The Poisson boundary of wreath products
Fuente:
arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2023
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| _version_ | 1866908876658442240 |
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| author | Frisch, Joshua Silva, Eduardo |
| author_facet | Frisch, Joshua Silva, Eduardo |
| contents | We give a complete description of the Poisson boundary of wreath products $A\wr B= \bigoplus_{B} A\rtimes B$ of countable groups $A$ and $B$, for probability measures $μ$ with finite entropy where lamp configurations stabilize almost surely. If, in addition, the projection of $μ$ to $B$ is Liouville, we prove that the Poisson boundary of $(A\wr B,μ)$ is equal to the space of limit lamp configurations, endowed with the corresponding hitting measure. In particular, this answers an open question asked by Kaimanovich, and Lyons-Peres, for $B=\mathbb{Z}^d$, $d\ge 3$, and measures $μ$ with a finite first moment. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2310_10160 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | The Poisson boundary of wreath products Frisch, Joshua Silva, Eduardo Group Theory Dynamical Systems Probability We give a complete description of the Poisson boundary of wreath products $A\wr B= \bigoplus_{B} A\rtimes B$ of countable groups $A$ and $B$, for probability measures $μ$ with finite entropy where lamp configurations stabilize almost surely. If, in addition, the projection of $μ$ to $B$ is Liouville, we prove that the Poisson boundary of $(A\wr B,μ)$ is equal to the space of limit lamp configurations, endowed with the corresponding hitting measure. In particular, this answers an open question asked by Kaimanovich, and Lyons-Peres, for $B=\mathbb{Z}^d$, $d\ge 3$, and measures $μ$ with a finite first moment. |
| title | The Poisson boundary of wreath products |
| topic | Group Theory Dynamical Systems Probability |
| url | https://arxiv.org/abs/2310.10160 |