The global structure of the minimal spanning tree (MST) is expected to be universal for a large class of underlying random discrete structures. However, very little is known about the intrinsic geometry of MSTs of most standard models, and so far the scaling limit of the MST viewed as a metric measure space has only been identified in the case of the complete graph (Addario-Berry et al. in Ann Probab 45(5):3075–3144, 2017). In this work, we show that the MST constructed by assigning i.i.d. continuous edge weights to either the random (simple) 3-regular graph or the 3-regular configuration model on n vertices, endowed with the tree distance scaled by \(n^{-1/3}\) and the uniform probability measure on the vertices, converges in distribution with respect to Gromov–Hausdorff–Prokhorov topology to a random compact metric measure space. Further, this limiting space has the same law as the scaling limit of the MST of the complete graph identified in Addario-Berry et al. (2017) up to a scaling factor of \(6^{1/3}\). Our proof relies on a novel argument that proceeds via a comparison between a 3-regular configuration model and the largest component in the critical Erdős–Rényi random graph. The techniques of this paper can be used to establish the scaling limit of the MST in the setting of general random graphs with given degree sequences provided two additional technical conditions are verified.

预计最小生成树 (MST) 的全局结构对于一大类底层随机离散结构是通用的。然而，对于大多数标准模型的 MST 的内在几何学知之甚少，到目前为止，仅在完整图的情况下才确定了被视为度量度量空间的 MST 的缩放限制（Addario-Berry et al.在 Ann Probab 45(5):3075–3144, 2017)。在这项工作中，我们展示了通过将 iid 连续边权重分配给n个顶点上的随机（简单）3-正则图或 3-正则配置模型来构造的 MST ，赋予了由\(n^{ -1/3}\)和顶点上的均匀概率测度，关于 Gromov-Hausdorff-Prokhorov 拓扑的分布收敛到随机紧凑度量测度空间。此外，这个限制空间与 Addario-Berry 等人确定的完整图的 MST 的缩放限制具有相同的规律。(2017) 缩放因子为\(6^{1/3}\)。我们的证明依赖于一个新颖的论点，该论点通过 3-正则配置模型与关键 Erdős-Rényi 随机图中的最大分量之间的比较进行。如果验证了两个额外的技术条件，本文的技术可用于在具有给定度数序列的一般随机图的设置中建立 MST 的标度限制。