{"id":"33c5e05e-939c-4f26-8e0c-1b686b5fc319","arxiv_id":"2607.04900","paper_version":1,"verdict":"ACCEPT","confidence":"HIGH","novelty_score":6.5,"correctness_risk":"low","formal_verification":"none","parameter_count":0,"one_line_summary":"Unique P-relative equilibrium states exist for uniformly Hölder potentials on random Anosov systems with one-dimensional stable leaves and fibrewise mixing, and they enjoy quenched exponential decay of correlations.","lead":"The paper proves existence and uniqueness of relative equilibrium states for random Anosov maps under uniform fibre hyperbolicity and a fibrewise mixing condition, together with quenched exponential decay of correlations. This supplies a cone-contraction toolkit for transfer-operator cocycles on hyperbolic random diffeomorphisms.","discovery_kind":"extension","skeptic_critique":{"model":"grok-4.5","headline":"No significant objection identified","rationale":"The paper's central claim is that under the stated geometric hypotheses every uniformly Hölder random potential admits a unique P-relative equilibrium state with quenched exponential decay of correlations (under the stronger tail). The argument proceeds by constructing adapted projective cones, proving Hilbert-metric contraction of the random transfer-operator cocycle after random but a.s. finite mixing times, extracting a spectral decomposition, building the candidate measure υϕ, establishing the weak-Gibbs property, and proving uniqueness via absolute continuity of conditionals on SLY partitions. All steps are written in detail; the only non-standard input is the fibrewise mixing condition H2/H2', which is used precisely where the reader locates it and is verified in the examples. Because that condition is part of the hypothesis package rather than a hidden gap, and because the subsequent analytic estimates close, there is no load-bearing concern that would alter the ACCEPT verdict. The concrete test above simply reconfirms the diameter control on the simplest model; a positive outcome leaves the claims intact.","tokens_in":81439,"tokens_out":538,"duration_ms":5736,"concrete_test":"Independently verify the diameter bound of Lemma 6.5 for the model of Example 3.1 (random positive SL(2,N) matrices): compute the Hilbert diameter of L^N_ω C_ω after a uniform mixing time N = B(δ,ε) and check that it remains finite and independent of ω; if the diameter is infinite for some admissible word the cone-contraction step fails.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The reader's weakest assumption (H2 / H2') is correctly identified as the place where the argument is most delicate, but it is not a soft spot that undermines the central claims. Hypothesis H2 is used only to guarantee that the random return times to the set A of Lemma 6.8 are almost surely infinite and that the projective diameter of the cocycle after those returns remains finite (Lemma 6.5 + Theorem 6.9). The subsequent spectral decomposition, construction of υϕ, weak-Gibbs property, and uniqueness via SLY partitions then follow by standard cone-contraction and absolute-continuity arguments that are fully written out. The three examples (random toral automorphisms, random Anosov diffeomorphisms on T^{2}, constant Anosov) verify that H2' holds in non-vacuous settings, so the hypotheses are not empty. No internal inconsistency or missing estimate appears in the chain from cone contraction to Theorems A and B.","agreement_with_reader":"agree"},"referee_report":{"model":"grok-4.5","summary":"The paper develops thermodynamic formalism for regular random dynamical systems that are uniformly hyperbolic on fibres (via deterministic invariant cone fields) with one-dimensional stable direction and a fibrewise topological mixing condition whose mixing time may depend on the base point (Hypothesis H, strengthened by an exponential tail in H'). For uniformly Hölder random potentials the authors construct adapted projective cones for the random Perron–Frobenius cocycle, prove Hilbert-metric contraction, and obtain a quenched spectral decomposition. From this they construct a unique P-relative equilibrium state (Theorem A) and, under H', quenched exponential decay of correlations with Lp constants for every p<\\infty (Theorem B). The argument proceeds through geometric preliminaries, cone construction, spectral decomposition, weak-Gibbs estimates, and uniqueness via SLY partitions.","tokens_in":81649,"tokens_out":728,"duration_ms":6198,"significance":"If correct, the work fills a genuine gap: thermodynamic formalism for random hyperbolic diffeomorphisms beyond the SRB setting, with quenched (rather than annealed) statistical properties and with mixing times allowed to be random. The cone-contraction approach is a natural random analogue of classical Birkhoff methods and is carried through carefully; the three examples (random toral automorphisms, random Anosov maps on T^{2}, constant Anosov) show that the hypotheses are non-vacuous. The results are of clear interest to the random dynamical systems and thermodynamic formalism communities.","major_comments":[{"comment":"The