Cycling in the forest with Wilson's algorithm

Abstract

We consider a probability measure on cycle-rooted spanning forests (CRSFs) introduced by Kenyon. CRSFs are spanning subgraphs, each connected component of which has a unique cycle; they generalize spanning trees. A generalization of Wilson's celebrated CyclePopping algorithm for uniform spanning trees has been proposed for CRSFs, and several concise proofs have been given that the algorithm samples from Kenyon's distribution. In this survey, we flesh out all the details of such a proof of correctness, progressively generalizing a proof by Marchal for spanning trees. This detailed proof has several interests. First, it serves as a modern tutorial on Wilson's algorithm, suitable for applied probability and computer science audiences. Compared to uniform spanning trees, the more sophisticated motivating application to CRSFs brings forth connections to recent research topics such as loop measures, partial rejection sampling, and heaps of cycles. Second, the detailed proof \emph{\`a la} Marchal yields the law of the time complexity of the sampling algorithm, shedding light on practical situations where the algorithm is expected to run fast.