{"id":"0374d187-13bb-428c-9179-97820e813e7c","arxiv_id":"2506.04124","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":8.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"The top Lyapunov exponent of i.i.d. random cocycles with values in Mat_m(R) is Hölder continuous in the Wasserstein metric under finite exponential moments, quasi-irreducibility, and a spectral gap.","lead":"This paper proves that the top Lyapunov exponent, which measures exponential growth of random products of matrices, varies Hölder-continuously when the matrix distribution is perturbed, even for non-invertible and unbounded matrices. The result extends a classical theorem to a more realistic setting and gives rigorous asymptotics for random Schrödinger operators with unbounded potentials.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Proposition 6.9's contraction estimate relies on an unproved upper bound for ∫log‖∧²g‖dμⁿ; Lemma 6.3 does not cover exterior-square cocycles under the stated assumptions.","rationale":"The reader correctly singles out Proposition 6.9 as the weakest load-bearing ingredient: if the Markov operator is not uniformly contracting on Hölder observables of the projective space, the strong mixing estimate in Theorem 6.1 and the Hölder continuity proof in Section 7 collapse. My concern is more specific than the reader's: within Proposition 6.9 itself, the proof needs an asymptotic upper bound for the expected log-norm of the exterior-square product. Lemma 6.3 supplies uniform convergence only for expectations of log‖Aₙv‖ with a fixed vector v; it cannot, by itself, control ∫ log‖∧²Aₙ‖dμⁿ, which involves a maximum over two-dimensional directions. The standard way to obtain such a bound is to apply the same argument to the exterior-square cocycle, but that requires hypotheses on ∧²μ — quasi-irreducibility and second-order moment bounds — that are not stated in Theorem 2.1. Quasi-irreducibility of the original semigroup does not automatically imply quasi-irreducibility of its exterior square, so the gap is substantive rather than cosmetic. The example of SO(4) shows that irreducibility in Rᵐ and reducibility of ∧²Rᵐ can coexist; while that example is compact and has no spectral gap, it illustrates that the implication is not formal. For this reason I would keep the CONDITIONAL verdict: the main theorem is plausible and the surrounding argument is detailed, but Proposition 6.9 needs a completed proof of the exterior-power estimate before the central claim can be regarded as established. The application-level issues already noted by the reader (unproved Proposition 9.6 and the O(λ⁻ᵖ) rate in Proposition 9.5) remain secondary. No independent counterexample is claimed; the concern is that a necessary proof step is missing.","tokens_in":84,"tokens_out":15671,"duration_ms":276622,"concrete_test":"Independently re-derive the inequality ∫ log‖∧²g‖dμⁿ(g) ≤ n(L₁(μ)+L₂(μ)+ε) from the stated hypotheses of Theorem 2.1: μ ∈ Mₚ^C, quasi-irreducibility, and L₁(μ)>L₂(μ). In particular, try to derive it by applying Lemma 6.3 to the exterior-square cocycle ∧²μ on ∧²Rᵐ, and record which assumptions are used. If the derivation requires Θ²ₚ(μ)<∞ or 2-quasi-irreducibility of μ, then Proposition 6.9 is incomplete as stated. A concrete case to probe is m=4 with μ a small perturbation of a distribution on {cR : c>0, R∈SO(4)} chosen so that L₁(μ)>L₂(μ); check whether ∧²μ is quasi-irreducible while μ is quasi-irreducible, and whether the displayed bound still follows without extra moment assumptions.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The load-bearing spectral step is Proposition 6.9, where the proof needs to show ∫ log(‖∧²g‖/‖gv‖²)dμⁿ(g) ≤ −1 for large n. To get this, the proof uses the estimate ∫ log‖∧²g‖dμⁿ(g) ≤ n(L₁(μ)+L₂(μ)+ε), writing log‖∧²g‖ = log s₁(g)+log s₂(g) for the product g ∼ μⁿ. This estimate is not justified by Lemma 6.3: that lemma gives uniform convergence of expected logarithmic growth along a fixed projective vector, i.e. of ∫ log‖gv‖dμⁿ(g), whereas ‖∧²Aₙ‖ is a maximum over two-dimensional directions and its expectation is not controlled by a supremum of vector expectations. The natural repair is to apply Lemma 6.3 to the exterior-square cocycle ∧²μ on ∧²Rᵐ, but that would require ∧²μ to be quasi-irreducible and the moment bound Θ²ₚ(μ)<∞ to be available. Neither is assumed in Theorem 2.1: quasi-irreducibility of μ does not imply quasi-irreducibility of ∧²μ (for instance, the standard representation of SO(4) is irreducible while its exterior square is reducible), and Mₚ^C only bounds Θₚ, not Θ²ₚ. Without the missing estimate, the inequality ∫ e^{αX}dμⁿ ≤ 1−α+O(α²)<1 in Proposition 6.9 has no base point. Consequently Theorem 6.1 and the Hölder estimate in Section 7 lose their key contraction assumption, so the central claim is not fully supported as written.