Four-dimensional loop-erased random walk

Loop-Erased Random Walk

The loop - erased random walk and the uniform spanning tree on the four - dimensional discrete torus

Let x and y be points chosen uniformly at random from Z4n, the four-dimensional discrete torus with side length n. We show that the length of the loop-erased random walk from x to y is of order n(logn), resolving a conjecture of Benjamini and Kozma. We also show that the scaling limit of the uniform spanning tree on Z4n is the Brownian continuum random tree of Aldous. Our proofs use the techniq...

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...

Two-sided loop-erased random walk in three dimensions

The loop-erased random walk (LERW) in three dimensions is obtained by erasing loops chronologically from simple random walk. In this paper we show the existence of the two-sided LERW which can be considered as the distribution of the LERW as seen by a point in the “middle” of the path.

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.