Loop-erased random walk on the Sierpinski gasket

TL;DR

This paper rigorously analyzes loop-erased random walks on the Sierpinski gasket, establishing the existence of a self-avoiding fractal path with a Hausdorff dimension greater than one, using a novel recursive approach.

Contribution

It introduces a new loop-erasing procedure on fractals and proves the existence and properties of the scaling limit of the walk.

Findings

Existence of a scaling limit for loop-erased random walks on the Sierpinski gasket.

The limiting path is almost surely self-avoiding with Hausdorff dimension > 1.

Recursive relations derived from fractal self-similarity enable analysis.

Abstract

We consider a model of loop-erased random walks on the finite pre-Sierpinski gasket which permits rigorous analysis. We prove the existence of the scaling limit and show that the path of the limiting process is almost surely self-avoiding, while having Hausdorff dimension strictly greater than 1. This result means that the path has infinitely fine creases, while having no self-intersection. Our loop-erasing procedure is formulated by a `larger-scale-loops-first' rule. It enables us to obtain exact recursion relations, making use of `self-similarity' of a fractal structure.