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proves local Hölder continuity of the top Lyapunov exponent for i.i.d. random linear cocycles with values in Mat_m(R), under three assumptions: finite exponential moments, quasi-irreducibility of the generated semigroup, and a positive gap L_1(μ)>L_2(μ). The topology is the Wasserstein distance W_p on the space M_p^C of measures with bounded exponential moments. Theorem 2.1 is the central result; Corollaries extend it to higher Lyapunov exponents and to locally constant Bernoulli cocycles, Theorem 2.2 gives stable exponential large-deviation estimates, and Section 9 applies the machinery to Schrödinger cocycles with unbounded potentials. The proof follows the spectral method of Nagaev/Bougerol/Le Page, with Sections 5–7 carrying the main argument. The paper is written as a self-contained derivation under explicit assumptions and does not fit free parameters, although it relies on several published results by the same authors and collaborators.","tokens_in":35010,"tokens_out":7254,"duration_ms":69611,"significance":"If Theorem 2.1 is correct after repair, it is a substantial extension of Le Page's Hölder regularity theory to non-invertible and non-compact random cocycles, in the Wasserstein topology rather than a parametric family. The treatment of non-invertible matrices and of non-compact support is a genuine novelty, and the paper also provides stable large-deviation estimates and applications to Schrödinger cocycles with unbounded potentials. The manuscript is transparent about its hypotheses and gives many of the key estimates explicitly. The main theorem would be a valuable addition to the Lyapunov regularity literature.","major_comments":[{"comment":"The proof of Proposition 6.9 needs the estimate ∫ log||∧²g|| dμ^n(g) ≤ n(L_1(μ)+L_2(μ)+ε), but this estimate is not a consequence of Lemma 6.3 as the text claims. Lemma 6.3 supplies uniform convergence of ∫(1/n)log||gv|| dμ^n(g) for fixed projective vectors v; the quantity log||∧²g|| is a supremum over two-dimensional directions and cannot be bounded by a supremum of vector expectations after integration. Applying Lemma 6.3 to the exterior-square cocycle ∧²μ would require quasi-irreducibility of ∧²μ and a bound on Θ²_p(μ), neither of which is assumed in Theorem 2.1. The estimate is probably true by subadditivity of log||∧²A_n|| together with Oseledets a.s. convergence and the moment bounds in M_p^C, but that argument is absent. Since Proposition 6.9 supplies the contraction κ_α(μ^n)≤σ_0 on which Theorem 6.1 and the Hölder estimate of Section 7 depend, this is a load-bearing gap that must be fixed.","section":"Section 6, Proposition 6.9"},{"comment":"The proof establishes W_p(µ̃_β, µ̃_0)≲β^p and then invokes Theorem 2.1 to conclude an error O(λ^{-p}). However, Theorem 2.1 only provides Hölder continuity with an unspecified exponent θ>0; in the proof of Theorem 2.1 the exponent is α/p with α<p/2, which is strictly smaller than 1/2. Consequently the displayed rate O(λ^{-p}) does not follow from the stated theorem. The asymptotic formula should either be weakened to O(λ^{-θ}) for some θ>0, or the proof must provide a uniform lower bound on the Hölder exponent along this family.","section":"Section 9.2, Proposition 9.5"},{"comment":"Proposition 9.6 is a numbered proposition in the applications section, but no proof is given anywhere in the manuscript. The asymptotic formula for L_1(∧^m A_{λ,E}) is therefore an unproved assertion. It should either be proved with analogues of Lemmas 9.3 and 9.4 for exterior powers and the preceding Wasserstein estimates, or it should be explicitly stated as a conjecture/removed from the paper.","section":"Section 9.3, Proposition 9.6"}],"minor_comments":[{"comment":"The displayed stationarity condition contains a typo: the set written as {ˆv ∈ Prob(P(Rm)) : g v=0} should be a subset of P(Rm), not Prob(P(Rm)), and the condition should read gv=0 for v∈P(Rm).","section":"Definition 2.2"},{"comment":"The statement of Proposition 5.5, '|L_1(µ)| ≤ C−1 p', is garbled; from the proof the