Random walks and quadratic number fields
Fuente:
arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2025
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| _version_ | 1866909941955035136 |
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| author | Borda, Bence |
| author_facet | Borda, Bence |
| contents | We establish a novel type of connection between random walks and analytic number theory. Working with a random walk on the circle group $\mathbb{R}/\mathbb{Z}$ in which each step is a random integer multiple of a given quadratic irrational $α$, we show that the rate of convergence to uniformity in the quadratic Wasserstein metric (also known as the periodic $L^2$ discrepancy) is governed by deep arithmetic invariants of the ring of algebraic integers of the real quadratic field $\mathbb{Q}(α)$, such as fundamental units and special values of zeta functions. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_03884 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Random walks and quadratic number fields Borda, Bence Probability Number Theory 60G50, 11K38, 11E45, 11R42 We establish a novel type of connection between random walks and analytic number theory. Working with a random walk on the circle group $\mathbb{R}/\mathbb{Z}$ in which each step is a random integer multiple of a given quadratic irrational $α$, we show that the rate of convergence to uniformity in the quadratic Wasserstein metric (also known as the periodic $L^2$ discrepancy) is governed by deep arithmetic invariants of the ring of algebraic integers of the real quadratic field $\mathbb{Q}(α)$, such as fundamental units and special values of zeta functions. |
| title | Random walks and quadratic number fields |
| topic | Probability Number Theory 60G50, 11K38, 11E45, 11R42 |
| url | https://arxiv.org/abs/2512.03884 |