Extreme Local Extrema of Two-Dimensional Discrete Gaussian Free Field

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 - 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.