Local limit theorems for random isometries of the plane
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arXiv
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| Hauptverfasser: | , |
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| Format: | Preprint |
| Veröffentlicht: |
2026
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| _version_ | 1866917219316793344 |
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| author | Drogin, Reuben Hernández, Felipe |
| author_facet | Drogin, Reuben Hernández, Felipe |
| contents | We consider a random walk $(Y_N)_{N\geq 0}$ on $\mathbb{R}^2$ generated by successively applying independent random isometries, drawn from a fixed measure $μ$, to the point $0$. When the support of $μ$ is finite and includes an irrational rotation satisfying a Diophantine condition, we establish a local central limit theorem (LCLT) for $Y_N$ down to super-polynomially small scales. When $μ$ includes rotations satisfying a further algebraic condition, we prove that a LCLT holds down to the scale $\exp(-cN^{1/3}/(\log N)^2)$. Due to group-theoretic obstructions, this is sharp for symmetric $μ$, up to the $\log$ factor. Lastly for a special class of asymmetric $μ$, we obtain an LCLT down to the much finer scale $\exp(-cN^{1/2})$. The proofs relate the fine-scale distribution of $Y_N$ to a question about the values of integer polynomials on the unit circle. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2601_16295 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Local limit theorems for random isometries of the plane Drogin, Reuben Hernández, Felipe Probability Group Theory Number Theory We consider a random walk $(Y_N)_{N\geq 0}$ on $\mathbb{R}^2$ generated by successively applying independent random isometries, drawn from a fixed measure $μ$, to the point $0$. When the support of $μ$ is finite and includes an irrational rotation satisfying a Diophantine condition, we establish a local central limit theorem (LCLT) for $Y_N$ down to super-polynomially small scales. When $μ$ includes rotations satisfying a further algebraic condition, we prove that a LCLT holds down to the scale $\exp(-cN^{1/3}/(\log N)^2)$. Due to group-theoretic obstructions, this is sharp for symmetric $μ$, up to the $\log$ factor. Lastly for a special class of asymmetric $μ$, we obtain an LCLT down to the much finer scale $\exp(-cN^{1/2})$. The proofs relate the fine-scale distribution of $Y_N$ to a question about the values of integer polynomials on the unit circle. |
| title | Local limit theorems for random isometries of the plane |
| topic | Probability Group Theory Number Theory |
| url | https://arxiv.org/abs/2601.16295 |