We describe recent advances in the study of Schramm-Loewner Evolution (SLE), a canonical model of conformally invariant, non-crossing random paths in the plane, and of Liouville Quantum Gravity (LQG), a canonical model of random surfaces in 2D quantum gravity. The latter is expected to be the universal, conformally invariant, continuum limit of random planar maps, as weighted by critical statistical models. SLE multifractal spectra have natural analogues on random planar maps and in LQG. An example is extreme nesting in the Conformal Loop Ensemble (CLE), as derived by Miller, Watson and Wilson, and extreme nesting in the $O(n)$-loop model on a random planar map, as derived recently via combinatorial methods. Their respective large deviations functions are Legendre transforms of two functions, that are shown to be conjugate of each other via a continuous Knizhnik-Polyakov-Zamolodchikov (KPZ) transform inherent to LQG.

Joint work with Gaetan Borot and Jérémie Bouttier.