论文标题
在重尾政权中最小跨越树的几何形状：新的普遍性课程
Geometry of the minimal spanning tree in the heavy-tailed regime: new universality classes
论文作者
论文摘要
在过去的十年中，由统计物理学家提出的最佳图表中最佳路径的行为的一个众所周知的开放问题，并得到大量数值证据的支持[31,32,38,70] [31,32,38,70]如下：对于带有学位的大量随机图模型的大量随机图模型，在（3,4,4）$中的距离（3,4,4）的距离（3,4,4）<超临界制度的缩放，例如$ n^{（τ-3）/（τ-1）} $。本文的目的是朝着这一猜想的证明取得进展。 我们考虑了一个超临界的不均匀随机图模型，该模型具有学位指数$τ\ in（3，4）$，与Aldous的乘法融合密切相关，并表明通过分配I.I.D构建的MST。连续的重量与其巨型组件的边缘连续，并以$ n^{ - （τ-3）/（τ-1）} $缩放的树距离缩放，相对于Gromov-Hausdorff拓扑结合了随机紧凑的真实树。此外，几乎可以肯定的是，这个极限空间中的每个点都具有一（叶），或两个或无穷大（轮毂），在这个空间中，叶子和集线器的集合都稠密，并且该空间的Minkowski维度等于$（τ-1）/（τ-1）/（τ-3）$。 从渐近的意义上讲，乘法结合描述了各种近临界随机图过程的组件大小的演变。我们期望本文中的限制空间是用于为其他重型尾部随机图模型构建的MST缩放限制的候选。
A well-known open problem on the behavior of optimal paths in random graphs in the strong disorder regime, formulated by statistical physicists, and supported by a large amount of numerical evidence over the last decade [31,32,38,70] is as follows: for a large class of random graph models with degree exponent $τ\in (3,4)$, the distance between two typical points on the minimal spanning tree (MST) on the giant component in the supercritical regime scales like $n^{(τ-3)/(τ-1)}$. The aim of this paper is to make progress towards a proof of this conjecture. We consider a supercritical inhomogeneous random graph model with degree exponent $τ\in(3, 4)$ that is closely related to Aldous's multiplicative coalescent, and show that the MST constructed by assigning i.i.d. continuous weights to the edges in its giant component, endowed with the tree distance scaled by $n^{-(τ-3)/(τ-1)}$, converges in distribution with respect to the Gromov-Hausdorff topology to a random compact real tree. Further, almost surely, every point in this limiting space either has degree one (leaf), or two, or infinity (hub), both the set of leaves and the set of hubs are dense in this space, and the Minkowski dimension of this space equals $(τ-1)/(τ-3)$. The multiplicative coalescent, in an asymptotic sense, describes the evolution of the component sizes of various near-critical random graph processes. We expect the limiting spaces in this paper to be the candidates for the scaling limit of the MST constructed for a wide array of other heavy-tailed random graph models.
