Indistinguishability in One-or-Two-Ended Forests on Unimodular Random Graphs

TL;DR

The paper introduces a new measure-theoretic approach to prove the indistinguishability of components in various random forests and related models on unimodular graphs, simplifying existing proofs and extending results.

Contribution

It presents a novel proof technique based on ancestry chain conditioning, applicable to multiple models, and establishes tail triviality of Markov chains on unimodular graphs.

Findings

New proof for the wired uniform spanning forest.

Indistinguishability results for river models and coalescing processes.

Tail triviality of Markov chains on unimodular graphs.

Abstract

We provide a new approach for proving the indistinguishability of connected components of random one-or-two-ended oriented forests on unimodular random graphs. In particular, this approach leads to a new and simpler proof for the wired uniform spanning forest, which is the only one-ended model previously studied in the literature. This approach can also be used for proving the indistinguishability of `level-sets' in this setting, where the previously available methods do not work. The approach leads to new indistinguishability results for a variety of models, including, for instance, river models, coalescing renewal process models, coalescing simple random walks and coalescing Markov chains. These models are unified as `coalescing Markov trajectories' (CMT), under some general conditions, where the out-going edges of the vertices are chosen randomly and independently. These models and results are also extended to models based on point-maps on Bernoulli/Poisson point processes, like Howard's model and the strip point-map. The proof technique is based on conditioning on the `ancestry chain' of the root. It leverages measure-theoretic results on the completion of certain invariant or tail sigma-fields, which are of independent interest. First, non-tail events are ruled out regardless of the forest model. The next step, which is model dependent, is to reduce the indistinguishability of the components (resp. level-sets) to the ergodicity (resp. tail triviality) of the ancestry chain of the root. A result of independent interest is the tail triviality of Markov chains on unimodular random graphs, under suitable conditions, which completes the last step of the proof of the indistinguishability of level-sets of CMTs on unimodular graphs.