We study the random geometry of first passage percolation on the complete graph equipped with independent and identically distributed positive edge weights. We consider the case where the lower extreme values of the edge weights are highly separated. This model exhibits strong disorder and a crossover between local and global scales. Local neighborhoods are related to invasion percolation that display self-organised criticality. Globally, the edges with relevant edge weights form a barely supercritical Erdős–Rényi random graph that can be described by branching processes. This near-critical behaviour gives rise to optimal paths that are considerably longer than logarithmic in the number of vertices, interpolating between random graph and minimal spanning tree path lengths. Crucial to our approach is the quantification of the extreme-value behavior of small edge weights in terms of a sequence of parameters \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(s_n)_{n\ge 1}$$\end{document}(sn)n≥1 that characterises the different universality classes for first passage percolation on the complete graph. We investigate the case where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$s_n\rightarrow \infty $$\end{document}sn→∞ with \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$s_n=o(n^{1/3})$$\end{document}sn=o(n1/3), which corresponds to the barely supercritical setting. We identify the scaling limit of the weight of the optimal path between two vertices, and we prove that the number of edges in this path obeys a central limit theorem with mean approximately \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$s_n\log {(n/s_n^3)}$$\end{document}snlog(n/sn3) and variance \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$s_n^2\log {(n/s_n^3)}$$\end{document}sn2log(n/sn3). Remarkably, our proof also applies to n-dependent edge weights of the form \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$E^{s_n}$$\end{document}Esn, where E is an exponential random variable with mean 1, thus settling a conjecture of Bhamidi et al. (Weak disorder asymptotics in the stochastic meanfield model of distance. Ann Appl Probab 22(1):29–69, 2012). The proof relies on a decomposition of the smallest-weight tree into an initial part following invasion percolation dynamics, and a main part following branching process dynamics. The initial part has been studied in Eckhoff et al. (Long paths in first passage percolation on the complete graph I. Local PWIT dynamics. Electron. J. Probab. 25:1–45, 2020. 10.1214/20-EJP484); the current paper focuses on the global branching dynamics.

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完整图 II 上第一通道渗透中的长路径。全局分支动态

我们研究了具有独立且同分布的正边权重的完整图上的第一通道渗透的随机几何。我们考虑边缘权重的下极值高度分离的情况。该模型表现出强烈的无序性以及局部和全局尺度之间的交叉。当地社区与显示自组织临界的入侵渗透有关。在全局范围内，具有相关边权重的边形成了一个几乎不超临界的 Erdős-Rényi 随机图，可以通过分支过程来描述。这种近乎临界的行为产生了在顶点数量上比对数长得多的最优路径，在随机图和最小生成树路径长度之间进行插值。我们调查 \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength 的情况{\oddsidemargin}{-69pt} \begin{document}$$s_n\rightarrow \infty $$\end{document}sn→∞ with \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \ usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$s_n=o(n^{1/ 3})$$\end{document}sn=o(n1/3)，对应于勉强超临界设置。我们确定了两个顶点之间最优路径权重的缩放限制，我们证明了这条路径中的边数服从中心极限定理，其均值约为 \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{ amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$s_n\log {(n/s_n^3)}$$\end{document}snlog( n/sn3) 和方差 \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \ setlength{\oddsidemargin}{-69pt} \begin{document}$$s_n^2\log {(n/s_n^3)}$$\end{document}sn2log(n/sn3)。值得注意的是，我们的证明也适用于 \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs 形式的 n 相关边权重} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$E^{s_n}$$\end{document}Esn，其中 E 是指数随机变量，均值为 1，因此稳定Bhamidi 等人的一个猜想。（距离随机平均场模型中的弱无序渐近。Ann Appl Probab 22(1):29–69, 2012）。该证明依赖于将最小权重树分解为入侵渗透动力学之后的初始部分和分支过程动力学之后的主要部分。Eckhoff 等人已经研究了初始部分。（完整图 I. 局部 PWIT 动力学的第一通道渗透中的长路径。电子。J.普罗巴布。25:1–45, 2020. 10.1214/20-EJP484)；目前的论文侧重于全球分支动态。