arXiv:2603.19717v3

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

Francois Baccelli, Ali Khezeli

math.PR60B9960G5537A2037A50

Paper metadata

Original submission: March 20, 2026
Latest revision: July 24, 2026 · version 3
Claimed MSC: 60B99, 60G55, 37A20, 37A50
arXiv categories: math.PR
Paper license: arXiv non-exclusive distribution license ↗
Comments: 96 pages, 4 figures

Abstract

Indistinguishability is a form of ergodicity, introduced by Lyons and Schramm for percolation clusters, that has become a fundamental notion in the theory of random infinite graphs. We prove the indistinguishability of connected components for a broad family of random one-ended or two-ended oriented forests on unimodular graphs using a new approach. We unify these models by introducing `coalescing Markov trajectories' (CMTs), which encompass a wide range of classical coalescing models, including river models and coalescing random walks. We also establish the indistinguishability of level-sets, a problem that has not previously been studied and lies beyond the scope of existing techniques. Using the latter, we prove that the clusters of the stationary voter model on a unimodular graph are indistinguishable. The core of the proof approach is the reduction of the indistinguishability of the components (respectively, level-sets) to the ergodicity (respectively, tail triviality) of the ancestry chain of the root. This shows a structural property that the ancestry chain is the fundamental object governing indistinguishability. For CMTs, the proof is completed by proving the tail triviality of Markov chains on unimodular random graphs, which is of independent interest. The flexibility of the approach is illustrated by further applications: It yields indistinguishability results for some point-map models on Bernoulli and Poisson point processes, including Howard's model and the strip point-map. It also yields a new and substantially simpler proof of indistinguishability for the wired uniform spanning forest, which is the only one-ended model for which indistinguishability was previously studied in the literature.

AI-generated audit

Audit summary

Audited against arXiv v3

Not a correctness certificate. A “Correct” result may include yellow typos or minor formal corrections that do not affect substantive soundness. It means this audit found no unresolved substantive error under the stated criteria; it does not replace expert scrutiny or formal verification.

Current report

Detailed mathematical audit

Generated August 18, 2026

01Statements4 reported findingsCorrect

The central result clusters are supported. Theorems 1.4–1.6 prove component and level-set indistinguishability through ergodicity/tail triviality of ancestor chains. Proposition 1.2 identifies forest clusters with the finest stationary voter partition. Theorems 1.8, 1.10, 1.12 and Proposition 1.11 classify disconnected components and connectivity decay. Theorems 1.14, 1.16, and 1.17 give the deterministic-graph analogues under their additional hypotheses.

Theorems 1.4–1.6Correct

Indistinguishability of components and level sets

Pages 6–11 · Theorems 1.4–1.6 · arXiv:2603.19717v3

For the equivariant point maps and coalescing Markov trees covered by the printed unimodularity, balance, and irreducibility assumptions, Theorems 1.4–1.6 show that every rerooting-invariant component property is almost surely shared by all components, and likewise for the corresponding level sets. The ancestor-chain ergodicity and tail-triviality conclusions used in these statements are included with their precise hypotheses; the deterministic and unimodular cases are not conflated. Proposition 1.1's bijection transfers the relevant invariant sigma-fields exactly.

Proposition 1.2Correct

Voter-model cluster identification

Pages 6–7 and 28–31 · Proposition 1.2 · arXiv:2603.19717v3

Coalescing dual trajectories determine whether two sites eventually share the same ancestor. The finest stationary multicolor voter model assigns equal colors exactly to trajectories that coalesce, so its clusters coincide almost surely with the forest components. Stationarity follows from equivariance of the graphical construction. The identification is equality of equivalence relations, not only equality in distribution, which is sufficient for the later indistinguishability transfer.

Theorems 1.8, 1.10, 1.12 and Proposition 1.11Correct

Disconnectedness, component count, connectivity decay, and one-endedness

Pages 12–15 · stated results · arXiv:2603.19717v3

In Model 1, if the forest is disconnected then mass transport rules out finite multi-ended exceptional components, and Green-function decay forces each component to be one-ended. The number of components has the zero-one alternatives stated in Theorem 1.10. Proposition 1.11's two-point connection probability tends to zero in the disconnected regime, and this decay is then used—not assumed—to establish the one-ended conclusion in Theorem 1.12.

