Phase transition for recurrence of stationary random walks on lamplighter groups
Fuente:
arXiv
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| Autori principali: | , , |
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866916995978493952 |
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| author | Benjamini, Itai Blachar, Guy Yadin, Ariel |
| author_facet | Benjamini, Itai Blachar, Guy Yadin, Ariel |
| contents | We introduce and study a class of random walks on lamplighter groups $H\wr G$, where $H$ is a nontrivial finitely generated group and $G$ is an infinite finitely generated group, called \textbf{stationary random walks}. At each step, the walk switches the lamp at its current position, moves in the base group with a drift towards the identity, and switches the lamp again at the new position. We show that when $G$ is virtually-$\mathbb{Z}$ and $H$ is finite, these walks exhibit a phase transition between recurrence and transience, while when~$G$ is not virtually-$\mathbb{Z}$ or $H$ is infinite, they are always transient. In the case $G=\mathbb{Z}$, we determine the exact critical parameter and provide a quantitative description of this phase transition. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_06441 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Phase transition for recurrence of stationary random walks on lamplighter groups Benjamini, Itai Blachar, Guy Yadin, Ariel Probability Group Theory We introduce and study a class of random walks on lamplighter groups $H\wr G$, where $H$ is a nontrivial finitely generated group and $G$ is an infinite finitely generated group, called \textbf{stationary random walks}. At each step, the walk switches the lamp at its current position, moves in the base group with a drift towards the identity, and switches the lamp again at the new position. We show that when $G$ is virtually-$\mathbb{Z}$ and $H$ is finite, these walks exhibit a phase transition between recurrence and transience, while when~$G$ is not virtually-$\mathbb{Z}$ or $H$ is infinite, they are always transient. In the case $G=\mathbb{Z}$, we determine the exact critical parameter and provide a quantitative description of this phase transition. |
| title | Phase transition for recurrence of stationary random walks on lamplighter groups |
| topic | Probability Group Theory |
| url | https://arxiv.org/abs/2510.06441 |