The Vlasov-Poisson Dynamics as the Mean Field Limit of Extended Charges

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 \({N \to \infty}\). 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 \({r_N \to \infty}\) and \({r_N/N \to 0}\), to a Poisson point process with intensity measure \({Z{(\rm dx)}{\rm e}^{-\alpha h} {\rm d}h}\), where \({\alpha:= 2/\sqrt{g}}\) 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.