Topics in loop measures and the loop-erased walk

ar X iv : m at h / 06 11 77 5 v 1 [ m at h . PR ] 2 5 N ov 2 00 6 Colored loop - erased random walk on the complete graph ∗

Starting from a sequence regarded as a walk through some set of values, we consider the associated loop-erased walk as a sequence of directed edges, with an edge from i to j if the loop erased walk makes a step from i to j. We introduce a coloring of these edges by painting edges with a fixed color as long as the walk does not loop back on itself, then switching to a new color whenever a loop i...

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.

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.

The infinite two-sided loop-erased random walk

The loop-erased random walk (LERW) in Zd, d ≥ 2, 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.

Loop-Erased Random Walk