Uniqueness of the critical and supercritical Liouville quantum gravity metrics

Abstract

We show that for each ${\mathbf c}_{\mathrm M} \in [1,25)$, there is a unique metric associated with Liouville quantum gravity (LQG) with matter central charge ${\mathbf c}_{\mathrm M}$. An earlier series of works by Ding-Dub\'edat-Dunlap-Falconet, Gwynne-Miller, and others showed that such a metric exists and is unique in the subcritical case ${\mathbf c}_{\mathrm M} \in (-\infty,1)$, which corresponds to coupling constant $\gamma \in (0,2)$. The critical case ${\mathbf c}_{\mathrm M} = 1$ corresponds to $\gamma=2$ and the supercritical case ${\mathbf c}_{\mathrm M} \in (1,25)$ corresponds to $\gamma \in \mathbb C$ with $|\gamma| = 2$. Our metric is constructed as the limit of an approximation procedure called Liouville first passage percolation, which was previously shown to be tight for $\mathbf c_{\mathrm M} \in [1,25)$ by Ding and Gwynne (2020). In this paper, we show that the subsequential limit is uniquely characterized by a natural list of axioms. This extends the characterization of the LQG metric proven by Gwynne and Miller (2019) for $\mathbf c_{\mathrm M} \in (-\infty,1)$ to the full parameter range $\mathbf c_{\mathrm M} \in (-\infty,25)$. Our argument is substantially different from the proof of the characterization of the LQG metric for $\mathbf c_{\mathrm M} \in (-\infty,1)$. In particular, the core part of the argument is simpler and does not use confluence of geodesics.