The phase transition for planar Gaussian percolation models without FKG

We develop techniques to study the phase transition for planar Gaussian percolation models that are not (necessarily) positively correlated. These lack property of positive associations (also known as ‘FKG inequality’), and hence many classical arguments in theory do apply. More precisely, we consider a smooth stationary centred field f and, given level ℓ∈R, connectivity properties excursion set {f≥−ℓ}. prove existence at critical ℓcrit=0 under only symmetry (very mild) correlation decay assumptions, which satisfied by random plane wave instance. As consequence, all nonzero lines bounded almost surely, although our result does settle boundedness zero (‘no criticality’). To show main result: (i) general sharp threshold criterion, inspired works Chatterjee, states ‘sharp thresholds equivalent delocalisation location’; (ii) crossing events large scales—at this step obtain but without being able locate threshold—and (iii) identify threshold, adapt Tassion’s RSW replacing FKG inequality sprinkling procedure. Although some specific setting, steps very hope may be adapted analyse other FKG.