Extremes of the supercritical Gaussian Free Field

TL;DR

This paper proves that the maximum of the discrete Gaussian Free Field in dimensions 3 and higher follows a Gumbel distribution, using the Stein-Chen method to establish the result for both infinite and zero boundary conditions.

Contribution

It demonstrates the Gumbel domain of attraction for the maximum of the DGFF in higher dimensions, applying Stein-Chen method in this context for the first time.

Findings

Maximum of DGFF in dimensions ≥3 follows Gumbel distribution

Results apply to both infinite-volume and zero boundary conditions

Uses Stein-Chen method for probabilistic approximation

Abstract

We show that the rescaled maximum of the discrete Gaussian Free Field (DGFF) in dimension larger or equal to 3 is in the maximal domain of attraction of the Gumbel distribution. The result holds both for the infinite-volume field as well as the field with zero boundary conditions. We show that these results follow from an interesting application of the Stein-Chen method from Arratia et al. (1989).