central chain (adapted cones → Hilbert contraction of the random PF cocycle → quenched spectral decomposition → candidate measure υϕ → weak Gibbs → uniqueness via SLY partitions) is load-bearing and appears complete. Hypothesis (H2)/(H2') is the most delicate assumption, but it is used only to guarantee almost-surely infinite returns to a set of finite projective diameter (Lemmas 6.5–6.8, Theorem 6.9); the subsequent estimates are standard and fully written. No load-bearing gap that would require a major revision was found.","section":null}],"minor_comments":[{"comment":"Notation for the corrected potential is inconsistent: the body works with ϕ̄ = ϕ − ϕJs while the proofs of Theorems A–B in §9 switch to eϕ = ϕ + ϕJs. A single convention stated once would help the reader.","section":null},{"comment":"Several long technical arguments are deferred to Appendices A–B (Propositions 6.3 and 6.5). A short roadmap at the beginning of §6 indicating which estimates are essential for the spectral decomposition would improve readability.","section":null},{"comment":"In Definition 5.7 and the subsequent norm (Definition 5.9) the parameters a, a1, b, c, κ, κ1, ν are introduced gradually; a single summary table or paragraph listing the admissible range of all cone parameters would make the construction easier to track.","section":null},{"comment":"Typographical: occasional missing spaces after punctuation and a few duplicated words (e.g. near the end of the proof of Lemma 5.10) should be cleaned in copy-editing.","section":null}],"recommendation":"accept","confidential_remarks":"The manuscript is long and technical but the argument is carefully organised and the main claims appear sound. I see no reason to request a major rewrite; the paper is ready for acceptance after routine copy-editing."},"author_rebuttal":null,"desk_editor":{"model":"grok-4.5","letter":"This is the paper that finally does thermodynamic formalism for genuine random Anosov diffeomorphisms, not just expanding maps or symbolic systems. Under uniform fibre hyperbolicity (deterministic cones, dim Es = 1) and a fibrewise mixing condition that can have random waiting times, every uniformly Hölder potential has a unique P-relative equilibrium state; with an exponential tail on the mixing times you also get quenched exponential decay with Lp constants. That is Theorems A and B, and the claims hold under the stated hypotheses.\n\nWhat is new is the adaptation of projective cones to the random hyperbolic setting. They build cones from leafwise log-Hölder densities and unstable holonomies, prove the random Perron–Frobenius cocycle contracts the Hilbert metric, extract a quenched spectral decomposition, construct the candidate measure, prove the weak Gibbs property, and get uniqueness via absolute continuity of conditionals on an SLY partition. The geometric preliminaries (stable/unstable manifolds, holonomies, rectangles) are taken from the literature and used cleanly. The three examples (random toral automorphisms, random Anosov on T^{2}, constant Anosov) show the hypotheses are non-empty.\n\nThe softest point is Hypothesis H2 (and its tail version H2'): the first time every local unstable manifold becomes dense has positive probability of being bounded (or exponential tails). That is what guarantees finite projective diameter after random returns. It is not a hidden circularity; it is an explicit mixing assumption, and the rest of the argument is standard cone-contraction once you have it. The one-dimensional stable restriction is also explicit and necessary for their cone construction; they note the dual case works by time reversal. No load-bearing gaps appear in the chain.\n\nThis is for people who work on random hyperbolic systems or transfer-operator methods. The proofs are long but complete and the citation pattern is honest. I would send it to peer review without hesitation; a serious referee will check the cone estimates and the SLY uniqueness argument, but the paper deserves that time. Worth engaging if the subfield is relevant to you.","headline":"Solid, carefully written paper that fills a real gap: unique relative equilibrium states and quenched decay for random Anosov diffeomorphisms via adapted cones.","tokens_in":82254,"tokens_out":520,"would_cite":true,"duration_ms":8937,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["28D20","37D20","37D35","37H05"],"pacs":[],"model":"grok-4.5","headline":"Random Anosov maps with random mixing times admit unique relative equilibrium states and quenched exponential decay of correlations for Hölder potentials.","keywords":["thermodynamic formalism","random dynamical systems","Anosov maps","equilibrium states","projective cones","Hilbert metric","quenched decay of correlations","Perron–Frobenius cocycle"],"falsifier":"Exhibit a random Anosov system satisfying the cone hyperbolicity and one-dimensional stable hypotheses for which the fibrewise mixing time is almost surely infinite, or for which two distinct P-relative equilibrium states exist for some Hölder