[Paper Review] Conformal symmetries in the extremal process of two-dimensional discrete Gaussian Free Field

Marek Biskup, Oren Louidor|arXiv (Cornell University)|Oct 17, 2014

Stochastic processes and statistical mechanics4 citations

TL;DR

This paper establishes that the extremal process of the 2D discrete Gaussian Free Field converges to a Poisson point process with intensity measure $ Z^D(dx) imes e^{-\alpha h} dh $, where $ Z^D $ is a random measure transforming conformally under analytic maps. The key result is that $ Z^D $ satisfies a Gibbs-Markov property and transforms under conformal maps via $ |f'(x)|^4 $, identifying it uniquely as critical Liouville Quantum Gravity measure associated with the continuum GFF.

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 $(Z^{f(D)}\\circ f)(d x)\\overset{\ ext{law}}=|f'(x)|^4\\, Z^D(d x)$. 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.

Motivation & Objective

• To understand how conformal symmetries of the 2D continuum Gaussian Free Field manifest in the scaling limit of extremal values of the discrete Gaussian Free Field.
• To characterize the random intensity measure $ Z^D $ that arises in the limit of the extremal process in bounded domains $ D \subset \mathbb{C} $.
• To determine the unique laws of $ Z^D $-measures through their transformation rules under conformal maps and restrictions to subdomains.
• To establish that $ Z^D $ corresponds to the critical Liouville Quantum Gravity measure associated with the continuum GFF.

Proposed method

• Prove convergence of the extremal process of the DGFF on lattice approximations of a domain $ D $ to a Poisson point process with intensity $ Z^D(dx) \otimes e^{-\alpha h} dh $, where $ \alpha = 2/\sqrt{g} $, $ g = 2/\pi $.
• Establish the Gibbs-Markov property: $ Z^{D'} \overset{\text{law}}{=} Z^D|_{D'} $ for subdomains $ D' \subset D $, ensuring consistency across domains.
• Derive the conformal transformation rule: $ (Z^{f(D)} \circ f)(dx) \overset{\text{law}}{=} |f'(x)|^4 Z^D(dx) $ for analytic bijections $ f $, linking $ Z^D $ to conformal invariance.
• Use a derivative martingale representation to characterize $ Z^D $ uniquely via its scaling and transformation behavior.
• Apply coupling arguments and convergence of harmonic measures to justify the continuum limit of the DGFF and its extremal statistics.
• Leverage the convergence of lattice harmonic measures and Brownian motion approximations to prove the scaling limit of the extremal process.

Experimental results

Research questions

• RQ1 How do conformal symmetries of the 2D continuum Gaussian Free Field emerge in the scaling limit of the extremal process of the discrete Gaussian Free Field?
• RQ2 What is the precise transformation rule of the limiting intensity measure $ Z^D $ under analytic maps between domains?
• RQ3 How are the random measures $ Z^D $ related across nested domains, and what property governs their consistency?
• RQ4 Can the law of $ Z^D $ be uniquely characterized by its transformation under conformal maps and restriction to subdomains?
• RQ5 Is the limiting measure $ Z^D $ identifiable with a known object in probability theory, such as critical Liouville Quantum Gravity?

Key findings

• The joint process of spatial positions and centered values of extreme local maxima in the DGFF converges to a Poisson point process with intensity $ Z^D(dx) \otimes e^{-\alpha h} dh $, where $ \alpha = 2/\sqrt{g} $, $ g = 2/\pi $.
• The random measure $ Z^D $ satisfies a Gibbs-Markov property: $ Z^{D'} \overset{\text{law}}{=} Z^D|_{D'} $ for any subdomain $ D' \subset D $, ensuring consistency across domains.
• Under analytic bijections $ f $, the measure transforms as $ (Z^{f(D)} \circ f)(dx) \overset{\text{law}}{=} |f'(x)|^4 Z^D(dx) $, revealing the conformal symmetry of the extremal process.
• The laws of $ Z^D $ are uniquely determined by the Gibbs-Markov property and the $ |f'|^4 $ transformation rule, identifying $ Z^D $ as the critical Liouville Quantum Gravity measure associated with the continuum GFF.
• The total mass of $ Z^D $, $ Z^D(D) $, appears in the Laplace transform of the centered maximum: $ \mathbb{E}[e^{-\alpha^{-1} e^{-\alpha t} Z^D(D)}] $, which characterizes the limit law of the maximum value.
• The convergence holds for a large class $ \mathfrak{D} $ of bounded open sets $ D \subset \mathbb{C} $, not just squares, generalizing earlier results from [7].