Convergence in law of the maximum of the two-dimensional discrete Gaussian free field

TL;DR

This paper proves that the distribution of the maximum value of a two-dimensional discrete Gaussian free field, after appropriate recentering, converges as the domain size grows, providing insights into its extreme value behavior.

Contribution

It establishes the convergence in law of the recentered maximum of the 2D Gaussian free field, a significant step in understanding its extreme value distribution.

Findings

Maximum of the field converges in distribution as N grows.

Recentered maximum exhibits a limiting law.

Provides a rigorous foundation for extreme value analysis of GFF.

Abstract

We consider the two-dimensional Gaussian Free Field on a box of side length $N$, with Dirichlet boundary data, and prove the convergence of the law of the recentered maximum of the field.