随机几何图投影中交叉数与应力的极限定理
Limit theorems for the number of crossings and stress in projections of the random geometric graph
AI总结：
本文研究高维随机几何图平面投影中边交叉数的极限行为，证明其点过程在 Kantorovich-Rubinstein 距离下收敛到 Poisson 点过程，并建立交叉数与应力之间的多元中心极限定理。
摘要：
我们考虑通过将某个紧凸集 $W\subset \mathbb{R}^d$（$d\geq 3$）上的随机几何图投影到平面上而生成的随机图绘制中的边交叉数量。这些交叉的位置构成一个点过程的支撑。我们证明，如果交叉的期望数量收敛到一个正但有限的值，则该点过程在 Kantorovich-Rubinstein 距离下收敛到 Poisson 点过程。我们进一步证明，当随机几何图中的期望顶点度收敛到一个正有限值时，交叉数量与称为应力的第二个变量之间存在多元中心极限定理。
英文摘要：
We consider the number of edge crossings in a random graph drawing generated by projecting a random geometric graph on some compact convex set $W\subset \mathbb{R}^d$, $d\geq 3$, onto a plane. The positions of these crossings form the support of a point process. We show that if the expected number of crossings converges to a positive but finite value, this point process converges to a Poisson point process in the Kantorovich-Rubinstein distance. We further show a multivariate central limit theorem between the number of crossings and a second variable called the stress that holds when the expected vertex degree in the random geometric graph converges to a positive finite value.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
