On the Number of Steps of CyclePopping in Weakly Inconsistent U(1)-Connection Graphs

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.