Uniqueness of the critical and supercritical Liouville quantum gravity metrics

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