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Averaging over the circles the gaussian free field in the Poincar{é} disk

Jean-Marc Derrien

TL;DR

This work studies the Gaussian free field on the unit disk by focusing on circle averages as tractable observables. It leverages the hyperbolic (Poincaré) geometry of the disk to exploit isometries, simplify Green functions, and relate circle averages at radius $\rho$ to Euclidean radii via $r=\tanh(\rho)$. The paper builds the Sobolev framework $H_0^1({\mathbb D})$, defines the GFF as a Gaussian isometry with covariance given by the inner product, and analyzes the circle-averaged field to reveal Brownian-motion-like behavior along the hyperbolic radius, including precise covariance calculations and continuity results. These ingredients provide a robust, geometry-aware approach to first properties of the GFF on ${\mathbb D}$ and offer a clear path to understanding its covariance structure through hyperbolic distance.

Abstract

The gaussian free field on the unit disk $D$ can be seen as a two-dimensional version of the Brownian bridge on the interval [0,1]. It is intrinsically associated with the Sobolev space $H_0^1 (D)$. To define the latter, we can choose any metric conformally equivalent to the Euclidean metric on $D$. This note is an introduction to the gaussian free field on the unit disk whose aim is to highlight some of the conveniences offered by hyperbolic geometryon $D$ to describe the first properties of this probabilistic object.

Averaging over the circles the gaussian free field in the Poincar{é} disk

TL;DR

This work studies the Gaussian free field on the unit disk by focusing on circle averages as tractable observables. It leverages the hyperbolic (Poincaré) geometry of the disk to exploit isometries, simplify Green functions, and relate circle averages at radius to Euclidean radii via . The paper builds the Sobolev framework , defines the GFF as a Gaussian isometry with covariance given by the inner product, and analyzes the circle-averaged field to reveal Brownian-motion-like behavior along the hyperbolic radius, including precise covariance calculations and continuity results. These ingredients provide a robust, geometry-aware approach to first properties of the GFF on and offer a clear path to understanding its covariance structure through hyperbolic distance.

Abstract

The gaussian free field on the unit disk can be seen as a two-dimensional version of the Brownian bridge on the interval [0,1]. It is intrinsically associated with the Sobolev space . To define the latter, we can choose any metric conformally equivalent to the Euclidean metric on . This note is an introduction to the gaussian free field on the unit disk whose aim is to highlight some of the conveniences offered by hyperbolic geometryon to describe the first properties of this probabilistic object.
Paper Structure (13 sections, 90 equations)

This paper contains 13 sections, 90 equations.

Table of Contents

  1. Introduction
  2. Un peu de géométrie dans le plan complexe
  3. Un peu de géométrie différentielle dans le plan complexe
  4. Un peu de géométrie dans le plan complexe euclidien
  5. Moyenner sur les cercles les fonctions régulières dans le plan complexe euclidien
  6. Un peu de géométrie dans le disque de Poincaré
  7. La métrique hyperbolique sur le disque unité
  8. Les isométries du disque de Poincaré
  9. Moyenner sur les cercles les fonctions régulières dans le disque de Poincaré
  10. L'espace de Sobolev $H_0^1 ({\mathbb D})$
  11. Moyenner sur les cercles les éléments de $H_0^1 ({\mathbb D})$ dans le disque de Poincaré
  12. Le champ libre gaussien sur le disque unité
  13. Moyenner sur les cercles le champ libre gaussien dans le disque de Poincaré