Intrinsic metrics (a.k.a. chemical distance) and random walks on percolation models have been attracting a lot of mathematical attention. The case of (low-dimensional) critical percolation, however, has remained poorly understood. In this talk, I will explain how to construct the scaling limits of the intrinsic metric and the random walk on 2D critical percolation clusters. More generally, for each CLE_\kappa, \kappa \in ]4,8[, we construct the canonical shortest-path metric and diffusion process on its gasket. We show that the metrics are uniquely characterised by their Markovian property, and that they are scale-covariant and conformally covariant.

This talk is based on joint works with Valeria Ambrosio, Irina Đanković, Maarten Markering, and Jason Miller.