The maximal potential energy of biased random walks on trees
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arXiv
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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2014
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| _version_ | 1866909649293279232 |
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| author | Hu, Yueyun Shi, Zhan |
| author_facet | Hu, Yueyun Shi, Zhan |
| contents | The biased random walk on supercritical Galton--Watson trees is known to exhibit a multiscale phenomenon in the slow regime: the maximal displacement of the walk in the first $n$ steps is of order $(\log n)^3$, whereas the typical displacement of the walk at the $n$-th step is of order $(\log n)^2$. Our main result reveals another multiscale property of biased walks: the maximal potential energy of the biased walks is of order $(\log n)^2$ in contrast with its typical size, which is of order $\log n$. The proof relies on analyzing the intricate multiscale structure of the potential energy. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_1403_6799 |
| institution | arXiv |
| publishDate | 2014 |
| record_format | arxiv |
| spellingShingle | The maximal potential energy of biased random walks on trees Hu, Yueyun Shi, Zhan Probability The biased random walk on supercritical Galton--Watson trees is known to exhibit a multiscale phenomenon in the slow regime: the maximal displacement of the walk in the first $n$ steps is of order $(\log n)^3$, whereas the typical displacement of the walk at the $n$-th step is of order $(\log n)^2$. Our main result reveals another multiscale property of biased walks: the maximal potential energy of the biased walks is of order $(\log n)^2$ in contrast with its typical size, which is of order $\log n$. The proof relies on analyzing the intricate multiscale structure of the potential energy. |
| title | The maximal potential energy of biased random walks on trees |
| topic | Probability |
| url | https://arxiv.org/abs/1403.6799 |