A combinatorial proof of Aldous–Broder theorem for general Markov chains

Abstract

AbstractAldous–Broder algorithm is a famous algorithm used to sample a uniform spanning tree of any finite connected graph , but it is more general: given an irreducible and reversible Markov chain on started at , the tree rooted at formed by the first entrance steps in each node (different from the root) has a probability proportional to , where the edges are directed toward . In this article we give proofs of Aldous–Broder theorem in the general case, where the kernel is irreducible but not assumed to be reversible (this generalized version appeared recently in Hu, Lyons, and Tang [5]). We provide two new proofs: an adaptation of the classical argument, which is purely probabilistic, and a new proof based on combinatorial arguments. On the way we introduce a new combinatorial object that we call the golf sequences.