This work introduces and compares approaches for estimating rare-event probabilities related to the number of edges in the random geometric graph on a Poisson point process. In the one-dimensional setting, we derive closed-form expressions for a variety of conditional probabilities related to the number of edges in the random geometric graph and develop conditional Monte Carlo algorithms for estimating rare-event probabilities on this basis. We prove rigorously a reduction in variance when compared to the crude Monte Carlo estimators and illustrate the magnitude of the improvements in a simulation study. In higher dimensions, we use conditional Monte Carlo to remove the fluctuations in the estimator coming from the randomness in the Poisson number of nodes. Finally, building on conceptual insights from large-deviations theory, we illustrate that importance sampling using a Gibbsian point process can further substantially reduce the estimation variance.

这项工作介绍并比较了估计与Poisson点过程中随机几何图中的边数有关的罕见事件概率的方法。在一维设置中，我们推导了与随机几何图中的边数有关的各种条件概率的闭式表达式，并在此基础上开发了用于估计稀有事件概率的条件蒙特卡罗算法。与原始的蒙特卡洛估算器相比，我们严格证明了方差的减小，并说明了模拟研究中改进的幅度。在更高的维度上，我们使用条件蒙特卡洛（Monte Carlo）消除来自节点泊松数的随机性的估计量波动。最后，基于大偏差理论的概念性见解，