{"id":"57ee70e7-508b-4e23-b4e4-eb35fd481858","arxiv_id":"2509.06797","paper_version":1,"verdict":"REJECT","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"high","formal_verification":"none","parameter_count":0,"one_line_summary":"Ergodic cocycles satisfying asymptotic past and future independence converge to the Gromov boundary and have positive drift when integrable.","lead":"This paper proves that certain ergodic random dynamical systems on hyperbolic and CAT(0) spaces converge to a point at infinity almost surely, and that their drift is positive under an integrability condition. It generalizes known random-walk theorems to time-inhomogeneous processes used in geometric group theory.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Lemma 6.4's a.s. hitting of a fixed loxodromic power is unsupported and false for transient random walks, so the convergence proof collapses.","rationale":"The reader's weakest_assumption identifies exactly the load-bearing gap: Lemma 6.4 needs an almost-sure hitting statement for fixed loxodromic powers. I checked whether Proposition 5.6 plus apafi could imply it. Proposition 5.6 is about the semigroup generated by the support of the one-step distribution; it has no pathwise content. Apafi is a measure-class independence condition between past and future Mackey ranges; in the i.i.d. random walk case it holds, yet the required hitting event has probability <1 for transient walks. Therefore the proof of Lemma 6.4 contains a genuine logical gap, not merely an omitted detail. Since Lemma 6.4 is used to prove Corollary 6.5 and then Theorem 6.2, the main convergence theorem is not proven by the arguments in the paper. The Hadamard-space results depend on Theorem 6.2 via hyperbolic models and inherit the gap. I do not see a way to derive the missing assertion from the stated hypotheses; the random walk counterexample is within the paper's scope, so the issue is internal, not a disagreement with external consensus. Other parts of the paper—boundary embedding, Mackey-range ergodicity—may be valid, but the headline theorem's proof collapses at this point. Hence the reader's REJECT verdict is appropriate, and my stress test does not change it.","tokens_in":39960,"tokens_out":3302,"duration_ms":40293,"concrete_test":"Take Ω = {a,b,a^{-1},b^{-1}}^ℤ with Bernoulli(1/4) measure, T the shift, and f(ω)=ω_0. Then χ(n,ω)=ω_{-1}...ω_{-n} is the usual simple random walk S_n on F_2. Fix g=a and k≥1. Compute or estimate q_k = P(∃n≥0 : S_n = a^k). For transient nearest-neighbor random walk on F_2, the Green's function gives q_k < 1 for every k; equivalently, the hitting probability of any fixed vertex is less than 1. If q_k<1, then Lemma 6.4's claim that such hitting occurs for almost every ω is false. This directly isolates the unsupported step in the proof of Lemma 6.4 and hence the failure of the main theorem's proof.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central convergence Theorem 6.2 depends on Lemma 6.4, which asserts that for a fixed loxodromic g, for almost every ω there is n with χ(n,ω)=g^k for some k≥k0. The proof justifies this by citing apafi and Proposition 5.6. Proposition 5.6 only says supp(f_*P) generates G as a semigroup: each group element is a finite word in the support, but it says nothing about individual sample paths realizing a specified word at some time. In the random walk case—which is explicitly included and where Theorem 1.1 claims to recover Maher–Tiozzo—χ(n,ω) is the product of i.i.d. increments. For a transient admissible random walk on F_2, e.g. simple random walk with support {a,b,a^{-1},b^{-1}}, the probability that the walk ever equals a fixed element a^k is strictly less than 1. Thus the event ⋃_n Ω_n is not conull, contradicting the assertion in Lemma 6.4. Without Lemma 6.4, Corollary 6.5 (support equals the full limit set) and the proof of Theorem 6.2, which needs two support points to apply the shadow argument, fail. The Hadamard-space theorems inherit this gap through Theorem 7.1. The claimed convergence may be true, but the proof as written does not establish it.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies discrete random dynamical systems generated by an ergodic measure-preserving transformation and a measurable map f from the probability space to a countable group G acting by isometries on a (possibly non-proper) Gromov-hyperbolic or CAT(0) space. Under the 'asymptotic past and future independence' (apafi) condition of Bader–Furman, it claims almost sure convergence of the backward cocycle to the Gromov or visual boundary, positive drift for integrable cocycles, and analogous statements for Hadamard spaces via hyperbolic models introduced by Petyt–Spriano–Zalloum. The paper also develops ergodic properties of Mackey ranges, including strong approximate transitivity and coarse metric ergodicity, and proves an equivariant embedding of the boundary of the hyperbolic models into the visual boundary of the CAT(0) space.","tokens_in":40323,"tokens_out":9965,"duration_ms":109968,"significance":"If the main theorems were established, they would give a genuinely general boundary convergence theorem for ergodic cocycles without moment assumptions, recovering the Maher–Tiozzo random walk theorem and extending it to CAT(0) spaces. The paper has several strengths: it is clearly written, it properly attributes external results, it contains no fitted parameters or hidden normalizations, and it contributes original technical tools, notably the adaptation of BCFS24 to almost geodesic hyperbolic spaces, the proof of Theorem 1.3, and the coarse metric ergodicity result for Mackey ranges. However, the proof of the central convergence theorem relies on a false hitting-time assertion in Lemma 6.4. This is a load-bearing gap: it invalidates Corollary 6.5, the proof of Theorem 6.2, and consequently the main hyperbolic and CAT(0) convergence and drift theorems as stated.","major_comments":[{"comment":"In the proof of Lemma 6.4, the paper asserts that by apafi and Proposition 5.6, for almost every ω there exists n such that χ(n,ω)=g^k for some k≥k0. Proposition 5.6 only shows that supp(f_*P) generates G as a semigroup; it gives no information about individual sample paths realizing a specified word at some time. In the random walk case, which the paper explicitly includes, χ(n,ω) is the product of i.i.d. increments. For a transient admissible random walk on a free group, for instance simple random walk on F_2 with support {a,b,a^{-1},b^{-1}}, the event that the walk ever equals a fixed element a^k has probability strictly less than 1. Hence the set ⋃_n Ω_n defined in the proof is not conull, and the assertion is false.","section":"§6.2, Lemma 6.4"},{"comment":"Corollary 6.5 is deduced directly from Lemma 6.4 and asserts that the support of μ+_{p+(ω)} is the full limit set Λ(G). Since Lemma 6.4 is unsupported, Corollary 6.5 is not established. The proof of Theorem 6.2 then explicitly uses Corollary 6.5 to choose two points ξ,η in the support of μ+_{p+(T^{-n}ω)} and applies the shadow argument of Proposition 6.6. Without a valid support statement, the contraction in Proposition 6.3 cannot be converted into convergence of the cocycle in X. The error therefore propagates to Theorem 1.1 and, through Theorem 7.1, to Theorems 1.2, 1.4, and 1.5.","section":"§6.2, Corollary 6.5 and Theorem 6.2"},{"comment":"The paper states that Theorem 1.1 recovers the Maher–Tiozzo random walk convergence theorem. That recovery is not justified by the present proof. For random walks the support of the stationary measure is known to be the entire limit set, but the argument supplied here, via Lemma 6.4, is invalid. A correct proof would need a different mechanism to show that all loxodromic fixed points lie in the support under apafi. The Hadamard-space results inherit the same gap because Theorem 7.1 applies Theorem 6.2 to the hyperbolic model; no independent support argument is given there.","section":"§1.1 and §7"}],"minor_comments":[{"comment":"The proof refers twice to 'Proposition 8.4', but the stated result is Corollary 8.4. Please correct the cross-reference.","section":"§8.1, proof of Theorem 8.1"},{"comment":"There is a typo in the citation to [Arn98]: 'Corolloary' should be 'Corollary'.","section":"§6.2, Lemma 6.4"},{"comment":"The phrase 'freedom to chose a pair of points ξ,η in the support' is imprecise. For the argument one needs, for a sequence of times, points in the support of μ+_{p+(T^{-n}ω)} that can be chosen measurably and stay away from the limit point. This is secondary to the support issue but should be clarified in a revision.","section":"§6.2, proof of Theorem 6.2"},{"comment":"The reduction from separable to non-separable spaces via [GST20, Remark 4] is sketched in one sentence. Since the paper claims to drop separability, a fuller explanation of the equivariant quasi-isometric embedding Y↪X would improve readability.","section":"§2.4"}],"recommendation":"reject","confidential_remarks":"The paper is well-structured and contains interesting auxiliary results, especially the boundary embedding for non-proper CAT(0) spaces and the coarse metric ergodicity of Mackey ranges. However, the main convergence proof relies on a false assertion in Lemma 6.4, and this gap destroys the proofs of the central theorems. Unless the author can provide a valid replacement for the support lemma, the paper does not establish its claimed results. I would not encourage a minor revision; a major new idea would be needed."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Short version: the reader's suspicion is correct. Lemma 6.4 is not a minor gap; it is the step that makes Theorem 6.2 work, and the assertion about hitting a fixed loxodromic power is false in the random-walk case the paper claims to recover. That said, the paper is not a throwaway. The proof of Theorem 1.3, extending the Petyt–Spriano–Zalloum boundary embedding to non-proper CAT(0) spaces, is a solid contribution on its own. Section 5's development of Mackey ranges, the SAT property, and coarse metric ergodicity is new and, as far as I can tell, correct; the adaptation of the BCFS24 apparatus to almost-geodesic hyperbolic spaces in Section 2 is careful and useful. The author is also honest about overlap with BF25.