Rare Events in Random Geometric Graphs

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