Theorems 1.14, 1.16, and 1.17Correct

Deterministic-graph analogues

Pages 15–17 · Theorems 1.14, 1.16, and 1.17 · arXiv:2603.19717v3

For a deterministic transitive graph, balance supplies the stationary root bias and weak irreducibility makes the ancestor chain communicate in the sense needed for steadiness. Theorem 1.16 proves steadiness rather than inserting it as an implicit assumption. The component and level-set conclusions of Theorems 1.14 and 1.17 then follow by the same invariant-sigma-field bijections, with the cycle-free and end hypotheses retained.

02Proofs3 reported findingsCorrect

Tail triviality, voter duality, and connectivity decay. The proof constructs equivariant bijections between forest components and between their level sets and verifies that they preserve the rerooting-invariant sigma-fields. Mass transport makes the ancestral Markov chain stationary from the correct biased root and excludes a finite exceptional class of components. Theorem 5.4 proves ergodicity of that chain, while Theorem 5.11 proves tail triviality under the stronger lattice assumptions; these supply Steps C3 and L3 instead of assuming indistinguishability. Pulling invariant events through the bijections then gives the component and level-set conclusions in Theorems 1.4–1.6.

Ancestor-chain and mass-transport analysisCorrect and complete

Tail triviality, voter duality, and connectivity decay

Pages 22–62 · Sections 3–10 · arXiv:2603.19717v3

The proof constructs equivariant bijections between forest components and between their level sets and verifies that they preserve the rerooting-invariant sigma-fields. Mass transport makes the ancestral Markov chain stationary from the correct biased root and excludes a finite exceptional class of components. Theorem 5.4 proves ergodicity of that chain, while Theorem 5.11 proves tail triviality under the stronger lattice assumptions; these supply Steps C3 and L3 instead of assuming indistinguishability. Pulling invariant events through the bijections then gives the component and level-set conclusions in Theorems 1.4–1.6.

Sections 3–4Correct and complete

Equivariant bijections and coalescing duality

Pages 22–38 · Sections 3–4 · arXiv:2603.19717v3

The component-to-level-set maps are defined intrinsically from the oriented ancestor relation and are measurable and equivariant. Their inverses are checked on every one- and two-ended case. The graphical construction of coalescing chains uses common randomness, so equality of voter colors is equivalent to eventual coalescence. These facts transfer invariant properties without changing null sets or root bias.

Sections 5–10Correct and complete

Markov-chain, mass-transport, and end-structure arguments

Pages 39–62 · Sections 5–10 · arXiv:2603.19717v3

Stationarity is obtained from the balance identity before ergodic theorems are applied. Shift-invariant events are pulled back to rerooting-invariant forest events, which unimodular ergodicity makes trivial; the tail proof adds the stated lattice assumptions. Separate mass transports rule out isolated exceptional ends, and the Green-kernel estimates make remote connection probabilities vanish. The one-ended and two-ended branches are treated separately, and the deterministic results invoke steadiness only after it has been proved.

03Novelty0 reported findingsNo non-novelty findings

No non-novelty findings.

Detailed audit reportFull reasoning, manuscript locations, and references.

Open report PDF ↗

Author response

Challenge an audit finding

Local workflow preview

A listed author may submit formal evidence that an audit is inaccurate. The response would be considered in a fresh AI re-evaluation; it would not edit the audit automatically.

Paper: arXiv:2603.19717v3
Authors listed: Francois Baccelli, Ali Khezeli
Audit date: August 18, 2026

01Establish identityMatch an authenticated scholarly identity to this paper.
02Submit evidenceIdentify the finding and give a formal mathematical response.
03Re-evaluateA separate agent checks the response and records a disposition.

Email fallback for papers without a linked ORCID

A production fallback could send a one-time link only when the submitted address matches an independently maintained author-contact allowlist for this paper. MathAudit must return the same message for every address so the form cannot reveal which contacts are on that list.

Structured response preview

This form remains unavailable until production identity verification succeeds. Nothing entered here is submitted.

Audit finding challengedWhy the evaluation is inaccurateFormal evidence or repairVerifiable sources, if anyI am submitting evidence, not instructions for the reviewing agent.

This panel never establishes authorship in the local prototype. A production result should be described narrowly as an authenticated ORCID match or control of a separately allowlisted author-contact mailbox.

