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.
: In a very recent posting, Chatterjee, Dembo and Ding @cite_37 show that an analogous result to holds for very general Gaussian fields.
@cite_37
1984642139
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 long-standing conjecture. The proof combines a recent observation of Bolthausen, Deuschel and Zeitouni with elements from (Bramson 1978) 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.