Abstract: The large-scale geometry of 2D critical percolation is now well-understood thanks to the works of Schramm, Smirnov, and others. The scaling limit of the percolation clusters belongs to a class of random fractals called the conformal loop ensemble (CLE) gaskets. However, some more refined features such as the intrinsic metric (a.k.a. chemical distance) and the random walk on the percolation clusters (a.k.a. the ant in the labyrinth) are not captured by these results. Our recent works establish the scaling limits of these objects. More generally, for each CLE_\kappa in the non-simple regime (\kappa \in ]4,8[) we show that there is a unique geodesic metric and a unique diffusion process on the CLE gasket that is determined by its local geometry. For \kappa=6, we show that it is the scaling limit of the chemical distance metric resp. the random walk on critical percolation. (For the other values of \kappa, they are the conjectural scaling limits of FK and loop O(n) models.)

In the introductory part of the talk, I will give an introduction to CLE and some of their basic properties. Then I will present the definition of the geodesic metric and the diffusion process and sketch the main steps of their construction. If time allows, I will give a brief introduction to resistance forms.

The presented results are based on joint work with Valeria Ambrosio, Irina Đanković, Maarten Markering, and Jason Miller.