Liouville quantum gravity and the Gaussian free field

Nathanaël Berestycki, Scott Sheffield, Xin Sun

Published 2014-10-20Version 1

Given an instance $h$ of the Gaussian free field on a planar domain $D$ and a constant $\gamma \in (0,2)$, one can use various regularization procedures to make sense of the {\em Liouville quantum gravity measure} $$\mu_h := e^{\gamma h(z)} dz.$$ It is known that the field $h$ a.s. determines the measure $\mu_h$. We show that the converse is true: namely, given $\mu_h$, it is a.s. possible to recover $h$. We then use this result to show that the path of a Liouville Brownian motion also determines the free field $h$.