Loop-erased random walk branch of uniform spanning tree in topological polygons

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.

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...

Loop-Erased Random Walk

Loop Erased Walks and Uniform Spanning Trees

The uniform spanning tree has had a fruitful history in probability theory. Most notably, it was the study of the scaling limit of the UST that led Oded Schramm [Sch00] to introduce the SLE process, work which has revolutionised the study of two dimensional models in statistical physics. But in addition, the UST relates in an intrinsic fashion with another model, the loop erased random walk (or...

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...