\n\nThe problem: Lemma 6.4 asserts that for a fixed loxodromic g, almost every trajectory has some prefix χ(n,ω) equal to g^k with k≥k0. Proposition 5.6 only gives that the semigroup generated by supp(f_*P) is G; it says nothing about individual sample paths realizing a particular word. For a transient admissible random walk on F_2, the probability that the walk ever equals a fixed a^k is strictly less than 1, so the event ⋃Ω_n is not conull. The random walk case is explicitly within the paper's scope, so this is not a technicality. Without Lemma 6.4, Corollary 6.5 (support equals the full limit set) and the shadow argument in Theorem 6.2 lack the two support points they need. The Hadamard-space theorems inherit the gap through Theorem 7.1, and the drift theorems depend on convergence.\n\nThe convergence statement may be true and could be provable by other means—for instance, by showing the support of µ+ω contains enough points without fixing g in advance. But the proof as written does not establish it. The paper deserves a serious referee: the auxiliary results are publishable, and the main question is important enough that an expert should examine whether the lemma can be replaced. I would not cite the main theorems in their current form.","headline":"The paper does real work on Mackey ranges and CAT(0) boundaries, but the main convergence proof leans on a hitting lemma that is false for transient random walks.","tokens_in":40780,"tokens_out":3226,"would_cite":false,"duration_ms":35354,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["37A30","37A50","20F65","60B15","37H99"],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper proves that ergodic cocycles on non-elementary hyperbolic actions converge almost surely to the Gromov boundary under the asymptotic past–future independence condition.","keywords":["ergodic cocycles","Gromov-hyperbolic spaces","Hadamard spaces","Mackey range","asymptotic past and future independence","boundary convergence","positive drift","contracting elements"],"falsifier":"Take the simple random walk on the free group F_2 with a finitely supported non-elementary measure, which is transient and apafi. Fix a loxodromic g and compute the probability that the backward cocycle χ(n,ω) ever equals some fixed positive power of g; transience makes this probability less than 1, contradicting the hitting assumption in Lemma 6.4. This does not by itself disprove the convergence theorem, but it shows the written proof's pivotal support claim fails for a standard example.","tokens_in":39894,"feed_emoji":"🎯","tokens_out":6758,"duration_ms":65978,"temperature":0.7,"pith_summary":"The paper proves that for ergodic random dynamical systems on a group acting non-elementarily on a possibly non-proper hyperbolic space, the backward cocycle converges almost surely to a point of the Gromov boundary, provided the system satisfies the asymptotic past and future independence condition (apafi). No integrability or properness is needed. When the cocycle has finite first moment, its drift is strictly positive. The same conclusions hold for Hadamard spaces with independent contracting elements, via curtain-based hyperbolic models. This extends boundary convergence and positive-speed laws from random walks to time-dependent ergodic cocycles.","feed_headline":"Ergodic cocycles converge to the boundary almost surely","feed_subtitle":"No moment assumptions are needed; with finite first moment the drift is positive, in hyperbolic and Hadamard spaces alike.","key_machinery":"The asymptotic past and future independence condition (apafi) is the load-bearing stochastic assumption: past and future Mackey-range boundaries are weakly independent, forming a G-boundary system. The Mackey ranges B±, defined as the spaces of ergodic components of one-sided trajectory spaces under the shift, replace the Poisson–Furstenberg boundary of a random walk. For Hadamard spaces, the curtain-based hyperbolic models X_L convert contracting isometries into loxodromic ones, and the boundary embedding ∂_L: ∂Grom X_L → ∂∞ X transfers hyperbolic convergence back to the visual boundary.","core_discovery":"The central claim is Theorem 1.1: under apafi, for any basepoint o, the backward cocycle χ(n,ω)o converges P-almost surely to a random boundary point ξ+(ω) in the Gromov boundary. The proof constructs past and future Mackey ranges B− and B+ from the cocycle, shows they form a G-boundary system, uses them to produce forward-invariant measures that are Dirac measures, and then exploits weak convexity of shadows to turn boundary contraction into actual convergence in the space. Theorem 1.4 adds that integrability forces the drift λ_X(φ) to be strictly positive. The Hadamard analogues, Theorems 1.2 and 1.5, follow by