A U(1)-connection graph $G$ is a graph in which each oriented edge is endowed with a unit complex number, the latter being conjugated under orientation flip. We consider cycle-rooted spanning forests (CRSFs), a particular kind of spanning subgraphs of $G$ that have recently found computational applications as randomized spectral sparsifiers. In this context, CRSFs are drawn from a determinantal measure. Under a condition on the connection, Kassel and Kenyon gave an elegant algorithm, named CyclePopping, to sample from this distribution. The algorithm is an extension of the celebrated algorithm of Wilson that uses a loop-erased random walk to sample uniform spanning trees. In this paper, we give an alternative, elementary proof of correctness of CyclePopping for CRSF sampling; we fill the gaps of a proof sketch by Kassel, who was himself inspired by Marchal's proof of the correctness of Wilson's original algorithm. One benefit of the full proof \`a la Marchal is that we obtain a concise expression for the law of the number of steps to complete the sampling procedure, shedding light on practical situations where the algorithm is expected to run fast. Furthermore, we show how to extend the proof to more general distributions over CRSFs, which are not determinantal. The correctness of CyclePopping is known even in the non-determinantal case from the work of Kassel and Kenyon, so our merit is only to provide an alternate proof. One interest of this alternate proof is again to provide the distribution of the time complexity of the algorithm, in terms of a Poisson point process on the graph loops, or equivalently as a Poisson process on pyramids of cycles, a combinatorial notion introduced by Viennot. Finally, we strive to make the connections to loop measures and combinatorial structures as explicit as possible, to provide a reference for future extensions of the algorithm and its analysis.

翻译：U(1)-连接图$G$是一种每条有向边都赋予单位复数（该复数在方向翻转时取共轭）的图。本文考虑循环根生成森林（CRSFs）——一种特殊的生成子图，其近期在随机化谱稀疏化中展现出计算应用价值。在此背景下，CRSFs由行列式测度生成。在连接条件约束下，Kassel与Kenyon提出了一种优雅的算法（命名为CyclePopping）用于从该分布中采样。该算法是Wilson经典算法（利用环擦除随机游走采样均匀生成树）的推广。本文给出了CRSF采样中CyclePopping算法正确性的另一种初等证明，填补了Kassel（其本人受Marchal对Wilson原始算法正确性证明的启发）在证明纲要中留下的空白。采用Marchal式完整证明的一个优势在于，我们获得了采样过程完成步数律的简洁表达式，揭示了算法在预期快速运行的实际情况。此外，我们展示了如何将该证明推广至CRSFs上更一般的非行列式分布。尽管Kassel与Kenyon已证实在非行列式情形下CyclePopping算法的正确性，本研究的贡献仅在于提供另一种证明。这种替代证明的另一价值在于，通过图环上的泊松点过程（或等价地通过Viennot提出的组合结构——环锥体上的泊松过程）给出了算法时间复杂度分布。最后，我们力求尽可能清晰地阐明算法与环测度及组合结构之间的联系，为算法及其分析的未来扩展提供参考。