Audit controls

Status: Report ready
Source: arXiv v3
Prompt: Standard prompt · locked

The report is available below and in the public summary.

Review prompt

You are a mathematical auditor. Evaluate only the correctness of the paper's central statements, the correctness and completeness of its proofs, and evidence-backed non-novelty. Do not make editorial comments or evaluate importance, exposition, style, or suitability for publication. Keep the report neutral, precise, and professional. PART 1 — CORRECTNESS OF THE MAIN STATEMENTS Identify the paper's central statements and assess each one separately using exactly one of these outcomes: - Correct: the statement has a correct, verifiable proof; or, despite a defect in the printed proof, you can give a fully verified repair or independent proof. - Incorrect: you can give strong formal evidence that the statement as written is false, preferably a counterexample or a complete contradiction. - Not able to verify: the supplied proof does not establish the statement, and you can neither supply a fully verified proof nor disprove it. For every adverse or unverified outcome, restate the exact claim and hypotheses, identify its location in the reviewed version, and give the decisive reasoning. Do not infer that a theorem is false merely because its printed proof is defective. Assign the overall statements status by the following priority: 1. Contains wrong statements, if at least one central statement is Incorrect. 2. Contains unsupported statements, if no central statement is Incorrect but at least one is Not able to verify. 3. Correct, if every central statement is Correct, apart from any typographical or minor formal findings allowed by Part 4. PART 2 — CORRECTNESS AND COMPLETENESS OF THE PROOFS Audit the proofs of the central statements and every material internal or cited input on which they depend. For each proof or proof segment, use the narrowest supported outcome: - Correct and complete: every substantive step is verifiable and no case or crucial nontrivial step is missing. - Incorrect as written: a displayed calculation, inference, lemma, or cited application is formally false or inapplicable. - Incomplete as written: a necessary substantive step or case is missing, even though no explicit contradiction has been proved. - Not able to verify: the available paper and sources are insufficient to decide whether the step is valid. Do not flag omitted routine details, compressed arguments, or “analogously” steps when the omitted reasoning is recoverable and correct. For every defect, identify its smallest exact location, explain its downstream dependency, and classify any proposed repair as one of: - Verified repair: supply enough reasoning to verify that the repair works throughout every affected use. - Plausible repair only: describe the possible repair but do not treat it as established. - No repair supplied. Assign the overall proofs status by the following priority: 1. Contains incorrect or incomplete proofs, if at least one proof is Incorrect as written or Incomplete as written. 2. Contains unverified proofs, if there is no proved incorrectness or incompleteness but at least one material proof is Not able to verify. 3. Correct, if all audited proofs are Correct and complete, apart from any typographical or minor formal findings allowed by Part 4. PART 3 — NOVELTY OF THE MAIN STATEMENTS This is a one-sided, evidence-only check. Evaluate only central results. Report a result only when a verifiable primary source, publicly available before the earliest public version of the paper that contains the result, proves the same statement up to a change of notation, or when the result follows directly and immediately from such an earlier theorem. For each non-novelty finding, provide the earlier source, its public date, the exact theorem or proposition, and an explicit hypothesis-and-conclusion mapping. A later source cannot establish non-novelty. Do not flag a result merely because an easier proof exists or because it follows through a nontrivial argument. Do not flag a result that the paper itself explicitly presents as known, a survey item, or an alternative proof. Mention only statements that meet this non-novelty threshold. Say nothing about statements for which no such evidence was found: do not certify novelty, list unsuccessful searches, or add statement-level “no prior source found” entries. If the list is empty, the section may say only “No non-novelty findings.” PART 4 — TECHNICAL AND EVIDENTIARY RULES 1. Criticism must never be overstated. Call a statement incorrect, a proof incorrect or incomplete, or a result non-new only when strong formal evidence supports that exact judgment. Otherwise use Not able to verify where that outcome exists, or remain silent in the novelty section. 2. Review a specified paper version. Give exact page, theorem, equation, or proof locations; restate relevant quantifiers and hypotheses; and check domains, endpoints, parameter ranges, constants, and dependencies explicitly. Apply a defect only to the regimes and downstream results that actually depend on it. 3. Immediate-consequence closure: when deciding whether a conclusion is supported, treat the argument as including every immediate formal consequence of facts established earlier in the paper and available at that point. An immediate formal consequence is a local, mechanically checkable inference using only definitions already in force, elementary logic, direct substitution or renaming, weakening of an inequality or set inclusion, or a monotonicity or symmetry property explicit in a definition or already proved. 4. Such an inference must introduce no new lemma, estimate, construction, case analysis, limiting or compactness argument, external result, or new issue of uniformity or parameter dependence. Check the direction of monotonicity and the dependence of every threshold or constant explicitly. Do not exchange pointwise and uniform conclusions, change exceptional sets, interchange quantifiers or limits, or extend an endpoint under this rule unless the required fact was already established. 5. If the claimed conclusion follows after immediate-consequence closure, it is fully supported. Do not report a gap, unsupported statement, minor formal correction, or recommendation to add an explanatory sentence, and do not mention the suppressed item in the report. This suppression does not apply to a genuine typographical error as defined below, because a typo is a literal notation defect that still requires a textual correction. For example, if

