Poisson-Dirichlet Statistics for the extremes of the two-dimensional discrete Gaussian Free Field

TL;DR

This paper proves that the extreme values of the two-dimensional discrete Gaussian free field follow Poisson-Dirichlet statistics, extending previous methods to account for boundary effects in this complex model.

Contribution

It applies a novel approach based on one-step replica symmetry breaking to establish Poisson-Dirichlet statistics for the 2D discrete Gaussian free field, considering boundary effects.

Findings

Poisson-Dirichlet statistics for the 2D discrete Gaussian free field

Extension of spin glass methods to boundary-sensitive models

Validation of the approach for log-correlated Gaussian fields

Abstract

In a previous paper, the authors introduced an approach to prove that the statistics of the extremes of a log-correlated Gaussian field converge to a Poisson-Dirichlet variable at the level of the Gibbs measure at low temperature and under suitable test functions. The method is based on showing that the model admits a one-step replica symmetry breaking in spin glass terminology. This implies Poisson-Dirichlet statistics by general spin glass arguments. In this note, this approach is used to prove Poisson-Dirichlet statistics for the two-dimensional discrete Gaussian free field, where boundary effects demand a more delicate analysis.