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Equivalence between Liouville quantum gravity and scaling limits of random planar maps

Prove the conjectured equivalence between Liouville quantum gravity (defined via the Liouville path integral/LCFT) and the scaling limit of random planar maps for general ensembles and topologies, beyond the special cases already established.

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Background

A central conjecture in two-dimensional quantum gravity posits that Liouville theory describes the scaling limit of random planar maps. This equivalence has been proved in specific cases such as uniform planar maps, with the general case remaining out of reach.

The authors note the conjectural status and partial progress, highlighting the need for a fully general proof across broader classes of planar maps and surfaces.

References

Liouville theory is conjecturally equivalent to the scaling limit of random planar maps, i.e.\ probability measures on finite triangulations of a fixed topological surface, see \Cref{triang}.

— Two Decades of Probabilistic Approach to Liouville Conformal Field Theory (2509.21053 - Rhodes et al., 25 Sep 2025) in Section 2 (A brief history of Liouville CFT)