论文标题
远程渗透图直径的急剧渐近顺序
A sharp leading order asymptotic of the diameter of a long range percolation graph
论文作者
论文摘要
许多现实世界网络都表现出所谓的小世界现象：它们的典型距离远小于其尺寸。该现象的一种数学模型是$ d $二维盒$ \ \ {0，1，\ cdots，n \}^d $上的远程渗透图，其中远距离站点之间的边缘是在远处的概率之间独立添加的，概率是eucklidean距离的力量。一个自然的问题是，以图理论距离测量的尺寸$ n $的盒子直径是如何用$ n $的。过去对这个问题进行了深入研究，答案取决于连接概率中的指数$ s $。在这项工作中，我们专注于Coppersmith，Gamarnik和Sviridenko在一项工作中较早研究的关键制度$ S = D $，并通过利用由于模型中的大量独立性而获得的高度浓度来改善其与急剧的前导渐近渐近造剂的界限。
Many real-world networks exhibit the so-called small-world phenomenon: their typical distances are much smaller than their sizes. One mathematical model for this phenomenon is a long-range percolation graph on a $d$-dimensional box $\{0, 1, \cdots, N\}^d$, in which edges are independently added between far-away sites with probability falling off as a power of the Euclidean distance. A natural question is how the resulting diameter of the box of size $N$, measured in graph-theoretical distance, scales with $N$. This question has been intensely studied in the past and the answer depends on the exponent $s$ in the connection probabilities. In this work we focus on the critical regime $s = d$ studied earlier in a work by Coppersmith, Gamarnik, and Sviridenko and improve the bounds obtained there to a sharp leading-order asymptotic, by exploiting the high degree of concentration due to the large amount of independence in the model.
