H. Vanneuville & S. Muirhead: The phase transition for planar Gaussian percolation without FKG

Abstract: In this talk, we present techniques to study the phase transition of planar percolation models that do not satisfy the FKG inequality. More precisely, the models that we consider are the following: given a stationary centered Gaussian field f on the plane and some level u \in \mathbb R, we study the connectivity properties of the set {f < u}. If the covariance of the field is not pointwise positive then the model does not satisfy the FKG inequality. We prove that the critical level is the self-dual level u=0 under only symmetry and (very mild) correlation-decay assumptions, which are satisfied for the important example of the random plane wave that we will introduce. However, we do not settle the boundedness of the components of {f<0}. As a result, these models are examples of planar models for which the critical point can be computed but for which we do not manage to prove that there is ‘no percolation at criticality’.
We are inspired a lot by works of Chatterjee on superconcentration and by the famous paper of Harris from 1960, as well as by Tassion’s RSW theory (even if in these two last works FKG is crucial; to our knowledge, Harris’s paper is even the first paper where this inequality is proven).
Although many arguments are specific to the Gaussian setting, some steps are very general and we hope that our techniques may be adapted to analyse other models without FKG.

This is joint work with Alejandro Rivera and the paper will hopefully be posted on Arxiv within a few weeks.


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