Abstract: Random fractals arise naturally as the scaling limit of large discrete models at criticality. These

fractals usually exhibit strong self similarity and spacial independence. In this talk, we will explain how

these additional properties should give the existence of a natural occupation measure on the fractal set,

defined to be the limit of the properly rescaled Lebesgue measure restricted to small neighborhoods of

the fractal. Moreover, the occupation measure is also the scaling limit of the normalized counting measure

over the corresponding discrete set. In two dimension, when putting an independent Liouville quantum

gravity background over such a planar fractal, the quantum version of the occupation measure still exists,

where the scaling dimension is related to the Euclidean one via the famous KPZ relation due to Knizhnik-

Polyakov-Zamolodchikov and Duplantier-Sheffield. The quantum occupation measure is supposed to be

the scaling limit of the normalized counting measure of the corresponding discrete set on certain random

planar maps. The picture described above is expected to be true in great generality yet it is only established

for a few models to various extents. In this talk, we report a fairly complete picture for planar percolation on the regular and random triangular lattice.