Percolation of the excursion sets of planar symmetric shot noise fields

TL;DR

This paper proves phase transitions in the connectivity of excursion sets of planar symmetric shot noise fields, depending on level or intensity, with implications for understanding spatial connectivity in stochastic models.

Contribution

It establishes the existence of phase transitions in the connectivity of shot noise fields, including cases with symmetric log-concave marks and kernels, extending previous knowledge.

Findings

Phase transition with respect to level for symmetric log-concave mark distributions.

Phase transition with respect to intensity for non-log-concave cases.

Applicable to Gaussian, uniform, and Laplace mark distributions.

Abstract

We prove the existence of phase transitions in the global connectivity of the excursion sets of planar symmetric shot noise fields. Our main result establishes a phase transition with respect to the level for shot noise fields with symmetric log-concave mark distributions, including Gaussian, uniform, and Laplace marks, and kernels that are positive, symmetric, and have sufficient tail decay. Without the log-concavity assumption we prove a phase transition with respect to the intensity of positive marks.