This course will introduce some of the topological ideas and Dirichlet form techniques that have proved useful in recent work regarding the scaling limits of random walks on various models of random trees and other graphs at criticality, including critical Galton-Watson trees, the critical Erdos-Renyi random graph and the uniform spanning tree in two dimensions. In each of these examples, the random structure considered exhibits some kind of interesting fractal behaviour in the limit, which gives rise to anomalous features (such as sub-diffusivity) for the limiting diffusion.

The slides for this course can be downloaded here.