Near-maxima of the two-dimensional Discrete Gaussian Free Field

TL;DR

This paper analyzes the near-maximum behavior of the 2D Discrete Gaussian Free Field, revealing its convergence to a product of critical Liouville Quantum Gravity and Rayleigh law, and explaining the universality of the limiting measure.

Contribution

It establishes the joint convergence of the near-extremal process, extremal process, and field values of the DGFF to a universal limit involving cLQG and Rayleigh law, clarifying the structure of near-maxima.

Findings

Near-extremal level sets converge to a product of cLQG and Rayleigh law.

The extremal process's intensity measure is given by the cLQG measure.

The limit near-extremal process is measurable w.r.t. the continuum GFF.

Abstract

We consider the Discrete Gaussian Free Field (DGFF) in domains $D_N\subseteq\mathbb Z^2$ arising, via scaling by $N$, from nice domains $D\subseteq\mathbb R^2$. We study the statistics of the values order $\sqrt{\log N}$ below the absolute maximum. Encoded as a point process on $D\times\mathbb R$, the scaled spatial distribution of these near-extremal level sets in $D_N$ and the field values (in units of $\sqrt{\log N}$ below the absolute maximum) tends, as $N\to\infty$, in law to the product of the critical Liouville Quantum Gravity (cLQG) $Z^D$ and the Rayleigh law. The convergence holds jointly with the extremal process, for which $Z^D$ enters as the intensity measure of the limiting Poisson point process, and that of the DGFF itself; the cLQG defined by the limit field then coincides with $Z^D$. While the limit near-extremal process is measurable with respect to the limit continuum GFF, the limit extremal process is not. Our results explain why the various ways to "norm" the lattice cLQG measure lead to the same limit object, modulo overall normalization.