Local metrics of the Gaussian free field

Understand

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 $h$.