We consider the discrete Gaussian Free Field in a square box in ( Z ^2 ) of side length N with zero boundary conditions and study the joint law of its properly-centered extreme values (h) and their scaled spatial positions (x) in the limit as ( N ). Restricting attention to extreme local maxima, i.e., the extreme points that are maximal in an r N -neighborhood thereof, we prove that the associated process tends, whenever ( r_N ) and ( r_N N 0 ), to a Poisson point process with intensity measure ( Z ( dx) e ^ - h d h ), where ( := 2 g ) with g: = 2 π and where Z(dx) is a random Borel measure on [0, 1]2. In particular, this yields an integral representation of the law of the absolute maximum, similar to that found in the context of Branching Brownian Motion. We give evidence that the random measure Z is a version of the derivative martingale associated with the continuum Gaussian Free Field.
@cite_7: We consider the discrete two-dimensional Gaussian free field on a box of side length @math , with Dirichlet boundary data, and prove the convergence of the law of the centered maximum of the field.© 2015 Wiley Periodicals, Inc.
@cite_47: We consider the maximum of the discrete two dimensional Gaussian free field in a box, and prove the existence of a (dense) deterministic subsequence along which the maximum, centered at its mean, is tight. The method of proof relies on an argument developed by Dekking and Host for branching random walks with bounded increments and on comparison results specific to Gaussian fields.
@cite_33: We consider the maximum of the discrete two-dimensional Gaussian free field (GFF) in a box and prove that its maximum, centered at its mean, is tight, settling a longstanding conjecture. The proof combines a recent observation by Bolthausen, Deuschel, and Zeitouni with elements from Bramson's results on branching Brownian motion and comparison theorems for Gaussian fields. An essential part of the argument is the precise evaluation, up to an error of order 1, of the expected value of the maximum of the GFF in a box. Related Gaussian fields, such as the GFF on a two-dimensional torus, are also discussed. © 2011 Wiley Periodicals, Inc.
@cite_20: We consider the lattice version of the free eld in two dimensions (also called harmonic crystal). The main aim of the paper is to discuss quantitatively the entropic repulsion of the random surface in the presence of a hard wall. The basic ingredient of the proof is the analysis of the maximum of the eld which requires a multiscale analysis reducing the problem essentially to a problem on a eld with a tree structure. 2000 MSC: 60K35, 60G15, 82B41
Let us now move to the subject of interest in the present paper, which is the behavior of the extreme values of the DGFF in the limit as @math . A particular aspect of this, the maximum has been studied very intensely. Indeed, Bolthausen, Deuschel and Giacomin @cite_20 showed that @math . More recently, building on Bolthausen, Deuschel and Zeitouni @cite_47 , Bramson and Zeitouni @cite_33 proved that the family @math is tight when @math is as in . Finally, in a very recent development, Bramson, Ding and Zeitouni @cite_7 proved the following result: