{"id":"95c14eb9-ba74-4f6e-9973-0ee2c8ae82ab","arxiv_id":"2607.04002","paper_version":1,"verdict":"ACCEPT","confidence":"HIGH","novelty_score":6.5,"correctness_risk":"low","formal_verification":"none","parameter_count":0,"one_line_summary":"Under unimodularity plus Green-function integrability, transient random graphs have the finite collision property; this classifies voter-model stationary measures on Gilbert, Delaunay, Gabriel and long-range percolation graphs.","lead":"Two independent random walks on certain random geometric graphs and long-range percolation clusters collide infinitely often in low dimensions and only finitely often in high dimensions. This decides the stationary measures of the voter model on those graphs.","discovery_kind":"extension","skeptic_critique":{"model":"grok-4.5","headline":"No significant objection identified","rationale":"The central claim (Theorem 2 + applications) rests on a transparent mass-transport bound that equates the expected number of collisions to E[deg(ρ)·G(ρ,ρ)]. The hypothesis is stronger than pure transience, but the paper never pretends otherwise and supplies the needed moment estimates for every model it treats. The transfer lemmas, Palm-to-unimodular reductions, and heat-kernel/resistance comparisons are standard and carefully checked. Because the open gap is already acknowledged and does not affect the theorems as stated, the reader's ACCEPT verdict stands without adjustment.","tokens_in":24938,"tokens_out":497,"duration_ms":4855,"concrete_test":"Independently recompute the energy of the unit flow Θ constructed in the proof of Lemma 9 (the comparison with Bernoulli percolation) for the Gilbert-graph renormalization of §3.3.3; confirm that the multiplicative constant C^{2}L remains finite for the chosen M and that the resulting moment E_{0}[R_eff(0↔∞)^{1+δ}] is still finite under the exponential tail of Lemma 8. If the constant diverges or the moment fails, the reduction of Theorem 3(2) to Theorem 2 collapses.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The reader's weakest-assumption note correctly flags that Theorem 2 uses the stronger integrability E[deg(ρ)·G(ρ,ρ)]<∞ rather than mere a.s. transience + E[deg(ρ)]<∞, an open gap the authors themselves record on p. 2. That gap does not undermine the paper's actual claims: the mass-transport argument of §2.3.2 is short and correct under the stated hypothesis, and every application (Gilbert/Delaunay/Gabriel via Lemmas 6–9 + heat-kernel tails; long-range via Theorem 7) verifies precisely this stronger moment. No internal inconsistency, missing step, or incorrect transfer appears in the argument that yields the finite-collision property and the subsequent characterization of I_e.","agreement_with_reader":"agree"},"referee_report":{"model":"grok-4.5","summary":"The paper studies discrete- and continuous-time collision properties of simple random walks on unimodular random rooted graphs. It recalls that recurrence plus E[deg(\rho)]<\\infty implies the infinite-collision property (Theorem 1, adapting Hutchcroft–Peres), and proves that the stronger integrability E[deg(\rho)·G(\rho,\rho)]<\\infty implies the finite-collision property almost surely (Theorem 2, via mass transport). These criteria are applied to the infinite components of the supercritical Gilbert graph, the Delaunay triangulation and the Gabriel graph in R^d (Theorem 3), and to long-range percolation clusters on Z^d in the regimes where an infinite cluster exists (Theorem 4). The collision properties, together with triviality of the random-walk tail \rho-algebra, completely characterize the extremal stationary measures of the voter model on these graphs (Corollary 1).","tokens_in":25136,"tokens_out":711,"duration_ms":10223,"significance":"If correct, the work supplies a clean, usable criterion for the finite-collision property on unbounded-degree unimodular graphs and thereby settles the structure of I_e for several natural geometric and long-range models that lie outside the classical bounded-degree or vertex-transitive setting. The mass-transport argument of Theorem 2 is short and transparent; the geometric applications rest on carefully transferred heat-kernel and resistance estimates (Barlow, Crawford–Sly, Rousselle) via an explicit comparison lemma (Lemma 9). The resulting complete description of stationary measures for the voter model is a concrete payoff that will be of interest both to interacting-particle-systems and to random-geometry communities.","major_comments":[],"minor_comments":[{"comment":"p. 2, after Theorem 2: the open question whether E[deg(\rho)]<\\infty plus a.s. transience already implies finite collisions is correctly flagged; a one-sentence remark that the present applications all verify the stronger moment would help the reader.","section":"Introduction"},{"comment":"Lemma 9 and the subsequent geometric constructions (Sections 3.3.3–3.3.4) are dense; a short schematic diagram of the good-box renormalization and the path \theta\toΘ would improve readability.","section":"§3.3.2"},{"comment":"Equation (11) equating continuous and discrete Green functions is standard but could be given a one-line reference or derivation for non-specialists.","section":"§2.3.2"},{"comment":"In the long-range section the restriction s\notin[d+2,2d] is explained by the lack of quantitative tails on the random time T_x; a pointer to the available heat-kernel bounds of Can–Croydon–Kumagai would be useful even if they are not yet strong enough for the argument.","section":"§4.2"}],"recommendation":"accept","confidential_remarks":"The manuscript is technically solid and the novelty relative to Hutchcroft–Peres is clear. The future arXiv date and the slightly unusual author-order presentation are cosmetic; I see no citation or priority issues. Suitable for a strong probability journal."},"author_rebuttal":null,"desk_editor":{"model":"grok-4.5","letter":"The new piece is Theorem 2: under the unimodular mass-transport principle and the single integrability E[deg(ρ)·G(ρ,ρ)]<\\infty you get both discrete and continuous finite collisions a.s. Combined with the Hutchcroft–Peres infinite-collision result under recurrence, this immediately classifies the extremal stationary measures of the voter model on the infinite Gilbert, Delaunay and Gabriel graphs (d=2 vs d≥3) and on long-range percolation clusters in the regimes where heat-kernel tails are known.\n\nThe mass-transport argument itself is short and correct; the continuous-time version is a routine adaptation of the same idea. The geometric applications are careful rather than flashy: they transfer everything to the Palm/unimodular version (Lemma 2), control the root degree by exponential moments (Lemma 3), then reduce the Green-function moment to effective-resistance tails via a comparison lemma (Lemma 9) that couples good boxes to supercritical Bernoulli percolation and imports Barlow and Crawford–Sly heat-kernel estimates. The long-range side does the same with the Crawford–Sly tails. Recurrence of the supercritical Gilbert graph in d=2 is proved from scratch by a Nash–Williams cutset argument that works. Citations are external and appropriate; no circularity.\n\nThe only soft spot is exactly the one the authors flag on page 2: the integrability they use is strictly stronger than a.s. transience plus E[deg]<\\infty, and it is open whether the weaker pair already forces finite collisions. Every application verifies the stronger moment, so the theorems as stated stand. The open question about discrete vs continuous collisions on bounded-degree graphs is left open, which is honest.\n\nThis is solid specialist work for people who care about random walks on continuum percolation or about the structure of I_e for the voter model. It deserves a serious referee; I would accept it for peer review and would cite the classification results myself.","headline":"Clean finite-collision criterion for unimodular graphs plus complete voter-model classification on standard unbounded-degree geometric and long-range models; the stronger Green integrability is flagged by the authors and does not break the claims.","tokens_in":25714,"tokens_out":506,"would_cite":true,"duration_ms":4795,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["60K35","60G50","82C22","05C81"],"pacs":[],"model":"grok-4.5","headline":"On unimodular random graphs, a Green-function moment decides whether two walks collide only finitely often, settling the voter model on geometric and long-range graphs.","keywords":["unimodular random graphs","random-walk collisions","Green function","voter model","Gilbert graph","Delaunay triangulation","long-range percolation","mass transport"],"falsifier":"Construct (or disprove the existence of) a unimodular random rooted graph that is almost surely transient, has finite