A random walk on a non-intersecting two-sided random walk trace is subdiffusive in low dimensions
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- by Daisuke Shiraishi PDF
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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 $d=3$.References
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Additional Information
- Daisuke Shiraishi
- Affiliation: Research Institute for Mathematical Sciences, Kyoto University, Kyoto, Japan
- MR Author ID: 916346
- Email: shiraishi@acs.i.kyoto-u.ac.jp
- Received by editor(s): June 30, 2011
- Received by editor(s) in revised form: October 29, 2011
- Published electronically: March 16, 2018
- © Copyright 2018 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 370 (2018), 4525-4558
- MSC (2010): Primary 82B41; Secondary 82D30
- DOI: https://doi.org/10.1090/tran/5737
- MathSciNet review: 3787377