## The Annals of Probability

- Ann. Probab.
- Volume 42, Number 3 (2014), 1020-1053.

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

Robert J. Adler, Elina Moldavskaya, and Gennady Samorodnitsky

#### 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.

#### Article information

**Source**

Ann. Probab., Volume 42, Number 3 (2014), 1020-1053.

**Dates**

First available in Project Euclid: 26 March 2014

**Permanent link to this document**

https://projecteuclid.org/euclid.aop/1395838123

**Digital Object Identifier**

doi:10.1214/12-AOP794

**Mathematical Reviews number (MathSciNet)**

MR3189065

**Zentralblatt MATH identifier**

1301.60041

**Subjects**

Primary: 60G15: Gaussian processes 60F10: Large deviations

Secondary: 60G60: Random fields 60G70: Extreme value theory; extremal processes 60G17: Sample path properties

**Keywords**

Gaussian process excursion set large deviations exceedence probabilities connected component optimal path energy of measures

#### Citation

Adler, Robert J.; Moldavskaya, Elina; Samorodnitsky, Gennady. On the existence of paths between points in high level excursion sets of Gaussian random fields. Ann. Probab. 42 (2014), no. 3, 1020--1053. doi:10.1214/12-AOP794. https://projecteuclid.org/euclid.aop/1395838123