passing to a hyperbolic model X_L and using a new equivariant homeomorphism bet","pith_inferences":["If the boundary embedding for curtain models extends to other wall-based hyperbolic models that detect strongly contracting directions, the same argument should give convergence and positive drift for actions on injective spaces or hierarchically hyperbolic spaces.","The apafi condition is plausibly close to necessary: without weak past–future independence, past-oriented stationary measures need not be diffuse, and the support argument for attracting fixed points breaks down; constructing a non-apafi ergodic cocycle with oscillating sample paths would delineate the boundary of the theorem.","The Busemann-cocycle proof of positive drift is set up for second-moment arguments; adding moment assumptions on displacement should yield central limit and iterated-logarithm rates for apafi cocycles.","Because convergence requires no integrability, apafi alone rules out a zero-drift non-convergent regime; checking whether apafi also forbids sublinear but unbounded displacement would sharpen the dichotomy between convergence and escape rate."],"forward_implications":["Almost-sure convergence to the Gromov boundary holds for every ergodic apafi cocycle on a non-elementary hyperbolic action, covering non-proper settings such as arc graphs, extension graphs, and R-trees.","For integrable cocycles the drift is positive, so sample paths escape to infinity at a positive linear rate; in Hadamard spaces they track a geodesic ray with sublinear error.","In Hadamard spaces with contracting elements, the hitting measure is supported on curtain-visibility points in B_L, so any two distinct limit points subtend angle π.","The map from the Gromov boundary of each hyperbolic model to the visual boundary is an Isom(X)-equivariant homeomorphism onto B_L, giving a dictionary between model directions and true boundary points.","The Mackey-range actions are coarsely metrically ergodic, extending strong approximate transitivity from Poisson boundaries to ergodic cocycles."],"supporting_citations":[{"why":"Supplies the apafi condition and the boundary-system framework for ergodic cocycles.","marker":"[BF14]"},{"why":"Provides the curtain-based hyperbolic models X_L and the contracting-to-loxodromic conversion.","marker":"[PSZ24]"},{"why":"Provides the classification of G-maps to hyperbolic boundaries and the coarse metric ergodicity toolkit.","marker":"[BCFS24]"},{"why":"Gives the random-walk convergence and shadow-convexity results being generalized.","marker":"[MT18]"},{"why":"Presents the boundary-system and Poisson-transform theory for ergodic cocycles used in later sections.","marker":"[BF25]"},{"why":"Supplies the random dynamical system formalism, invariant measures, and disintegration tools.","marker":"[Arn98]"},{"why":"Gives the subadditive ergodic theorem used to define the drift.","marker":"[Kin68]"},{"why":"Gives the geodesic tracking theorem used for the sublinear tracking corollary.","marker":"[KM99]"}],"fun_headline_variants":["Cocycles converge to boundary without moment assumptions","Past-future independence forces cocycle boundary limits","No moments needed: cocycles hit Gromov boundary a.s.","Ergodic cocycles converge a.s. in hyperbolic and Hadamard","Finite first moment gives positive drift for ergodic cocycles"],"cache_read_input_tokens":2688,"weakest_assumption_plain":"The key lemma assumes that apafi guarantees every sample path eventually hits a fixed positive power of any given loxodromic element; transient random walks satisfy apafi but can violate this, and the proof of boundary convergence depends on that hitting event.","fun_headline_variants_meta":{"raw":{"variants":["Cocycles converge to boundary without moment assumptions","Past-future independence forces cocycle boundary limits","No moments needed: cocycles hit Gromov boundary a.s.","Ergodic cocycles converge a.s. in hyperbolic and Hadamard","Finite first moment gives positive drift for ergodic cocycles"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000312,"raw_usage":{"total_tokens":1567,"prompt_tokens":654,"completion_tokens":913,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":398,"completion_tokens_details":{"reasoning_tokens":835}},"tokens_in":398,"tokens_out":913,"duration_ms":8388,"temperature":1.0,"reasoning_tokens":835,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-04T23:07:03.435243+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Take the simple random walk on the free group F_2 with a finitely supported non-elementary measure, which is transient and apafi. Fix a loxodromic g and compute the probability that the backward cocycle χ(n,ω) ever equals some fixed positive power of g; transience makes this probability less than 1, contradicting the hitting assumption in Lemma 6.4. This does not by itself disprove the convergence theorem, but it shows the written proof's pivotal support claim fails for a standard example.","supporting_citations":[],"review_version":1}