Extreme Local Extrema of Two-Dimensional Discrete Gaussian Free Field

• DOI: 10.1007/s00220-015-2565-8
• Publication Date: 2016-01-28T21:49:47Z
• Authors: Marek Biskup; Oren Louidor
• Keywords: Gaussian free field

32 pages, version to appear in Commun. Math. Phys<br/>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 $r_N/N\to0$, to a Poisson point process with intensity measure $Z(dx)e^{-��h}dh$, where $��:= 2/\sqrt{g}$ with $g:=2/��$ 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.<br/>