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

Figures (4)

• Figure 1: The level-sets in the strip point-map on the Poisson point process in the plane (\ref{['ex:strip']}). On the left, part of the graph is shown together with one level-set. On the right, only 20 consecutive level-sets are shown.
• Figure 2: The dependency structure of the sections. A dotted arrow means that only the definitions of the sigma-fields (see Table \ref{['tab:symbols']}) are used from the preceding section.
• Figure 3: The relations between the sigma-fields mentioned in \ref{['tab:symbols']} (see \ref{['subsec:pathErgodic', 'subsec:tail-fixed']} for analogous sigma-fields for a fixed underlying graph). An arrow, say from $\mathcal{F}$ to $\mathcal{F}'$, means $\mathcal{F}\subseteq\mathcal{F}'$ mod $\mathbb P$ (\ref{['def:augmentation']}), possibly with some additional assumptions. Only $\mathcal{I}^{\mathrm{comp}}_{\mathrm{}}, \mathcal{T}^{\mathrm{branch}}_{\mathrm{}}, \mathcal{I}^{\mathrm{path}}_{\mathrm{}}$ and $\mathcal{I}^{\mathrm{}}_{\mathrm{}}$ are needed for the proof of indistinguishability of components. The label of an arrow navigates to the result which proves the inclusion (those without label are straightforward). Solid arrows are valid for all models, but the dashed arrows are proved only for Markov chains or CMTs in \ref{['sec:Markov', 'sec:drainage', 'sec:cmt-proof']} (some are also proved for $\mathop{\mathrm{WUSF}}\nolimits$ in \ref{['sec:wusf']}).
• Figure 4: The paths used in the proof of \ref{['lem:foil-connectivity']}.

Theorems & Definitions (168)

• Theorem 1.1: Indistinguishability in General Point-Maps
• Theorem 1.2: Indistinguishability in CMTs and Similar Models
• proof
• Theorem 1.3: Steps \ref{['step3comp']} and \ref{['step3foil']}: Ergodicity and Tail Triviality of Markov Chains on Unimodular Graphs
• Proposition 1.4
• Theorem 1.5: Disconnectedness Implies One-Endedness
• Definition 1.6: Collision Property
• Theorem 1.7: Number of Components
• Proposition 1.8: Connectivity Decay
• Theorem 1.9: One-Endedness