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

TL;DR

The paper develops a product replacement accumulator that outputs single group elements with near-uniform distribution after a provable mixing time. It combines a refined random-walk model on randomly changing Cayley graphs with a computer-assisted sum-of-squares proof to bound the spectral gap, achieving a bound of $t \ge \frac{23 k^2}{k-5}\big((k+1)\log|G| + \log(\epsilon^{-1})\big)$ for $k>5$. The authors also provide a detailed decomposition of the squared Laplacian, an induction mechanism to extend results to all $k\ge5$, and a robust framework for turning numerical certificates into rigorous proofs via interval arithmetic and Netzer–Thom bounds. Their approach yields a simple, memory-efficient sampler whose performance depends only on $|G|$ and $k$, offering advantages over other sampling methods in regimes where only a few elements need to be drawn.

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.

Theorems & Definitions (27)

• Theorem : \ref{['t:rate-of-convergence']}
• Remark 2.1
• Example 2.2
• Example 2.3
• Lemma 3.1
• proof
• Lemma 3.2
• Lemma 3.3
• Remark 3.4
• Lemma 4.1