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

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.