Loop-erased random walk on the Sierpinski gasket
Kumiko Hattori, Michiaki Mizuno

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.
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Taxonomy
TopicsStochastic processes and statistical mechanics · Theoretical and Computational Physics · Mathematical Dynamics and Fractals
