Extreme local extrema of two-dimensional discrete Gaussian free field

Marek Biskup; Oren Louidor

doi:10.1007/s00220-015-2565-8

paper

Extreme local extrema of two-dimensional discrete Gaussian free field

arXiv:1306.2602 · doi:10.1007/s00220-015-2565-8

Abstract

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 $N \to \infty$ . 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 $Z (d x) e^{- α h} d h$ , where $α := 2/ g$ with $g := 2/ π$ and where $Z (d x)$ 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.

32 pages, version to appear in Commun. Math. Phys

Extreme local extrema of two-dimensional discrete Gaussian free field

Abstract

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