Full extremal process, cluster law and freezing for the two-dimensional discrete Gaussian Free Field

TL;DR

This paper analyzes the extremal process of the 2D Discrete Gaussian Free Field, characterizing its scaling limit, cluster structure, and freezing phenomenon, with applications to Liouville measures and Gibbs measures.

Contribution

It provides a detailed description of the extremal process, cluster law, and freezing phenomenon for the 2D DGFF, advancing understanding of its extreme value structure.

Findings

Scaling limit of the extremal process is a Cox process with decorated clusters.

Controlled the scaling limit of the supercritical Liouville measure.

Established the freezing phenomenon in the DGFF's Gibbs measure.

Abstract

We study the local structure of the extremal process associated with the Discrete Gaussian Free Field (DGFF) in scaled-up (square-)lattice versions of bounded open planar domains subject to mild regularity conditions on the boundary. We prove that, in the scaling limit, this process tends to a Cox process decorated by independent, correlated clusters whose distribution is completely characterized. As an application, we control the scaling limit of the discrete supercritical Liouville measure, extract a Poisson-Dirichlet statistics for the limit of the Gibbs measure associated with the DGFF and establish the "freezing phenomenon" conjectured to occur in the "glassy" phase. In addition, we prove a local limit theorem for the position and value of the absolute maximum. The proofs are based on a concentric, finite-range decomposition of the DGFF and entropic-repulsion arguments for an associated random walk. Although we naturally build on our earlier work on this problem, the methods developed here are largely independent.