Extreme Local Extrema of Two-Dimensional Discrete Gaussian Free Field

We consider the discrete Gaussian Free Field in a square box in Z² 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 ^{- α h}d h, where α:=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.