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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
Department of Mathematics and Information Sciences, Tokyo Metropolitan University, Hachioji, Tokyo 192-0397, Japan |
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 and Continuous Dynamical Systems - S,
2017, 10
(2)
: 289-311.
doi: 10.3934/dcdss.2017014
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References:
| [1] |
K. B. Athreya and P. E. Ney,
Branching Processes Springer, 1972. |
| [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. |
| [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. |
| [4] |
K. Hattori,
Fractal geometry of self-avoiding processes, J. Math. Sci. Univ. Tokyo, 3 (1996), 379-397.
|
| [5] |
T. Hattori,
Random Walks and Renormalization Group Kyoritsu Publishing (in Japanese). |
| [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. |
| [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.
|
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [13] |
O. D. Jones,
Large deviations for supercritical multitype branching processes, J. Appl. Prob., 41 (2004), 703-720.
doi: 10.1017/S0021900200020490. |
| [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. |
| [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. |
| [16] |
G. F. Lawler,
A self-avoiding random walk, Duke Math. J., 47 (1980), 655-693.
doi: 10.1215/S0012-7094-80-04741-9. |
| [17] |
G. F. Lawler,
The logarithmic correction for loop-erased walk in four dimensions, J. Fourier Anal. Appl., (1995), 347-361.
|
| [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. |
| [19] |
N. Madras and G. Slade,
The Self-avoiding Walk Birkhäuser, 1993. |
| [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. |
| [21] |
M. Shinoda, E. Teufl and S. Wagner,
Uniform spanning trees on Sierpiński graphs, Lat. Am. J. Probab. Math. Stat., 11 (2014), 737-780.
|
show all references
References:
| [1] |
K. B. Athreya and P. E. Ney,
Branching Processes Springer, 1972. |
| [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. |
| [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. |
| [4] |
K. Hattori,
Fractal geometry of self-avoiding processes, J. Math. Sci. Univ. Tokyo, 3 (1996), 379-397.
|
| [5] |
T. Hattori,
Random Walks and Renormalization Group Kyoritsu Publishing (in Japanese). |
| [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. |
| [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.
|
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [13] |
O. D. Jones,
Large deviations for supercritical multitype branching processes, J. Appl. Prob., 41 (2004), 703-720.
doi: 10.1017/S0021900200020490. |
| [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. |
| [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. |
| [16] |
G. F. Lawler,
A self-avoiding random walk, Duke Math. J., 47 (1980), 655-693.
doi: 10.1215/S0012-7094-80-04741-9. |
| [17] |
G. F. Lawler,
The logarithmic correction for loop-erased walk in four dimensions, J. Fourier Anal. Appl., (1995), 347-361.
|
| [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. |
| [19] |
N. Madras and G. Slade,
The Self-avoiding Walk Birkhäuser, 1993. |
| [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. |
| [21] |
M. Shinoda, E. Teufl and S. Wagner,
Uniform spanning trees on Sierpiński graphs, Lat. Am. J. Probab. Math. Stat., 11 (2014), 737-780.
|
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