Random walks on percolation models (`the ant in the labyrinth') have been intensely studied in the probability literature. Many results have been proved for supercritical percolation, and recently for high-dimensional critical percolation. In this talk I will discuss critical percolation in 2D. We prove that the properly rescaled simple random walk converges to a diffusion process on the CLE_6 gasket. We also prove that the properly rescaled graph metric (`chemical distance') converges to a geodesic metric on the CLE_6 gasket. Some implications are the existence of the following exponents: chemical distance exponent, resistance growth exponent, walk dimension, spectral dimension.
More generally, we have constructed the canonical geodesic metric and the Brownian motion on each CLE_\kappa gasket in the dense regime \kappa \in ]4,8[.
This talk is based on joint works with Valeria Ambrosio, Irina Đanković, Maarten Markering, and Jason Miller.