On the existence of paths between points in high level excursion sets of Gaussian random fields

TL;DR

This paper investigates the probability and characteristics of paths connecting points within high-level excursion sets of Gaussian random fields, using large deviations theory to analyze connectivity and most likely paths.

Contribution

It provides the first asymptotic analysis of the probability that two points in an excursion set are connected, and characterizes the most probable connecting paths.

Findings

Asymptotic probabilities for connectivity in high-level excursion sets.

Characterization of most likely paths connecting points.

Insights into the structure of Gaussian random fields at high levels.

Abstract

The structure of Gaussian random fields over high levels is a well researched and well understood area, particularly if the field is smooth. However, the question as to whether or not two or more points which lie in an excursion set belong to the same connected component has constantly eluded analysis. We study this problem from the point of view of large deviations, finding the asymptotic probabilities that two such points are connected by a path laying within the excursion set, and so belong to the same component. In addition, we obtain a characterization and descriptions of the most likely paths, given that one exists.