Excursion Sets and Critical Points of Gaussian Random Fields

Abstract

Modeling the critical points of a Gaussian random field is an important challenge in stochastic geometry. In this thesis, we focus on stationary Gaussian random fields and study the locations and types of the critical points over high thresholds. Under certain conditions, we show that when the threshold tends to infinity and the searching area expands with a matching speed, both the locations of the local maxima and the locations of all the critical points above the threshold converge weakly to a Poisson point process. We then discuss the local behavior of the critical points by looking at the type of a critical point given there is another critical point close to it. We show if two critical points above u are very close one to each other, then they are most likely to be one local maximum and one saddle point with index N − 1. We will further discuss the modeling of the critical points when the threshold is high but not very high. The proposed model has a hierarchical structure that can capture the positions of the global maxima and other critical points simultaneously. The performance of the proposed model is evaluated by the comparisons between the L functions.