Uniqueness of the critical and supercritical Liouville quantum gravity metrics

Abstract

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