Level-set percolation for the Gaussian free field

A simulation of the 2-dimensional DGFF.

Existence of phase transition for the level-set percolation for the discrete Gaussian free field on \(\mathbb{Z^d}\) (DGFF) is a problem that received much attention in the past years, in particular it was studied in the 80’s by J. Bricmont, J.L. Lebowitz and C. Maes. They showed that in three dimension the DGFF has a nontrivial percolation behavior: sites on which \(\varphi_x\geq h\) percolate if and only if \(h<h_*\) with \(0\leq h_*<\infty\). Moreover, they generalized the lower bound for \(h_*\) in any dimension \(d\geq 3\), i.e. \(h_*(d)\geq 0\), but they were not able to extend the proof of existence of a non trivial transition for any \(d\geq 4.\) Recently P.-F. Rodriguez and A.-S. Sznitman proved that \(h_*(d)\) is finite for all \(d\geq 3\) as a corollary of a more general result concerning the stretched exponential decay of the connectivity function when \(h>h_{**}\) , where \(h_{**}\) is a second critical parameter that satisfies \(h_{**}\geq h_*\) . In this thesis we tried to get acquainted with some of the techniques developed in the domain, notably to control the large excursions of these fields and to understand the entropic repulsion phenomena, and to comprehend the results on level set percolation in dimension three and larger. In particular, the main goal is to present the two works of Bricmont, Lebowitz and Maes and of Rodriguez and Sznitman. Finally, in the last two chapters we also present a new and original (but incomplete) generalization of the proof (due to J. Bricmont, J.L. Lebowitz and C. Maes ) of the existence of a non trivial phase transition to any \(d\geq 3\).

Master Thesis (PDF)