The phase transition for planar Gaussian percolation models without positive associations

Abstract: Given a smooth stationary centred Gaussian field f on the plane and a level ℓ in ℝ, we study the connectivity properties of the set {f < ℓ}. We prove that the critical level is ℓc = 0 under only symmetry and (very mild) correlation-decay assumptions, which includes the important example of the random plane wave. Since these models are not necessarily positively associated (i.e. they do not enjoy the “Fortuin-Kasteleyn-Ginibre (FKG) inequality”), many classical arguments from percolation/statistical mechanics do not apply, and so these are a rare example of non-FKG models whose critical point can be rigorously computed.

Although many arguments are specific to the Gaussian setting we hope that our techniques may be adapted to analyse other non-FKG models, of which there are many important examples (e.g. anti-ferromagnetic Ising models, certain regimes of the FK model and O(n) loop models, random current models, Boolean models on non-Poisson point processes etc). This is joint work with Hugo Vanneuville and Alejandro Rivera and will appear on arXiv very soon.

probability

Audience: researchers in the topic

Probability Victoria Seminar (PVSeminar)

Series comments: All the information about the forthcoming and past events (including the seminars' Zoom links, where available) can be found at probvic.wordpress.com/pvseminar/. The URL of the seminar YouTube channel is: www.youtube.com/channel/UCsEpm-bMfCL2w8abq9s7w6w