Duplantier–Sheffield conjecture for scaling limits of random planar maps

Let $\mathbf{m}_n\in\mathbf{M}_n^p$ be sampled according to a statistical physics model in the $\gamma$-universality class. Let $\widehat{\mathbf{m}}_n\subset\mathbb{S}^2$ be any natural embedding, and let $\mu_n$ be the pushforward measure associated with this embedding. The Liouville measure is $\mu_\gamma=e^{\gamma h}$, where $h$ is an instance of the Gaussian Free Field on the sphere $\mathbb{S}^2$. Duplantier–Sheffield conjecture. The measures $\mu_n$ weakly converge as $n\to\infty$ towards a random measure closely related to the Liouville measure $\mu_\gamma$. If one keeps track of the root, it is asymptotically distributed according to $\mu_\gamma$. This conjecture proposes Liouville quantum gravity as the scaling limit of statistical-physics models on random planar maps in the $\gamma$-universality class; the source presents it as striking and does not provide a resolution.