距离随机平均场模型的直径
Diameter of the stochastic mean-field model of distance
摘要：
本文研究了具有指数平均$n$边长的$n$上的完全图$\cK_n$.将$C_{ij}$写为顶点$i,j\in [n]$之间的最小权重路径的权重，Janson证明了$\max_{i,j\in [n]} C_{ij}/\log{n}$以概率收敛到3。我们扩展了这一结果表明，$\max_{i,j\in [n]} C_{ij} - 3\log{n}$收敛分布的一个限制随机变量，可以通过一个最大化的过程中确定的一个限制无限的随机结构。有趣的是，Riordan和Wormald的工作中也发现了这个极限随机变量作为刚超临界Erdens-Rényi随机图的重中心图直径的弱极限.
英文摘要：
We consider the complete graph $\cK_n$ on $n$ vertices with exponential mean $n$ edge lengths. Writing $C_{ij}$ for the weight of the smallest-weight path between vertex $i,j\in [n]$, Janson showed that $\max_{i,j\in [n]} C_{ij}/\log{n}$ converges in probability to 3. We extend this result by showing that $\max_{i,j\in [n]} C_{ij} - 3\log{n}$ converges in distribution to a limiting random variable that can be identified via a maximization procedure on a limiting infinite random structure. Interestingly, this limiting random variable has also appeared as the weak limit of the re-centered graph diameter of the barely supercritical Erdős-Rényi random graph in work by Riordan and Wormald.
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