Conformal symmetries in the extremal process of two-dimensional discrete Gaussian Free Field

TL;DR

This paper investigates the extremal process of the 2D Discrete Gaussian Free Field, revealing conformal symmetry properties in the scaling limit and connecting it to Liouville Quantum Gravity.

Contribution

It proves the convergence of extremal processes to a Poisson point process with a conformally covariant intensity measure, linking discrete GFF maxima to Liouville Quantum Gravity.

Findings

Extremal process converges to a Poisson point process with a specific intensity measure.

The random measures exhibit conformal covariance under analytic maps.

Identification of the limiting measure with critical Liouville Quantum Gravity.

Abstract

We study the extremal process associated with the Discrete Gaussian Free Field on the square lattice and elucidate how the conformal symmetries manifest themselves in the scaling limit. Specifically, we prove that the joint process of spatial positions ($x$) and centered values ($h$) of the extreme local maxima in lattice versions of a bounded domain $D\subset\mathbb C$ converges, as the lattice spacing tends to zero, to a Poisson point process with intensity measure $Z^D(dx)\otimes e^{-\alpha h}d h$, where $\alpha$ is a constant and $Z^D$ is a random a.s.-finite measure on $D$. The random measures $\{Z^D\}$ are naturally interrelated; restrictions to subdomains are governed by a Gibbs-Markov property and images under analytic bijections $f$ by the transformation rule $(Z^{f(D)}\circ f)(d x)\overset{\text{law}}=|f'(x)|^4\, Z^D(d x)$. Conditions are given that determine the laws of these measures uniquely. These identify $Z^D$ with the critical Liouville Quantum Gravity associated with the Continuum Gaussian Free Field.