AI 中文总结
该研究针对分配问题的变体，证明两类场景下相关图为生成树，给出p=1、2时树的非交叉性，还提出随机欧氏分配问题中树的渐近分布猜想。
AI 中文摘要
最小匹配问题是在给定边权图G中寻找具有最小权重的独立边集M_★(G)；当G为二分图时，该问题简化为分配问题。我们考虑该问题的一个变体，其定义为基础图G的多个略微修改版本的最优匹配的并集：H_J(G)=∪_{U∈J} M_★(G_U)。我们建立两类结果：（1）在分配问题的两个不同场景中，我们证明所得图H_J以及某些相关图H̄_J是对应基础图G和Ḡ上的生成树。（2）在这些相同场景中，假设边权由平面点构型的p次欧氏距离幂给出，我们证明当p=1时，树H_J是非交叉的（即其平面嵌入无交叉边）；而值得注意的是，当p=2时，相关树H̄_J是非交叉的。最后，我们在统计力学中提出新的猜想，留待未来研究：在随机欧氏分配问题中（其中点在平面区域上独立同分布），我们猜想当p=2时，树H̄_J渐近服从自由和有线边界条件下的均匀生成树分布，对应两个场景。特别地，在第二个场景中该树的合适路径，以及第一个场景中其平面对偶图的合适路径，渐近服从κ=2的SLE_κ分布。
英文摘要
The \emph{Minimum Matching Problem} consists of finding an independent edge set of minimum weight $M_{\star}(G)$ in a given edge-weighted graph $G$. When $G$ is bipartite, this reduces to the \emph{Assignment Problem}. We consider a variant of this problem defined by taking the union of optimal matchings across various slightly modified versions of the base graph: $H_{\mathcal{J}}(G)=\bigcup_{U \in \mathcal{J}} M_{\star}(G_{U})$. We establish two families of results: (1) In two distinct settings for the Assignment Problem, we prove that the resulting graphs $H_{\mathcal{J}}$, as well as certain associated graphs $\bar{H}_{\mathcal{J}}$, are spanning trees on the relevant base graphs $G$ and $\bar{G}$. (2) In these same settings, assuming the edge weights are given by the $p$-th power of Euclidean distances for point configurations in the plane, we show that for $p=1$ the tree $H_{\mathcal{J}}$ is non-crossing (i.e., its planar embedding has no crossing edges), whereas, remarkably, for $p=2$ the associated tree $\bar{H}_{\mathcal{J}}$ is non-crossing. Finally, we introduce novel conjectures in Statistical Mechanics, to be explored in future work: in the Random Euclidean Assignment Problem (where points are i.i.d.\ on a planar domain), we conjecture that for $p=2$ the trees $\bar{H}_{\mathcal{J}}$ are asymptotically distributed as Uniform Spanning Trees with free and wired boundary conditions in the two respective settings. In particular, suitable paths on the tree in the second setting, and on its planar dual in the first setting, are asymptotically distributed as $\text{SLE}_κ$ with $κ=2$.
Comments116 pages