intended bound appears to be |L_1(µ)| ≤ C/p.","section":"Section 5.2, Proposition 5.5"},{"comment":"The subsection heading 'Proof of Theorem 2.4' refers to Corollary 2.4; the numbering should be corrected.","section":"Section 7, proof of Corollary 2.4"},{"comment":"The attribution to Barrientos and Malicet cites '[6, Proposition X]', where 'X' appears to be a placeholder rather than an actual proposition number.","section":"Corollary 2.4, footnote/reference"},{"comment":"In the proof of Lemma 9.2 the integral is written with |t−a|^{-p}, while the statement of the lemma and the final estimate use |t−a|^{-d}; clarify which exponent is meant in each occurrence.","section":"Lemma 9.2"}],"recommendation":"major_revision","confidential_remarks":"The paper fits the scope of a dynamics journal and the central idea is promising, but the proof gap in Proposition 6.9 is load-bearing and must be repaired before the main theorem can be accepted. The application sections also need the corrections listed above. I did not find evidence of circularity or parameter fitting; the reliance on prior peer-reviewed work by the same authors is standard. The unproved Proposition 9.6 and the overly strong rate in Proposition 9.5 should be addressed in the revision rather than left as stated."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The right headline: this is a serious extension of Le Page's theorem, and the main result is likely true, but the proof as written has a load-bearing gap in Proposition 6.9. I checked the stress-test concern, and it lands.\n\nWhat is genuinely new: the paper treats non-invertible, non-compact random cocycles in Mat_m(R) and proves local Hölder continuity in Wasserstein distance, where the projective action degenerates on singular matrices. The machinery for handling those singularities—truncation, induced kernel, isospectral reduction—is real work and the exposition in Sections 5–7 is careful. The literature placement is honest, and the reliance on earlier published results is not, by itself, a problem.\n\nNow the soft spots, in order.\n\nFirst, Proposition 6.9. The proof needs to show ∫ log(‖∧²g‖/‖gv‖²) dμⁿ(g) ≤ −1, and for the term ∫ log‖∧²g‖ dμⁿ it uses the bound n(L₁+L₂+ε). That bound does not follow from Lemma 6.3, which controls expectations of log‖gv‖ for vectors, not the maximum over two-dimensional directions. To get it you would need Lemma 6.3 applied to the exterior-square cocycle ∧²μ, and that requires ∧²μ to be quasi-irreducible and a bound on the second exterior moment Θ²_p(μ). Neither is assumed in Theorem 2.1. Quasi-irreducibility of μ does not imply quasi-irreducibility of ∧²μ; the SO(4) example is enough to show that. Without the missing estimate, the inequality ∫ e^{αX} dμⁿ < 1 has no base, and Theorem 6.1 along with the Hölder estimate lose their key contraction assumption. This is not cosmetic.\n\nSecond, the applications have smaller but real gaps. Proposition 9.6 is stated without proof. Proposition 9.5 claims O(λ^{-p}), which is stronger than what Theorem 2.1 supplies: the Hölder exponent in the proof is at most min(α/p, 1/3) with α < p/2, so the W_p distance bound ≲ β^p only gives error O(β^{p·h}) with h < 1, not O(β^p). The exponent needs correction or a separate argument.\n\nIf Proposition 6.9 can be repaired, either by adding exterior-square quasi-irreducibility to the hypotheses or by finding another contraction argument, this becomes a strong paper. As written, the central theorem is not fully supported. The authors deserve a serious referee, but the referee should focus on Section 6 first. I would not cite it until the gap is fixed; I would bring it to a reading group to discuss whether the repair is straightforward.","headline":"Real theorem, serious gap: the Hölder continuity claim is probably true, but Proposition 6.9's contraction estimate is not justified by the stated hypotheses.","tokens_in":35540,"tokens_out":3962,"would_cite":false,"duration_ms":41616,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["37H15","37A30","60B20"],"pacs":[],"model":"deepseek-v4-flash","headline":"The paper proves that the top Lyapunov exponent of a random matrix product is H\\\"older continuous in the Wasserstein metric around any measure that is quasi-irreducible and has a spectral gap.","keywords":["Lyapunov exponents","H\\\"older continuity","random matrix products","non-invertible cocycles","Wasserstein distance","quasi-irreducibility","spectral gap","Schr\\\"odinger