Extreme local extrema of two-dimensional discrete Gaussian free field

TL;DR

This paper investigates the extreme local maxima of the two-dimensional discrete Gaussian Free Field, revealing their asymptotic distribution as a Poisson point process with a random intensity measure, and connects it to the continuum Gaussian Free Field.

Contribution

It establishes the limiting distribution of extreme local maxima of the 2D discrete GFF as a Poisson process with a random intensity, extending understanding of extremal behavior in Gaussian fields.

Findings

Extreme local maxima form a Poisson point process in the limit.

The intensity measure involves a random Borel measure related to the derivative martingale.

Provides an integral representation for the law of the absolute maximum.

Abstract

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 $Z(dx)e^{-\alpha h}dh$, where $\alpha:= 2/\sqrt{g}$ with $g:=2/\pi$ 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.