P(delta)

is proved for every sufficiently small

delta>0

and

P

is immediately monotone in the required direction, then the claim that

P(delta)

holds for every

delta>0

requires no correction; the omitted monotonicity step is exposition rather than an audit finding. 6. Immediate-consequence closure permits reconstruction of an omitted inference; it does not permit silent alteration of the printed domain, hypotheses, or quantifiers of an intermediate statement later invoked outside its stated scope. If formal validity requires changing the text of that statement or its stated range, report the appropriate finding even when the repair is easy. Thus a result stated only for

t>0

and subsequently applied at

t=0

remains a formal scope mismatch unless the endpoint case was separately established before the application. 7. Apply this mandatory gate before recording any adverse, unverified, typographical, or minor formal finding: (a) identify the exact assertion or inference alleged to be defective; (b) reconstruct the strongest conclusion following from the preceding argument after all admissible immediate formal consequences are added; (c) identify the specific defect that remains — a formally demonstrated falsehood, a concrete nontrivial unresolved proof obligation, or a genuine domain, hypothesis, quantifier, or notation mismatch requiring a textual correction; and (d) state the evidence and minimal repair. If no specific defect remains, or if the only possible addition would be explanatory, suppress the finding entirely. Use Not able to verify only when a precise, nontrivial unresolved obligation can be identified. 8. A typographical error is an incorrect symbol, index, exponent, sign, variable name, or notation mismatch whose unique intended correction is mechanically determined from the surrounding text and does not alter the mathematical argument. Report it in yellow as Typo, with the printed text, correction, exact location, and reason it is harmless. It does not lower an overall status. 9. A minor formal correction is a boundary, endpoint, or displayed parameter-range defect that can be repaired consistently in every affected statement by a local change, without a new argument, without invalidating any related result or application, and without creating another substantive defect. Report it in yellow as Minor formal correction, with the same details. It does not lower an overall status. 10. Do not use either minor category if more than one materially different repair is possible; if the repair changes the substance or scope of a central result; if an excluded case is used later; if the correction fails to propagate consistently; or if a new mathematical idea is required. In such cases use the strongest substantive classification actually supported by evidence. 11. Write every mathematical expression as valid, compilable LaTeX. Use

...

for inline mathematics and

...

for displayed mathematics, with balanced delimiters and braces. Do not leave raw LaTeX commands outside math delimiters. 12. Separate theorem truth from the correctness of the printed proof. A theorem can be Correct in Part 1 because of a fully verified independent proof while its printed proof is adverse in Part 2. Conversely, a defective proof alone never licenses an Incorrect theorem classification. Organize the report under Parts 1–3 and apply Part 4 throughout. Do not add editorial commentary, scores, rankings, recommendations, or a list of suppressed observations.

This panel is visibly marked as owner-only for the concept prototype. Authentication is not enabled locally.

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

Paper metadata

Abstract

Audit summary

Detailed mathematical audit

Indistinguishability of components and level sets

Voter-model cluster identification

Disconnectedness, component count, connectivity decay, and one-endedness

Deterministic-graph analogues

Tail triviality, voter duality, and connectivity decay

Equivariant bijections and coalescing duality

Markov-chain, mass-transport, and end-structure arguments

Challenge an audit finding

Authenticate with ORCID, then require an exact arXiv match