A note on the extremal process of the supercritical Gaussian Free Field

TL;DR

This paper proves that the extremal process of the supercritical Gaussian Free Field in dimensions three and higher converges to a Poisson point process, using the Stein-Chen method.

Contribution

It establishes the convergence of the extremal process of the supercritical Gaussian Free Field to a Poisson point process in higher dimensions.

Findings

Extremal process converges to a Poisson point process

Results apply to both infinite-volume and finite-box DGFF

Uses Stein-Chen method for proof

Abstract

We consider both the infinite-volume discrete Gaussian Free Field (DGFF) and the DGFF with zero boundary conditions outside a finite box in dimension larger or equal to 3. We show that the associated extremal process converges to a Poisson point process. The result follows from an application of the Stein-Chen method from Arratia et al. (1989).