Reduced Random Walks in the Hyperbolic Plane$\hspace{1pt}!\hspace{-3.8pt}?$
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arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2025
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| _version_ | 1866908340102103040 |
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| author | Defant, Colin Lee, Mitchell |
| author_facet | Defant, Colin Lee, Mitchell |
| contents | We study Lam's reduced random walk in a hyperbolic triangle group, which we view as a random walk in the upper half-plane. We prove that this walk converges almost surely to a point on the extended real line. We devote special attention to the reduced random walk in $PGL_2(\mathbb{Z})$ (i.e., the $(2,3,\infty)$ triangle group). In this case, we provide an explicit formula for the cumulative distribution function of the limit. This formula is written in terms of the interrobang function, a new function $!\hspace{-3.8pt}?\colon[0,1]\to\mathbb{R}$ that shares several of the remarkable analytic and arithmetic properties of Minkowski's question-mark function. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2504_19367 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Reduced Random Walks in the Hyperbolic Plane$\hspace{1pt}!\hspace{-3.8pt}?$ Defant, Colin Lee, Mitchell Probability Combinatorics Group Theory 20F55 (Primary) 05C81, 20F67, 60B15, 60J10, 11F06 (Secondary) We study Lam's reduced random walk in a hyperbolic triangle group, which we view as a random walk in the upper half-plane. We prove that this walk converges almost surely to a point on the extended real line. We devote special attention to the reduced random walk in $PGL_2(\mathbb{Z})$ (i.e., the $(2,3,\infty)$ triangle group). In this case, we provide an explicit formula for the cumulative distribution function of the limit. This formula is written in terms of the interrobang function, a new function $!\hspace{-3.8pt}?\colon[0,1]\to\mathbb{R}$ that shares several of the remarkable analytic and arithmetic properties of Minkowski's question-mark function. |
| title | Reduced Random Walks in the Hyperbolic Plane$\hspace{1pt}!\hspace{-3.8pt}?$ |
| topic | Probability Combinatorics Group Theory 20F55 (Primary) 05C81, 20F67, 60B15, 60J10, 11F06 (Secondary) |
| url | https://arxiv.org/abs/2504.19367 |