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A random walk on a non-intersecting two-sided random walk trace is subdiffusive in low dimensions


Author: Daisuke Shiraishi
Journal: Trans. Amer. Math. Soc. 370 (2018), 4525-4558
MSC (2010): Primary 82B41; Secondary 82D30
DOI: https://doi.org/10.1090/tran/5737
Published electronically: March 16, 2018
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Abstract: Let $ (\overline {S}^{1}, \overline {S}^{2})$ be the two-sided random walks in $ \mathbb{Z}^{d} \ (d=2,3)$ conditioned so that $ \overline {S}^{1}[0,\infty ) \cap \overline {S}^{2}[1, \infty ) = \emptyset $, which was constructed by the author in 2012. We prove that the number of global cut times up to $ n$ grows like $ n^{\frac {3}{8}}$ for $ d=2$. In particular, we show that each $ \overline {S}^{i}$ has infinitely many global cut times with probability one. Using this property, we prove that the simple random walk on $ \overline {S}^{1}[0,\infty ) \cup \overline {S}^{2}[0,\infty )$ is subdiffusive for $ d=2$. We show the same result for $ d=3$.


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Additional Information

Daisuke Shiraishi
Affiliation: Research Institute for Mathematical Sciences, Kyoto University, Kyoto, Japan
Email: shiraishi@acs.i.kyoto-u.ac.jp

DOI: https://doi.org/10.1090/tran/5737
Received by editor(s): June 30, 2011
Received by editor(s) in revised form: October 29, 2011
Published electronically: March 16, 2018
Article copyright: © Copyright 2018 American Mathematical Society

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