# Local metrics of the Gaussian free field

## TL;DR

This paper introduces the concept of local metrics for the Gaussian free field, establishing criteria for their equivalence and uniqueness, which underpin the mathematical foundation of Liouville quantum gravity.

## Contribution

It defines local metrics for the GFF, provides criteria for their equivalence and determination by the field, forming a basis for LQG metric studies.

## Key findings

- Criteria for bi-Lipschitz equivalence of local metrics
- Conditions for a local metric to be determined by the GFF
- Framework supporting the existence and uniqueness of LQG metrics

## 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.