Random walks on infinite percolation clusters are key examples of stochastic motion in a random environment. Simple random walk on a graph is obtained by letting a walker take independent steps chosen uniformly among the neighbors of the current position. Key questions are how far the walker moves away from its starting point, and whether it visits its starting point infinitely often. Random walks on infinite percolation clusters are obtained by letting the graph be the infinite cluster in percolation, a key model for a random geometric network in statistical physics showing fascinating behavior close to criticality. Random walk on a percolation cluster is a paradigm model for a stochastic process on a random network. Considerable progress has been made on random walks on percolation clusters, particularly in the supercritical phase, where the infinite cluster has full dimension and the random walker on it behaves quite similarly to that on the full lattice. A central question is what happens when the percolation model becomes critical. In high-dimensions, the incipient infinite cluster (IIC), a mathematical definition of an infinite critical cluster, has been defined. The key challenge in this proposal is to study how the fractal nature of the IIC slows down the motion on it. This shall be achieved by investigating the structure of the IIC as well as the motion on fractal media, using the lace expansion and relations between random walks and electrical networks. |