均匀随机交集图的稠密与非稠密极限
Dense and nondense limits for uniform random intersection graphs
AI总结：
本文研究了排列图、圆图和单位区间图三类均匀随机交集图的缩放极限，分别证明了稠密情形下收敛到确定性图限，非稠密情形下在Gromov-Prokhorov意义下收敛到带随机度量的区间，并给出了固定大小团数量的渐近结果。
摘要：
我们获得了在交集图的三个族中均匀绘制的随机图的缩放极限：排列图、圆图和单位区间图。前两个族通常生成稠密图，在这些情况下，我们证明了几乎必然收敛到一个显式的确定性图限。均匀单位区间图是非稠密的，我们证明了在距离归一化后，在Gromov-Prokhorov意义下的收敛：极限对象是区间$[0,1]$，赋予通过布朗游走定义的随机度量。还给出了这三个族中每个族中均匀随机图中大小为$k$（$k$固定）的团数量的渐近结果。在三种情况下，证明的一个重要成分是，对于每个类中的不可分解图（其中不可分解性的概念取决于类），定义图的组合对象（排列、匹配或区间）本质上是唯一的。
英文摘要：
We obtain the scaling limits of random graphs drawn uniformly in three families of intersection graphs: permutation graphs, circle graphs, and unit interval graphs. The two first families typically generate dense graphs, in these cases we prove a.s. convergence to an explicit deterministic graphon. Uniform unit interval graphs are nondense and we prove convergence in the sense of Gromov-Prokhorov after normalization of the distances: the limiting object is the interval $[0,1]$ endowed with a random metric defined through a Brownian excursion. Asymptotic results for the number of cliques of size $k$ ($k$ fixed) in a uniform random graph in each of these three families are also given. In all three cases, an important ingredient of the proof is that, for indecomposable graphs in each class (where the notion of indecomposability depends on the class), the combinatorial object defining the graph (permutation, matching, or intervals) is essentially unique.
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