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 two other recent papers, Ding @cite_43 and Ding and Zeitouni @cite_4 have studied the tails of the maximum. Specifically, for the upper tail they derived for some @math . For the lower tail, they got the estimates for some @math . The recent work of Bramson, Ding and Zeitouni @cite_7 controls the asymptotic form of the upper tail of @math including the multiplicative constant. Indeed, recasting Proposition 2.2 of @cite_7 into a simpler form, we get:
@cite_43 @cite_4 @cite_7
2592044494 2070582197 2097649415
We study the tail behavior for the maximum of discrete Gaussian free field on a 2D box with Dirichlet boundary condition after centering by its expectation. We show that it exhibits an exponential decay for the right tail and a double exponential decay for the left tail. In particular, our result implies that the variance of the maximum is of order 1, improving an (o( n) ) bound by Chatterjee (Chaos, concentration, and multiple valleys, 2008) and confirming a folklore conjecture. An important ingredient for our proof is a result of Bramson and Zeitouni (Commun. Pure Appl. Math, 2010), who proved the tightness of the centered maximum together with an evaluation of the expectation up to an additive constant. We consider in this paper the collection of near maxima of the discrete, two dimensional Gaussian free field in a box with Dirichlet boundary conditions. We provide a rough description of the geometry of the set of near maxima, estimates on the gap between the two largest maxima, and an estimate for the right tail up to a multiplicative constant on the law of the centered maximum. 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.