发表机构
IISER Bhopal（印度科学教育研究所博帕尔分校）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文研究带独立边权和顶点标记的伯努利随机图的极值生成树代价，发现基于尾部指数存在相变，并用鞅差方法证明缩放中心化后代价的L^2收敛。
AI 中文摘要
本文考虑一个具有非均匀边概率的伯努利随机图~G，其顶点数为~n，每个顶点具有独立的标记，每条边配备独立的正常权重。边的代价取决于权重以及端点的标记，我们估计包含所有顶点的生成树的最大和最小代价的增长。对于具有重尾的边权且顶点标记均匀分布于单位正方形的图，我们根据尾部衰减指数~s 获得了相变：如果~s 较大，则最大代价基本上由顶点位置决定；如果~s 较小，则边权对最大代价有决定性影响。我们对最小代价生成树得出了类似的结果，并利用基于鞅差的方法建立了适当缩放和中心化后的极值代价的~L^2 收敛性。
英文摘要
In this paper, we consider a Bernoulli random graph~\(G\) on~\(n\) vertices with non-uniform edge probabilities, where each vertex has an independent mark and each edge is equipped with an independent positive weight. The cost of an edge depends on the weight as well the marks of the endvertices and we estimate the growth of the maximum and minimum cost of a spanning tree containing all the vertices. For edge weights with heavy tails and vertex marks distributed uniformly in the unit square, we obtain a phase transition in terms of the tail decay exponent~\(s:\) If~\(s\) is large, then the maximum cost is essentially determined by the vertex locations and if~\(s\) is small, then the edge weights crucially influence the maximum cost. We derive a similar result for minimum cost spanning trees and use martingale difference based methods to establish the~\(L^2-\)convergence of the extremal cost, appropriately scaled and centred.
