Uniqueness of the critical and supercritical Liouville quantum gravity metrics

Understand

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).