Minimal spanning arborescence

TL;DR

This paper investigates the properties of minimal spanning arborescences in infinite graphs, introducing a novel recursive algorithm and stochastic process to analyze their geometric structure and limits.

Contribution

It introduces a new recursive algorithm and stochastic process, the loop contracting random walk, for studying minimal spanning arborescences in infinite graphs, extending classical methods.

Findings

Infinite volume limit exists almost surely in certain transient trees.

Limit is almost surely one-ended for nonamenable, unimodular graphs under certain conditions.

Introduces the loop contracting random walk, a new stochastic process related to Wilson's algorithm.

Abstract

We study the minimal spanning arborescence which is the directed analogue of the minimal spanning tree, with a particular focus on its infinite volume limit and its geometric properties. We prove that in a certain large class of transient trees, the infinite volume limit exists almost surely. We also prove that for nonamenable, unimodular graphs, the limit is almost surely one-ended assuming a certain sufficient condition that guarantees the existence of the limit. This object cannot be studied using well-known algorithms, such as Kruskal's or Prim's algorithm, to sample the minimal spanning tree which has been instrumental in getting analogous results about them (Lyons, Peres, and Schramm). Instead, we use a recursive algorithm due to Chu, Liu, Edmonds, and Bock, which leads to a novel stochastic process which we call the \emph{loop contracting random walk}. This is similar to the well-known and widely studied loop erased random walk, except instead of erasing loops we contract them. The full algorithm bears similarities with the celebrated Wilson's algorithm to generate uniform spanning trees and can be seen as a certain limit of the original Wilson's algorithm.