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arXiv · 2501.06020

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

Jean-Marc Derrien
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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.

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Jean-Marc Derrien. 2025-01-10. Averaging over the circles the gaussian free field in the Poincar{\'e} disk. https://arxiv.org/abs/2501.06020

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