# American Institute of Mathematical Sciences

April  2017, 10(2): 289-311. doi: 10.3934/dcdss.2017014

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

 Department of Mathematics and Information Sciences, Tokyo Metropolitan University, Hachioji, Tokyo 192-0397, Japan

* Corresponding author:Kumiko Hattori

Received  October 2015 Revised  November 2016 Published  January 2017

We show that the ‘erasing-larger-loops-first’ (ELLF) method, which was first introduced for erasing loops from the simple random walk on the Sierpiński 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 Sierpiński 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 $ν$ governing the short-time behavior of the scaling limit varies continuously in the parameter. The limit process is almost surely self-avoiding, while it has path Hausdorff dimension $1/ν$, which is strictly greater than $1$.

Citation: Kumiko Hattori, Noriaki Ogo, Takafumi Otsuka. A family of self-avoiding random walks interpolating the loop-erased random walk and a self-avoiding walk on the Sierpiński gasket. Discrete & Continuous Dynamical Systems - S, 2017, 10 (2) : 289-311. doi: 10.3934/dcdss.2017014
##### References:

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##### References:
$F_3$
$w, \tilde{w}, w_1, w_2, w_3$
Loopless paths from $O$ to $a$ on $F_1$
The loop-erasing procedure: (a) $w$, (b) $Q_1w$, (c) $LQ_1w=\hat{Q}_1w$, (d) $\tilde{L}w$
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