Averaging over the circles the gaussian free field in the Poincar{\'e} disk

TL;DR

This paper explores the Gaussian free field on the unit disk, emphasizing how hyperbolic geometry simplifies understanding its properties compared to Euclidean approaches.

Contribution

It introduces the Gaussian free field on the unit disk using hyperbolic geometry, highlighting advantages over Euclidean metrics for describing its properties.

Findings

Hyperbolic geometry offers convenient descriptions of the Gaussian free field.

The Gaussian free field on the disk relates to Sobolev spaces and conformal metrics.

Hyperbolic metrics facilitate the analysis of the field's properties.

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.