Extreme local extrema of two-dimensional discrete Gaussian free field

We consider the discrete Gaussian Free Field in a square box in ${\mathbb{Z}}^2$ 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 $N\to \mathrm{\infty }$. 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 $r_N\to \mathrm{\infty }$ and $r_N\mathrm{/}N\to 0$, to a Poisson point process with intensity measure $Z(dx)e^{-\alpha h}dh$,where$\alpha:= 2/\sqrt{g}$with$g:=2/\pi$andwhere$Z(dx)$ 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.