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arXiv · 1905.00379

Local metrics of the Gaussian free field

Ewain Gwynne·Jason Miller
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Abstract

We introduce the concept of a local metric of the Gaussian free field (GFF) $h$, which is a random metric coupled with $h$ in such a way that it depends locally on $h$ in a certain sense. This definition is a metric analog of the concept of a local set for $h$. We establish general criteria for two local metrics of the same GFF $h$ to be bi-Lipschitz equivalent to each other and for a local metric to be a.s. determined by $h$. Our results are used in subsequent works which prove the existence, uniqueness, and basic properties of the $\gamma$-Liouville quantum gravity (LQG) metric for all $\gamma \in (0,2)$, but no knowledge of LQG is needed to understand this paper.

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Ewain Gwynne, Jason Miller. 2019-05-01. Local metrics of the Gaussian free field. https://arxiv.org/abs/1905.00379

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