A family of self-avoiding random walks interpolating the loop-erased random walk and a self-avoiding walk on the Sierpinski gasket

TL;DR

This paper introduces a new family of self-avoiding walks on the Sierpinski gasket, interpolating between loop-erased random walks and standard self-avoiding walks, with proven scaling limits and varying exponents.

Contribution

It extends the erasing-larger-loops-first method to non-Markov walks, creating a continuous family of self-avoiding walks with proven scaling limits on the Sierpinski gasket.

Findings

The scaling limit exists and is almost surely self-avoiding.

The path Hausdorff dimension exceeds 1 and varies with the parameter.

The short-time behavior exponent varies continuously in the family.

Abstract

We show that the `erasing-larger-loops-first' (ELLF) method, which was first introduced for erasing loops from the simple random walk on the Sierpinski gasket, does work also for non-Markov random walks, in particular, self-repelling walks to construct a new family of self-avoiding walks on the Sierpinski gasket. The one-parameter family constructed in this method continuously connects the loop-erased random walk and a self-avoiding walk which has the same asymptotic behavior as the `standard' self-avoiding walk. We prove the existence of the scaling limit and study some path properties: The exponent governing the short-time behavior of the scaling limit varies continuously in the parameter. The limit process is almost surely self-avoiding, while its path Hausdorff dimension is the reciprocal of the exponent above, which is strictly greater than 1.