The phase transition for planar Gaussian percolation models without FKG

Abstract

We develop techniques to study the phase transition for planar Gaussian percolation models that are not (necessarily) positively correlated. These models lack the property of positive associations (also known as the 'FKG inequality'), and hence many classical arguments in percolation theory do not apply. More precisely, we consider a smooth stationary centred planar Gaussian field f and, given a level ∈ R, we study the connectivity properties of the excursion set {f ≥ − }. We prove the existence of a phase transition at the critical level crit = 0 under only symmetry and (very mild) correlation decay assumptions, which are satisfied by the random plane wave for instance. As a consequence, all non-zero level lines are bounded almost surely, although our result does not settle the boundedness of zero level lines ('no percolation at criticality').To show our main result: (i) we prove a general sharp threshold criterion, inspired by works of Chatterjee, that states that 'sharp thresholds are equivalent to the delocalisation of the threshold location'; (ii) we prove threshold delocalisation for crossing events at large scales -at this step we obtain a sharp threshold result but without being able to locate the threshold -and (iii) to identify the threshold, we adapt Tassion's RSW theory replacing the FKG inequality by a sprinkling procedure. Although some arguments are specific to the Gaussian setting, many steps are very general and we hope that our techniques may be adapted to analyse other models without FKG. Contents 1. Introduction 1 2. Proof of the main results 10 3. Sharp thresholds are equivalent to the delocalisation of the threshold location 20 4. Delocalisation of the threshold location 24 Appendix A. Basic properties of smooth Gaussian fields 30 Appendix B. Stratified Morse functions 31 Appendix C. RSW theory (by Laurin Köhler-Schindler) 33 References 39

Boundedness of the nodal domains of additive Gaussian fields

“…In order to prove Proposition 3.1 we use the hypercontractivity argument of [23] (itself based on an argument of Chatterjee [4]), except that we modify some details to allow us to lift the assumption that K(x) is non-increasing. This is similar to arguments that appeared in [18] in a related setting.…”

“…The advantage of this decomposition is that, in order to analyse the phase transition for f , it will suffice to study the extrema of the processes g i on large intervals. This avoids the need to invoke sharp threshold criteria (as in, for instance, [22,19,18] following the classical approach introduced in [14]).…”

“…• If > c , almost surely {f ≤ } has an unbounded connected component. More recently it has been shown that c = 0 under very mild conditions ( [18], and see also [22,19,21,10] for quantitative versions of this result under stronger conditions). The question of the absence of percolation at criticality (i.e.…”

Asymptotics for the critical level and a strong invariance principle for high intensity shot noise fields

“…The fact that c (f ) = 0 and the existence of a sharp threshold in the sense of (2.1) are closely related, but it is not known that they are equivalent in general. In [MRV20] a quantitative version of (2.1) was used to deduce that c (f ) = 0, and for our purposes it turns out that c (f ) = 0 alone is insufficient (see however Remark 2.10).…”

Percolation phase transition on planar spin systems

“…The question that emerges is: how to study the case where no monotonicity is present? In [35], the authors study a planar percolation model without positive association, which can give a direction for future research in this vein.…”