Stationary measures and random walks on $\tilde{A}_2$-buildings
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866912590298349568 |
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| author | Bars, Corentin Le |
| author_facet | Bars, Corentin Le |
| contents | We consider a non-elementary group action $G \curvearrowright X$ of a locally compact second countable group $G$ on a possibly exotic non-discrete affine building $X$ of type $\tilde{A}_2$. We prove that if $μ$ is an admissible symmetric probability measure on $G$, there is a unique $μ$-stationary measure supported on the chambers of the spherical building at infinity. We use this result to study random walks induced by the $G$-action, and we prove that if $μ$ has finite second moment, $(Z_n o)$ converges almost surely to a regular point of the boundary and the Lyapunov spectrum of the random walk is simple. Applied to Bruhat-Tits buildings, these results extend some classical theorems due to H.~Furstenberg. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2410_18821 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Stationary measures and random walks on $\tilde{A}_2$-buildings Bars, Corentin Le Group Theory Probability 20E42, 20G15, 20P05 We consider a non-elementary group action $G \curvearrowright X$ of a locally compact second countable group $G$ on a possibly exotic non-discrete affine building $X$ of type $\tilde{A}_2$. We prove that if $μ$ is an admissible symmetric probability measure on $G$, there is a unique $μ$-stationary measure supported on the chambers of the spherical building at infinity. We use this result to study random walks induced by the $G$-action, and we prove that if $μ$ has finite second moment, $(Z_n o)$ converges almost surely to a regular point of the boundary and the Lyapunov spectrum of the random walk is simple. Applied to Bruhat-Tits buildings, these results extend some classical theorems due to H.~Furstenberg. |
| title | Stationary measures and random walks on $\tilde{A}_2$-buildings |
| topic | Group Theory Probability 20E42, 20G15, 20P05 |
| url | https://arxiv.org/abs/2410.18821 |