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Equivalence of Liouville theory and scaling limits of random planar maps

Prove that Liouville conformal field theory is equivalent to the scaling limit of random planar maps on fixed topological surfaces, extending beyond cases proved for uniform planar maps.

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Background

In two-dimensional quantum gravity, Liouville theory is proposed as a continuum description of random geometries obtained from random planar maps. This equivalence has been established in specific settings (notably for uniform planar maps).

A general proof would solidify the connection between probabilistic Liouville field theory and discrete random geometry across a broad class of ensembles.

References

In particular, in the context of two-dimensional quantum gravity, 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)