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A combinatorial proof of Aldous-Broder theorem for general Markov chains
Luis Fredes, Jean-François Marckert · 2021-02-17 · via math.PR updates on arXiv.org

Aldous-Broder algorithm is a famous algorithm used to sample a uniform spanning tree of any finite connected graph $G$, but it is more general: given an irreducible and reversible Markov chain $M$ on $G$ started at $r$, the tree rooted at $r$ formed by the first entrance steps in each node (different from the root) has a probability proportional to $\prod_{e=(e^-,e^+)\in {\sf Edges}(t,r)} M_{e^{-},e^+}$, where the edges are directed toward $r$. In this paper we give proofs of Aldous-Broder theorem in the general case, where the kernel $M$ is irreducible but not assumed to be reversible (this generalized version appeared recently in Hu, Lyons and Tang )

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