mean degree at the root, yet infinite expected degree-times-Green-function, and check whether two independent walks still collide only finitely often.","tokens_in":25862,"feed_emoji":"🔀","tokens_out":721,"duration_ms":6318,"temperature":0.7,"pith_summary":"The paper continues the study of when two independent simple random walks on a random graph meet only finitely many times or infinitely often. Earlier work already showed that recurrence plus finite mean degree at the root forces infinitely many collisions almost surely. The new result is the matching finite-collision theorem: if the expectation of the degree times the Green function at the root is finite, then the walks collide only finitely often almost surely (in both discrete and continuous time). Because many natural random geometric graphs have unbounded degrees, classical bounded-degree arguments do not apply; the unimodular mass-transport principle supplies the missing control. The authors verify the moment condition for the supercritical Gilbert graph, the Delaunay triangulation, the Gabriel graph, and selected regimes of long-range percolation. As a direct corollary they obtain a complete description of the extremal stationary measures of the voter model on each of these graphs: only the two consensus measures when collisions are infinite, and the whole one-parameter family of Bernoulli product measures when collisions are finite and the walk tail is trivial.","feed_headline":"Green moment forces finite walk collisions on random graphs","feed_subtitle":"The criterion settles the voter model on geometric graphs and long-range percolation","key_machinery":"Mass-transport principle applied to the expected number of collisions: the transport function f(G,u,v)=p_n(u,v)p_n(v,u)deg(v) converts the double sum of return probabilities into an expectation of deg(ρ)·G(ρ,ρ), which is finite by hypothesis and therefore forces the number of collisions to be finite a.s.","core_discovery":"If (G,ρ) is a unimodular random rooted graph satisfying E[deg(ρ)·∑_n p_n(ρ,ρ)] < ∞, then two independent simple random walks on G collide only finitely often almost surely, both in discrete and in continuous time. Together with the known infinite-collision statement under recurrence, this dichotomy completely determines the set of extremal stationary measures of the voter model on the graphs under study.","pith_inferences":[],"forward_implications":[],"fun_headline_variants":["Green moment forces finite SRW collisions on unimodular graphs","Integrable Green yields only finitely many walk collisions a.s.","Root Green moment implies finite collisions of independent walks","Transience plus Green integrability: walks collide finitely often","Unimodular graphs: Green condition settles finite walk collisions"],"cache_read_input_tokens":16512,"weakest_assumption_plain":"The argument needs a finite expectation of degree times Green function at the root; mere almost-sure transience plus finite mean degree may not be enough, and the paper leaves open whether that weaker pair already implies finite collisions.","fun_headline_variants_meta":{"raw":{"variants":["Green moment forces finite SRW collisions on unimodular graphs","Integrable Green yields only finitely many walk collisions a.s.","Root Green moment implies finite collisions of independent walks","Transience plus Green integrability: walks collide finitely often","Unimodular graphs: Green condition settles finite walk collisions"]},"model":"grok-4.5","effort":"low","cost_usd":0.003662,"raw_usage":{"total_tokens":1129,"prompt_tokens":682,"num_sources_used":0,"completion_tokens":65,"cost_in_usd_ticks":36620000,"prompt_tokens_details":{"text_tokens":682,"audio_tokens":0,"image_tokens":0,"cached_tokens":256},"completion_tokens_details":{"audio_tokens":0,"reasoning_tokens":382,"accepted_prediction_tokens":0,"rejected_prediction_tokens":0}},"tokens_in":682,"tokens_out":65,"duration_ms":3840,"temperature":1.0,"reasoning_tokens":382,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-07-11T22:22:41.678763+00:00","model_set":{"reader":"grok-4.5"},"falsifier":"Construct (or disprove the existence of) a unimodular random rooted graph that is almost surely transient, has finite mean degree at the root, yet infinite expected degree-times-Green-function, and check whether two independent walks still collide only finitely often.","supporting_citations":[],"review_version":1}