Existence and uniqueness of the Liouville quantum gravity metric for $\gamma \in (0,2)$

Ewain Gwynne, Cambridge University
Fine Hall 214

We show that for each $\gamma \in (0,2)$, there is a unique metric associated with $\gamma$-Liouville quantum gravity (LQG).  More precisely, we show that for the Gaussian free field $h$ on a planar domain $U$, there is a unique random metric $D_h = ``e^{\gamma h} (dx^2 + dy^2)"$ on $U$ which is uniquely characterized by a list of natural axioms.

The $\gamma$-LQG metric can be constructed explicitly as the scaling limit of \emph{Liouville first passage percolation} (LFPP), the random metric obtained by exponentiating a mollified version of the Gaussian free field. Earlier work by Ding, Dub\'edat, 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. In the case when $\gamma = \sqrt{8/3}$, our metric coincides with the $\sqrt{8/3}$-LQG metric constructed in previous work by Miller and Sheffield.

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.