Existence and uniqueness of the Liouville quantum gravity metric.

We show that there is a unique metric associated with subcritical Liouville quantum gravity (LQG).  More precisely, we show that for the Gaussian free field h on a planar domain, there is a unique random metric D_h = ``e^{\gamma h} (dx^2 + dy^2)", which is characterized by a list of natural axioms.

This LQG metric can be constructed explicitly as the scaling limit of Liouville first passage percolation (LFPP), the random metric obtained by exponentiating a mollified version of the Gaussian free field. Earlier work by Ding, Dubedat, Dunlap, and Falconet (2019) showed that LFPP admits non-trivial subsequential limits. We show that the subsequential limit is unique and satisfies our list of axioms.

Based on four joint papers with Jason Miller, one joint paper with Julien Dubedat, Hugo Falconet, Josh Pfeffer, and Xin Sun, and one joint paper with Josh Pfeffer.