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arxiv: 2603.19717 · v2 · pith:JZQ4CHIRnew · submitted 2026-03-20 · 🧮 math.PR

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

Francois Baccelli , Ali Khezeli This is my paper

Pith reviewed 2026-05-21 11:20 UTC · model grok-4.3

classification 🧮 math.PR
keywords indistinguishabilityunimodular random graphscoalescing Markov trajectoriesancestry chainwired uniform spanning forestlevel setsone-ended forests
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The pith

A conditioning argument on ancestry chains establishes indistinguishability of components and level-sets in one- or two-ended oriented forests on unimodular random graphs.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper develops a method to prove that the connected components of random oriented forests with at most two ends are indistinguishable when observed from a typical point. The argument proceeds by conditioning on the ancestry chain of the root, first excluding non-tail events and then reducing the indistinguishability question to ergodicity or tail triviality of that chain. This yields a simpler proof for the wired uniform spanning forest and new results for coalescing Markov trajectories, which include river models, coalescing renewal processes, and coalescing random walks. The same technique also settles indistinguishability for level-sets, a case previous methods left open.

Core claim

For one-or-two-ended oriented forests on unimodular random graphs, the connected components are indistinguishable and so are the level-sets; the proof first rules out non-tail events for any such forest, then reduces the claim to ergodicity of the ancestry chain under the given forest law, and finally invokes tail triviality of Markov chains on unimodular graphs to finish the level-set case.

What carries the argument

The ancestry chain of the root, whose ergodicity or tail triviality decides whether components or level-sets remain distinguishable.

If this is right

  • The wired uniform spanning forest admits a shorter proof of component indistinguishability.
  • River models, coalescing renewal processes, coalescing simple random walks and coalescing Markov chains all satisfy component indistinguishability when outgoing edges are chosen independently.
  • Level-set indistinguishability holds for the same family of coalescing Markov trajectories.
  • The results extend to point-map constructions on Bernoulli and Poisson point processes, including Howard's model.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The tail-triviality result for Markov chains on unimodular graphs may apply directly to other tail events in percolation or random walks on the same graphs.
  • Similar ancestry-chain arguments could be tested on forests with more than two ends or on non-unimodular base graphs to map the boundary of the method.

Load-bearing premise

For each specific forest construction the indistinguishability of components reduces to ergodicity of the ancestry chain of the root.

What would settle it

An explicit CMT construction on a unimodular random graph in which the ancestry chain admits a non-trivial invariant event yet the components remain indistinguishable from almost every vertex.

Figures

Figures reproduced from arXiv: 2603.19717 by Ali Khezeli, Francois Baccelli.

Figure 1
Figure 1. Figure 1: The level-sets in the strip point-map on the Poisson point process in [PITH_FULL_IMAGE:figures/full_fig_p006_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: The dependency structure of the sections. A dotted arrow means that [PITH_FULL_IMAGE:figures/full_fig_p018_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: The relations between the sigma-fields mentioned in Table 1 (see Sec [PITH_FULL_IMAGE:figures/full_fig_p019_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: The paths used in the proof of Theorem 5.6. [PITH_FULL_IMAGE:figures/full_fig_p047_4.png] view at source ↗
read the original 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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 3 minor

Summary. The manuscript develops a new approach to proving indistinguishability of connected components (and level-sets) in random one- or two-ended oriented forests on unimodular random graphs. The method first rules out non-tail events uniformly, then reduces the target property to ergodicity (or tail triviality) of the ancestry chain of the root via conditioning; measure-theoretic results on completions of invariant/tail sigma-fields are applied to complete the argument. This yields a simpler proof for the wired uniform spanning forest, extends to level-set indistinguishability, and produces new results for models unified as coalescing Markov trajectories (CMT) under independent outgoing-edge conditions, including river models, coalescing renewal processes, simple random walks, and Markov chains, with further extensions to point-map constructions on point processes such as Howard's model.

Significance. If the central claims hold, the work supplies a unified and technically economical framework that covers a broad family of models previously handled case-by-case, while also treating level-sets where earlier techniques do not apply. The explicit model-dependent reductions for the CMT class, the auxiliary tail-triviality theorem for Markov chains on unimodular graphs, and the measure-theoretic lemmas on invariant sigma-fields are all of independent interest and strengthen the contribution.

minor comments (3)
  1. [§2.2] §2.2: the definition of the ancestry chain and the associated sigma-field should be accompanied by a short diagram or explicit recursive construction to improve readability for readers outside the immediate subfield.
  2. [§4.3] §4.3: while the independent-outgoing-edge condition is stated for CMT, a compact table listing which models satisfy the moment and irreducibility hypotheses used in the tail-triviality result would make the scope of the unification immediately visible.
  3. [Theorem 5.1] Theorem 5.1: the statement of tail triviality for Markov chains could usefully include a one-sentence remark on why the unimodularity hypothesis is essential, with a pointer to the relevant place in the proof.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive and accurate summary of our manuscript, as well as for recommending minor revision. We appreciate the recognition that the ancestry-chain conditioning technique provides a unified and technically economical framework for proving indistinguishability of components and level-sets across a range of models on unimodular random graphs, and that the auxiliary results on tail triviality and invariant sigma-fields are of independent interest.

Circularity Check

0 steps flagged

No significant circularity; derivation is self-contained

full rationale

The paper's central argument conditions on the ancestry chain of the root and reduces component indistinguishability to ergodicity (or level-set indistinguishability to tail triviality) of that chain under explicit model-dependent conditions on independent outgoing edges. It then invokes or proves measure-theoretic facts about invariant/tail sigma-fields and tail triviality of Markov chains on unimodular graphs under stated moment and irreducibility hypotheses. These steps are carried out directly in the manuscript without reducing any target quantity to a fitted parameter or self-citation chain; the cited or proved auxiliary results are independent of the indistinguishability claims and are presented as results of independent interest. No self-definitional, fitted-input, or load-bearing self-citation patterns appear in the derivation chain.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 1 invented entities

The paper rests on standard domain assumptions about unimodular random graphs and introduces the CMT framework as a unifying definition; no free parameters or new postulated physical entities are indicated in the abstract.

axioms (1)
  • domain assumption Unimodular random graphs admit a consistent choice of root and satisfy mass-transport principles.
    Invoked throughout the literature on unimodular graphs and used to define the ancestry chain.
invented entities (1)
  • coalescing Markov trajectories (CMT) no independent evidence
    purpose: Unifying framework for river models, coalescing walks, and point-map constructions.
    Defined in the abstract as models where outgoing edges are chosen randomly and independently; no independent evidence outside the paper is supplied.

pith-pipeline@v0.9.0 · 5852 in / 1294 out tokens · 57399 ms · 2026-05-21T11:20:32.089565+00:00 · methodology

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Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

  • IndisputableMonolith/Foundation/AbsoluteFloorClosure.lean reality_from_one_distinction unclear
    ?
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    Relation between the paper passage and the cited Recognition theorem.

    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... reduce the indistinguishability of the components (resp. level-sets) to the ergodicity (resp. tail triviality) of the ancestry chain of the root.

What do these tags mean?
matches
The paper's claim is directly supported by a theorem in the formal canon.
supports
The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends
The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
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The paper appears to rely on the theorem as machinery.
contradicts
The paper's claim conflicts with a theorem or certificate in the canon.
unclear
Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

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