Topics in loop measures and the loop-erased walk

Topics in loop measures and the loop-erased walk

These are lecture notes from an advanced graduate course given in the fall of 2016 at the University of Chicago.

Loop-Erased Random Walk

Scaling Limit of Loop-erased Random Walk

The loop-erased random walk (LERW) was first studied in 1980 by Lawler as an attempt to analyze self-avoiding walk (SAW) which provides a model for the growth of a linear polymer in a good solvent. The self-avoiding walk is simply a path on a lattice that does not visit the same site more than once. Proving things about the collection of all such paths is a formidable challenge to rigorous math...

Markov Chain Intersections and the Loop-erased Walk

Let X and Y be independent transient Markov chains on the same state space that have the same transition probabilities. Let L denote the “loop-erased path” obtained from the path of X by erasing cycles when they are created. We prove that if the paths of X and Y have infinitely many intersections a.s., then L and Y also have infinitely many intersections a.s. §

Convergence of loop-erased random walk in the natural parameterization

We prove that loop-erased random walk parametrized by renormalized length converges in the lattice size scaling limit to SLE2 parametrized by Minkowski content.