Institute of Mathematics

Talk

Modul:   MAT971  Stochastische Prozesse

Coincidence of critical parameters for percolation of Gaussian Free Field level-sets

Franco Severo talk

Date: 13.11.19  Time: 17.15 - 18.15  Room: ETH HG G 43

We consider level-sets of the Gaussian Free Field on Z^d, for d>2, above a given height parameter h \in R. As h varies, this defines a canonical site percolation model with slow (polynomial) decay of correlations. We prove that three natural critical parameters associated to this model (h_{**}, h_{*} and \bar{h}) respectively describing a strongly non-percolative regime, the emergence of an infinite cluster, and a strongly percolative regime, actually coincide. Combined with previous results, this equality has many implications regarding the geometry of GFF level-sets, both in the subcritical and supercritical regimes. The core of the proof is a new interpolation scheme that integrates out the long-range dependency of the GFF, thus allowing to import results from finite-range percolation models. In order to implement this strategy, we make extensive use of renormalization techniques.

Based on a joint work with Hugo Duminil-Copin, Subhajit Goswami and Pierre-François Rodriguez.