Extreme Local Extrema of Two-Dimensional Discrete Gaussian Free Field

We consider the discrete Gaussian Free Field in a square box in 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 . 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 and , to a Poisson point process with intensity measure , where with g: = 2/pi 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.