Scaling limit of the loop-erased random walk Green’s function

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...

Scaling Limit of Loop Erased Random Walk — a Naive Approach

We give an alternative proof of the existence of the scaling limit of loop-erased random walk which does not use Löwner’s differential equation.

Loop-Erased Random Walk

The scaling limit of loop-erased random walk in three dimensions

Loop-erased random walk is a model for a random simple (i.e. non-selfintersecting) path created by taking a random walk and, whenever it hits itself, deleting the resulting loop and continuing. We will explain why this model is interesting and why scaling limits are interesting, and then go on to describe the proof (that the limit exists), as time will permit.

Scaling Limits of the Uniform Spanning Tree and Loop-erased Random Walk on Finite Graphs

Let x and y be chosen uniformly in a graph G. We find the limiting distribution of the length of a loop-erased random walk from x to y on a large class of graphs that include the torus Zn for d ≥ 5. Moreover, on this family of graphs we show that a suitably normalized finite-dimensional scaling limit of the uniform spanning tree is a Brownian continuum random tree.