个显式$K_4$族表明，当独立边分布无需相同时，不存在通用常数成立。
英文摘要
Let $G$ be a finite connected multigraph whose edges receive independent weights from one atomless law, and let $\operatorname{MST}(G)$ be the resulting random minimum spanning tree. Its law is not pairwise negatively correlated: Lyons, Peres and Schramm exhibited two positively correlated edges, and we give such an example on a simple graph. We prove that positive correlation is nevertheless uniformly controlled: $\mathbf{P}(e,f\in T)\leq 8\mathbf{P}(e\in T)\mathbf{P}(f\in T)$, answering a question of R. Lyons recorded by Tang and Zhang. After conditioning on all other weights, Harris's inequality gives conditional negative correlation; two bottleneck distances and a sharp second-moment estimate control the remaining environmental covariance. For $K_n$ we prove pairwise negative correlation for every $n\geq 3$. The key finite identity is $\mathbf{E}[\mathrm{deg}(x)^2]=10(n-1)/n-4\mathbf{E}[L_n]$, where $L_n$ is the total weight of the minimum spanning tree under rate-one exponential weights. Known expansions for $\mathbf{E}[L_n]$ then give the rate of convergence to $10-4ζ(3)$ and the limits of both pair-correlation ratios. Finally, an explicit $K_4$ family shows that no universal constant survives when the independent edge laws need not be identical.
