The sharp phase transition for level set percolation of smooth planar Gaussian fields

TL;DR

This paper establishes a sharp phase transition for the connectivity of level sets in smooth planar Gaussian fields, extending known results to broader correlation decay conditions and employing innovative probabilistic techniques.

Contribution

It proves the existence and sharpness of the phase transition for level set percolation under polynomial decay of correlations, generalizing previous super-exponential decay results.

Findings

Phase transition occurs at the zero level for broad Gaussian fields.

Crossing probabilities decay exponentially in the sub-critical regime.

New quasi-independence results and sharp threshold techniques are developed.

Abstract

We prove that the connectivity of the level sets of a wide class of smooth centred planar Gaussian fields exhibits a phase transition at the zero level that is analogous to the phase transition in Bernoulli percolation. In addition to symmetry, positivity and regularity conditions, we assume only that correlations decay polynomially with exponent larger than two -- roughly equivalent to the integrability of the covariance kernel -- whereas previously the phase transition was only known in the case of the Bargmann-Fock covariance kernel which decays super-exponentially. We also prove that the phase transition is sharp, demonstrating, without any further assumption on the decay of correlations, that in the sub-critical regime crossing probabilities decay exponentially. Key to our methods is the white-noise representation of a Gaussian field; we use this on the one hand to prove new quasi-independence results, inspired by the notion of influence from Boolean functions, and on the other hand to establish sharp thresholds via the OSSS inequality for i.i.d. random variables, following the recent approach of Duminil-Copin, Raoufi and Tassion.