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

Kumiko Hattori; Noriaki Ogo; Takafumi Otsuka

doi:10.3934/dcdss.2017014

Article Contents

2017, Volume 10, Issue 2: 289-311. Doi: 10.3934/dcdss.2017014

This issue Previous Article Homoclinic tangencies to resonant saddles and discrete Lorenz attractors Next Article Stability index, uncertainty exponent, and thermodynamic formalism for intermingled basins of chaotic attractors

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
* Corresponding author:Kumiko Hattori

Received: October 2015

Revised: November 2016

Published: February 2018

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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 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 $1$.

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Mathematics Subject Classification: 60F99, 60G17, 28A80, 37F25, 37F35.

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Figure 1. $F_3$

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Figure 2. $w, \tilde{w}, w_1, w_2, w_3$

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Figure 3. Loopless paths from $O$ to $a$ on $F_1$

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Figure 4. The loop-erasing procedure: (a) $w$, (b) $Q_1w$, (c) $LQ_1w=\hat{Q}_1w$, (d) $\tilde{L}w$

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