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

TL;DR

This paper provides a combinatorial proof of the Aldous-Broder theorem for Markov chains that are irreducible but not necessarily reversible, extending its applicability beyond the classical reversible case.

Contribution

It offers the first combinatorial proof of the Aldous-Broder theorem for general irreducible Markov chains, broadening the theorem's scope.

Findings

Proves the Aldous-Broder theorem for non-reversible Markov chains

Extends the understanding of spanning trees in Markov chain contexts

Provides a combinatorial approach to a probabilistic theorem

Abstract

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 )