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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

Kumiko Hattori , , Noriaki Ogo and  Takafumi Otsuka

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 $1$.

Keywords: Loop-erased random walk, self-avoiding walk, self-repelling walk, scaling limit, displacement exponent, fractal dimension, Sierpinski gasket, fractal.
Mathematics Subject Classification: 60F99, 60G17, 28A80, 37F25, 37F35.
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:
[1]

K. B. Athreya and P. E. Ney, Branching Processes Springer, 1972.  Google Scholar

[2]

M. T. Barlow and E. A. Perkins, Brownian motion on the Sierpinski gasket, Probab. Theory Relat. Fields, 79 (1988), 543-623.  doi: 10.1007/BF00318785.  Google Scholar

[3]

B. Hambly, K. Hattori and T. Hattori, Self-repelling walk on the Sierpiński gasket, Probab. Theory Relat. Fields, 124 (2002), 1-25.  doi: 10.1007/s004400100192.  Google Scholar

[4]

K. Hattori, Fractal geometry of self-avoiding processes, J. Math. Sci. Univ. Tokyo, 3 (1996), 379-397.   Google Scholar

[5]

T. Hattori, Random Walks and Renormalization Group Kyoritsu Publishing (in Japanese). Google Scholar

[6]

K. Hattori and T. Hattori, Self-avoiding process on the Sierpinski gasket, Probab. Theory Relat. Fields, 88 (1991), 405-428.  doi: 10.1007/BF01192550.  Google Scholar

[7]

K. Hattori and T. Hattori, Displacement exponent of self-repelling walks and self-attracting walks on the Sierpinski gasket, J. Math. Sci. Univ. Tokyo, 12 (2005), 417-443.   Google Scholar

[8]

K. Hattori, T. Hattori and S. Kusuoka, Self-avoiding paths on the pre-Sierpinski gasket, Probab. Theory Relat. Fields, 84 (1990), 1-26.  doi: 10.1007/BF01288555.  Google Scholar

[9]

K. Hattori, T. Hattori and S. Kusuoka, Self-avoiding paths on the three-dimensional Sierpinski gasket, Publ. RIMS, 29 (1993), 455-509.  doi: 10.2977/prims/1195167053.  Google Scholar

[10]

T. Hattori and S. Kusuoka, The exponent for mean square displacement of self-avoiding random walk on Sierpinski gasket, Probab. Theory Relat. Fields, 93 (1992), 273-284.  doi: 10.1007/BF01193052.  Google Scholar

[11]

K. Hattori and M. Mizuno, Loop-erased random walk on the Sierpinski gasket, Stoch. Process. Appl., 124 (2014), 566-585.  doi: 10.1016/j.spa.2013.08.006.  Google Scholar

[12]

R. van der Hofstad and W. König, A survey of one-dimensional random polymers, J. Stat. Phys., 103 (2001), 915-944.  doi: 10.1023/A:1010309005541.  Google Scholar

[13]

O. D. Jones, Large deviations for supercritical multitype branching processes, J. Appl. Prob., 41 (2004), 703-720.  doi: 10.1017/S0021900200020490.  Google Scholar

[14]

G. Kozma, The scaling limit of loop-erased random walk in three dimensions, Acta Math., 199 (2007), 29-152.  doi: 10.1007/s11511-007-0018-8.  Google Scholar

[15]

T. Kumagai, Construction and some properties of a class of non-symmetric diffusion processes on the Sierpinski gasket, Asymptotic problems in probability theory: Stochastic models and diffusions on fractals (Sanda/Kyoto, 1990), Pitman Res. Notes Math. Ser. , Longman Sci. Texh. , Harlow, 283 (1980), 219–247.  Google Scholar

[16]

G. F. Lawler, A self-avoiding random walk, Duke Math. J., 47 (1980), 655-693.  doi: 10.1215/S0012-7094-80-04741-9.  Google Scholar

[17]

G. F. Lawler, The logarithmic correction for loop-erased walk in four dimensions, J. Fourier Anal. Appl., (1995), 347-361.   Google Scholar

[18]

G. F. Lawler, O. Schramm and W. Werner, Conformal invariance of planar loop-erased random walks and uniform spanning trees, Ann. Probab., 32 (2004), 939-995.  doi: 10.1214/aop/1079021469.  Google Scholar

[19]

N. Madras and G. Slade, The Self-avoiding Walk Birkhäuser, 1993.  Google Scholar

[20]

O. Schramm, Scaling limits of loop-erased random walks and uniform spanning trees, Israel J. Math. , 118 (2000), 221–288. doi: 10.1007/BF02803524.  Google Scholar

[21]

