The scaling limit conjecture for random planar maps to Liouville quantum gravity

Let $M_n$ be a random planar map with $n$ edges whose law is given, for each fixed map $\mathfrak m$ with $n$ edges, by

$$\mathbbm{P}[M_n=\mathfrak m] = c_0(\operatorname{det}\Delta_{\mathfrak m})^{- {\bf c}_{\operatorname{M}} /2},$$

where $\Delta_{\mathfrak m}$ is the graph Laplacian and $c_0>0$ is a normalizing constant. The parameter ${\bf c}_{\operatorname{M}}$ is the central charge of the conformal matter, and $\gamma$ is the corresponding LQG parameter determined by the relation in the source. Scaling limit conjecture. For ${\bf c}_{\operatorname{M}}\leq 1$, the planar maps $M_n$ converge in the scaling limit to a $\gamma$-LQG surface. This conjecture expresses the prediction that random planar maps coupled to conformal matter have Liouville quantum gravity as their continuum scaling limit; it is known for a number of natural models, including several maps decorated by statistical physics models, but the general statement remains open.