Sampling elements of a finite group: efficiency of the product replacement algorithm with an accumulator
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| Format: | Preprint |
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2025
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| _version_ | 1866908725142355968 |
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| author | Marcinkowski, Michał Mizerka, Piotr |
| author_facet | Marcinkowski, Michał Mizerka, Piotr |
| contents | Let $G$ be a finite group generated by $k$ elements. The well-known product replacement algorithm provides an effective method for sampling generating sets of $G$. We study a refinement of this algorithm that is designed to output individual elements of $G$. We show that after $O(k^2\log|G|)$ steps, the distribution of the output is close to uniform on $G$, which improves upon the best results known to date. The proof proceeds via spectral gap estimates and uses computer assisted calculations. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2512_18302 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Sampling elements of a finite group: efficiency of the product replacement algorithm with an accumulator Marcinkowski, Michał Mizerka, Piotr Group Theory Operator Algebras Probability Primary 05C81, Secondary 20F65, 20P05, 60J10, 68W20, 68Q87 Let $G$ be a finite group generated by $k$ elements. The well-known product replacement algorithm provides an effective method for sampling generating sets of $G$. We study a refinement of this algorithm that is designed to output individual elements of $G$. We show that after $O(k^2\log|G|)$ steps, the distribution of the output is close to uniform on $G$, which improves upon the best results known to date. The proof proceeds via spectral gap estimates and uses computer assisted calculations. |
| title | Sampling elements of a finite group: efficiency of the product replacement algorithm with an accumulator |
| topic | Group Theory Operator Algebras Probability Primary 05C81, Secondary 20F65, 20P05, 60J10, 68W20, 68Q87 |
| url | https://arxiv.org/abs/2512.18302 |