M. Shinoda, E. Teufl and S. Wagner, Uniform spanning trees on Sierpiński graphs, Lat. Am. J. Probab. Math. Stat., 11 (2014), 737-780.   Google Scholar

show all references

References:
[1]

K. B. Athreya and P. E. Ney, Branching Processes Springer, 1972.  Google Scholar

[2]

M. T. Barlow and E. A. Perkins, Brownian motion on the Sierpinski gasket, Probab. Theory Relat. Fields, 79 (1988), 543-623.  doi: 10.1007/BF00318785.  Google Scholar

[3]

B. Hambly, K. Hattori and T. Hattori, Self-repelling walk on the Sierpiński gasket, Probab. Theory Relat. Fields, 124 (2002), 1-25.  doi: 10.1007/s004400100192.  Google Scholar

[4]

K. Hattori, Fractal geometry of self-avoiding processes, J. Math. Sci. Univ. Tokyo, 3 (1996), 379-397.   Google Scholar

[5]

T. Hattori, Random Walks and Renormalization Group Kyoritsu Publishing (in Japanese). Google Scholar

[6]

K. Hattori and T. Hattori, Self-avoiding process on the Sierpinski gasket, Probab. Theory Relat. Fields, 88 (1991), 405-428.  doi: 10.1007/BF01192550.  Google Scholar

[7]

K. Hattori and T. Hattori, Displacement exponent of self-repelling walks and self-attracting walks on the Sierpinski gasket, J. Math. Sci. Univ. Tokyo, 12 (2005), 417-443.   Google Scholar

[8]

K. Hattori, T. Hattori and S. Kusuoka, Self-avoiding paths on the pre-Sierpinski gasket, Probab. Theory Relat. Fields, 84 (1990), 1-26.  doi: 10.1007/BF01288555.  Google Scholar

[9]

K. Hattori, T. Hattori and S. Kusuoka, Self-avoiding paths on the three-dimensional Sierpinski gasket, Publ. RIMS, 29 (1993), 455-509.  doi: 10.2977/prims/1195167053.  Google Scholar

[10]

T. Hattori and S. Kusuoka, The exponent for mean square displacement of self-avoiding random walk on Sierpinski gasket, Probab. Theory Relat. Fields, 93 (1992), 273-284.  doi: 10.1007/BF01193052.  Google Scholar

[11]

K. Hattori and M. Mizuno, Loop-erased random walk on the Sierpinski gasket, Stoch. Process. Appl., 124 (2014), 566-585.  doi: 10.1016/j.spa.2013.08.006.  Google Scholar

[12]

R. van der Hofstad and W. König, A survey of one-dimensional random polymers, J. Stat. Phys., 103 (2001), 915-944.  doi: 10.1023/A:1010309005541.  Google Scholar

[13]

O. D. Jones, Large deviations for supercritical multitype branching processes, J. Appl. Prob., 41 (2004), 703-720.  doi: 10.1017/S0021900200020490.  Google Scholar

[14]

G. Kozma, The scaling limit of loop-erased random walk in three dimensions, Acta Math., 199 (2007), 29-152.  doi: 10.1007/s11511-007-0018-8.  Google Scholar

[15]

T. Kumagai, Construction and some properties of a class of non-symmetric diffusion processes on the Sierpinski gasket, Asymptotic problems in probability theory: Stochastic models and diffusions on fractals (Sanda/Kyoto, 1990), Pitman Res. Notes Math. Ser. , Longman Sci. Texh. , Harlow, 283 (1980), 219–247.  Google Scholar

[16]

G. F. Lawler, A self-avoiding random walk, Duke Math. J., 47 (1980), 655-693.  doi: 10.1215/S0012-7094-80-04741-9.  Google Scholar

[17]

G. F. Lawler, The logarithmic correction for loop-erased walk in four dimensions, J. Fourier Anal. Appl., (1995), 347-361.   Google Scholar

[18]

G. F. Lawler, O. Schramm and W. Werner, Conformal invariance of planar loop-erased random walks and uniform spanning trees, Ann. Probab., 32 (2004), 939-995.  doi: 10.1214/aop/1079021469.  Google Scholar

[19]

N. Madras and G. Slade, The Self-avoiding Walk Birkhäuser, 1993.  Google Scholar

[20]

O. Schramm, Scaling limits of loop-erased random walks and uniform spanning trees, Israel J. Math. , 118 (2000), 221–288. doi: 10.1007/BF02803524.  Google Scholar

[21]

M. Shinoda, E. Teufl and S. Wagner, Uniform spanning trees on Sierpiński graphs, Lat. Am. J. Probab. Math. Stat., 11 (2014), 737-780.   Google Scholar

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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