The most common model for random behavior is the “drunkard’s walk” where at each time an individual chooses their step from some probability distribution. I will review this and then discuss what happens when one puts some constraints on the walker to try to avoid places already visited. We will see the relationship between the “fractal dimension” of the random path and the ambient dimension in which it lives.
It was predicted by theoretical physicists that lattice models from equilibrium statistical physics “at criticality” in two dimensions have limits that are conformally invariant. There has been an incredible amount of work in the last twenty years making these ideas precise and rigorous and I will survey this work. The starting point was the development of the Schramm-Loewner evolution (SLE) which I will define.
This talk will focus on two related models: loop measures and the loop-erased random walk which are closely related to uniform spanning trees and describe some relatively recent work in this area in dimensions two, three, and four.