cocycles"],"falsifier":"Compute $\\kappa_\\alpha(\\mu^n)$ for a concrete quasi-irreducible measure with $L_1>L_2$, for instance a finitely supported mixture of shear and rotation matrices in $\\mathrm{Mat}_2(\\mathbb{R})$; Proposition 6.9 predicts it is eventually $<1$ for some $\\alpha>0$. Numerically observing it stuck at $1$ for all $\\alpha$ would break the spectral step of the proof.","tokens_in":34434,"feed_emoji":"🎲","tokens_out":7330,"duration_ms":66795,"temperature":0.7,"pith_summary":"This paper proves a quantitative stability statement for the top Lyapunov exponent of random matrix products. For any probability measure on all $m \\times m$ matrices (not just invertible ones, and not necessarily compactly supported) with finite exponential moments, the paper shows that if the generated semigroup is quasi-irreducible and the top two Lyapunov exponents are separated, then small perturbations of the matrix distribution in the Wasserstein metric move the top exponent by at most a power of the perturbation size. The result matters because realistic models, including Schr\\\"odinger operators with unbounded random potentials, fall outside the invertible, compactly supported regime where such regularity was previously known.","feed_headline":"Lyapunov exponents proved H\\\"older-stable for non-invertible cocycles","feed_subtitle":"Extends classical results to non-compact, singular matrices, covering Schr\\\"odinger cocycles with unbounded potentials.","key_machinery":"The load-bearing object is the average contraction coefficient $\\kappa_\\alpha(\\mu)$, the $\\mu$-average of the $\\alpha$-H\\\"older norm of the projective action $\\hat g$ on $\\mathbb{P}(\\mathbb{R}^m)$. Proposition 6.9 shows that under quasi-irreducibility and a spectral gap, some convolution power satisfies $\\kappa_\\alpha(\\mu^n) \\le \\sigma_0 < 1$; this gives uniform strong mixing of the Markov operator $Q_\\mu$ on H\\\"older observables (Theorem 6.1) and, after a Wasserstein perturbation estimate for $Q_\\mu$ (Lemma 7.3), the H\\\"older modulus for $L_1$. Singularities of non-invertible matrices are handled by truncating the log-norm observable and inducing the transition kernel on a compact set away from the singular locus.","core_discovery":"On the paper's own terms: given a measure $\\mu$ with finite exponential moments of order $p$, quasi-irreducibility, and a spectral gap $L_1(\\mu) > L_2(\\mu)$, the map $\\nu \\mapsto L_1(\\nu)$ is H\\\"older continuous on a small $W_p$-neighborhood of $\\mu$ (Theorem 2.1). The proof supplies the same regularity for all higher Lyapunov exponents via exterior powers (Corollary 2.2), for locally constant $\\mathrm{SL}_m(\\mathbb{R})$ cocycles over Bernoulli shifts in the $L^1$ distance (Corollary 2.4), and yields stable exponential large-deviation estimates (Theorem 2.2). The closing section applies these results to Schr\\\"odinger cocycles whose potentials are sampled from Frostman measures, obtaining an asymptotic expansion of the Lyapunov exponent as the coupling grows.","pith_inferences":["Beyond the paper, the contraction coefficient $\\kappa_\\alpha$ could serve as a quantitative measure of randomness for non-invertible cocycles, giving explicit H\\\"older exponents for specific noise models.","Beyond the paper, the truncation-and-induction method may extend to constant-rank random cocycles, where the projective action is genuinely discontinuous and current results only give continuity.","Beyond the paper, the stable large-deviation bounds suggest a Berry-Esseen-type estimate for the log norm in the non-invertible regime, since the exponential rates now hold uniformly in a Wasserstein neighborhood.","Beyond the paper, the asymptotic separation $q\\log\\lambda$ plus a measure-dependent correction could be used to infer the Frostman dimension of the potential distribution from the Lyapunov exponent's subleading term."],"forward_implications":["All Lyapunov exponents, not just the top, are locally H\\\"older continuous as functions of the matrix-distribution measure (Corollary 2.2).","For locally constant $\\mathrm{SL}_m(\\mathbb{R})$ cocycles over Bernoulli shifts, the top exponent is H\\\"older continuous with respect to the $L^1$ distance between cocycle maps (Corollary 2.4).","Exponential large-deviation bounds for the log norm hold uniformly for all measures in a small Wasserstein neighborhood, with rate $c(\\varepsilon) \\sim \\varepsilon^2/\\log(1/\\varepsilon)$ (Theorem 2.2).","For Schr\\\"odinger cocycles with Frostman-distributed unbounded potentials, the Lyapunov exponent is locally H\\\"older jointly in the potential distribution and the energy, and satisfies $L_1(\\mu_{\\lambda,E,q}) = q\\log\\lambda + L_1(\\tilde\\mu_{0,E,q}) + O(\\lambda^{-p})$ as $\\lambda \\to \\infty$ (Propositions 9.1 and 9.5).","The sum of the first $m$ Lyapunov exponents of symplectic Schr\\\"odinger cocycles admits the asymptotic $m \\log \\lambda + \\int \\log|\\det(s-E)|\\, d\\mu(s) + O(\\lambda^{-p'})$ (Proposition 9.6)."],"supporting_citations":[{"why":"Supplies the abstract Birkhoff theorem for Markov processes and the characterization of stationary measures used to extend Furstenberg-Kifer theory to singular cocycles.","marker":"[26]"},{"why":"The classical H\\\"older-continuity theorem for invertible, compactly supported random matrices that this paper extends to non-invertible and non-compact cases.","marker":"[32]"},{"why":"Origin of the spectral method for Markov chains that underlies the strong-mixing argument.","marker":"[29]"},{"why":"Adapted the spectral method to random linear cocycles, providing the framework for the Markov-operator analysis.","marker":"[12]"},{"why":"Source of the projective-action inequalities and Oseledets-decomposition facts used in Lemmas 6.2, 7.1 and 7.2.","marker":"[19]"},{"why":"Simplified proof of Le Page's result whose structure, uniform convergence of expected logarithmic growth plus contraction, is followed here.","marker":"[4]"}],"fun_headline_variants":["Hölder continuity for Lyapunov exponents without invertibility","Lyapunov exponents Hölder-stable for non-compact cocycles","Hölder regularity for Lyapunov exponents of singular cocycles","Lyapunov exponents Hölder continuous for non-invertible cocycles"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The argument rests on the existence of a power of the measure whose expected projective contraction is uniformly below one; if no such contracting power exists, the uniform mixing and the H\\\"older estimate collapse.","fun_headline_variants_meta":{"raw":{"variants":["Hölder continuity for Lyapunov exponents without invertibility","Lyapunov exponents Hölder-stable for non-compact cocycles","Hölder regularity for Lyapunov exponents of singular cocycles","Lyapunov exponents Hölder continuous for non-invertible cocycles"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001195,"raw_usage":{"total_tokens":4874,"prompt_tokens":838,"completion_tokens":4036,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":454,"completion_tokens_details":{"reasoning_tokens":3957}},"tokens_in":454,"tokens_out":4036,"duration_ms":23432,"temperature":1.0,"reasoning_tokens":3957,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-07T10:47:13.619133+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Compute $\\kappa_\\alpha(\\mu^n)$ for a concrete quasi-irreducible measure with $L_1>L_2$, for instance a finitely supported mixture of shear and rotation matrices in $\\mathrm{Mat}_2(\\mathbb{R})$; Proposition 6.9 predicts it is eventually $<1$ for some $\\alpha>0$. Numerically observing it stuck at $1$ for all $\\alpha$ would break the spectral step of the proof.","supporting_citations":[{"cited_title":"Furstenberg and Yu","cited_arxiv_id":null,"evidence_quote":"Supplies the abstract Birkhoff theorem for Markov processes and the characterization of stationary measures used to extend Furstenberg-Kifer theory to singular cocycles."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"The classical H\\\"older-continuity theorem for invertible, compactly supported random matrices that this paper extends to non-invertible and non-compact cases."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Origin of the spectral method for Markov chains that underlies the strong-mixing argument."},{"cited_title":"Theory Related Fields 78 (1988), no","cited_arxiv_id":null,"evidence_quote":"Adapted the spectral method to random linear cocycles, providing the framework for the Markov-operator analysis."},{"cited_title":"3, Atlantis Press, 2016","cited_arxiv_id":null,"evidence_quote":"Source of the projective-action inequalities and Oseledets-decomposition facts used in Lemmas 6.2, 7.1 and 7.2."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Simplified proof of Le Page's result whose structure, uniform convergence of expected logarithmic growth plus contraction, is followed here."}],"review_version":1}