potential.","tokens_in":82349,"feed_emoji":"🔄","tokens_out":869,"duration_ms":8039,"temperature":0.7,"pith_summary":"The paper builds thermodynamic formalism for random Anosov systems: a base map drives a family of fibre diffeomorphisms that are uniformly hyperbolic, with one-dimensional stable directions, and that mix fibrewise on a time scale that can depend on the base point. For every uniformly Hölder random potential the authors produce a unique invariant measure that maximises relative free energy with respect to the base measure, and they show that this measure has exponential decay of correlations along almost every realisation of the base. The technical engine is a family of projective cones, adapted to stable leaves and unstable holonomies, on which the random transfer-operator cocycle contracts Hilbert metrics; the resulting spectral gap yields both the equilibrium state and the quenched mixing rates. Readers interested in random dynamics, climate-type models, or statistical properties of hyperbolic systems will care because the mixing-time hypothesis is weaker than the uniform mixing used in earlier work, yet still strong enough for uniqueness and exponential decay.","feed_headline":"Unique equilibrium states for random Anosov maps","feed_subtitle":"Fibrewise mixing with random times still yields uniqueness and quenched exponential decay","key_machinery":"Adapted projective cones for the random Perron–Frobenius cocycle, defined via averages on admissible stable leaves, leafwise Hilbert metrics, and unstable-holonomy control; the cocycle strictly contracts these cones, producing a quenched spectral decomposition.","core_discovery":"Under uniform fibre hyperbolicity given by deterministic cones, one-dimensional stable direction, and a fibrewise mixing condition whose first return time has positive probability of being bounded (Hypothesis H), every uniformly Hölder random potential admits a unique P-relative equilibrium state; under the exponential-tail strengthening of that mixing condition the same measure satisfies quenched exponential decay of correlations with constants in every Lp space.","pith_inferences":["The cone construction may extend to random systems whose hyperbolicity is only non-uniform, provided the stable leaves still admit a controlled holonomy.","The same spectral gap should yield quenched large-deviation principles and central-limit theorems for the relative equilibrium measures.","Allowing the stable dimension to be higher than one would require a genuine anisotropic Banach space rather than a projective cone, reopening the question of random anisotropic norms."],"forward_implications":["Uniqueness of relative equilibrium states holds for random compositions of Anosov maps on the torus driven by mixing subshifts of finite type.","Quenched exponential decay of correlations is available for Hölder observables along almost every base orbit, with Lp-integrable constants under the tail hypothesis.","The same cone-contraction method applies, after time reversal, when the unstable direction rather than the stable direction is one-dimensional.","Uniform fibrewise mixing recovers the stronger uniform-in-ω decay constants previously obtained for SRB measures."],"fun_headline_variants":["Unique P-relative equilibria for random Anosov maps","Fibrewise mixing yields unique random Anosov equilibria","Adapted cones prove unique equilibria in random Anosov systems","Quenched decay of correlations for random Anosov potentials","Hilbert contraction gives unique relative equilibria for random maps"],"cache_read_input_tokens":65664,"weakest_assumption_plain":"The fibrewise mixing condition that, with positive probability, every short local unstable manifold becomes dense after a uniformly bounded number of iterates (and the exponential tail on successive mixing times).","fun_headline_variants_meta":{"raw":{"variants":["Unique P-relative equilibria for random Anosov maps","Fibrewise mixing yields unique random Anosov equilibria","Adapted cones prove unique equilibria in random Anosov systems","Quenched decay of correlations for random Anosov potentials","Hilbert contraction gives unique relative equilibria for random maps"]},"model":"grok-4.5","effort":"low","cost_usd":0.003882,"raw_usage":{"total_tokens":1127,"prompt_tokens":624,"num_sources_used":0,"completion_tokens":80,"cost_in_usd_ticks":38820000,"prompt_tokens_details":{"text_tokens":624,"audio_tokens":0,"image_tokens":0,"cached_tokens":256},"completion_tokens_details":{"audio_tokens":0,"reasoning_tokens":423,"accepted_prediction_tokens":0,"rejected_prediction_tokens":0}},"tokens_in":624,"tokens_out":80,"duration_ms":3830,"temperature":1.0,"reasoning_tokens":423,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-07-11T11:42:47.241293+00:00","model_set":{"reader":"grok-4.5"},"falsifier":"Exhibit a random Anosov system satisfying the cone hyperbolicity and one-dimensional stable hypotheses for which the fibrewise mixing time is almost surely infinite, or for which two distinct P-relative equilibrium states exist for some Hölder potential.","supporting_citations":[],"review_version":1}