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arXiv · 2512.18302

Sampling elements of a finite group: efficiency of the product replacement algorithm with an accumulator

Michał Marcinkowski·Piotr Mizerka
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Abstract

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.

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Michał Marcinkowski, Piotr Mizerka. 2025-12-20. Sampling elements of a finite group: efficiency of the product replacement algorithm with an accumulator. https://arxiv.org/abs/2512.18302

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