{"id":"ed7fa878-86f7-4480-a0ba-20f0c5c1f29f","arxiv_id":"2608.09881","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":8.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Extremal rooted Gibbs-DLR measures in planar directed polymers form a closed, totally ordered, coalescing family and generate a unique canonical Busemann process with L1 continuity.","lead":"Directed polymers in a random planar environment are shown to have a well-ordered family of infinite-volume Gibbs states, with a canonical Busemann process attached to each tilt. The paper proves stability and large-deviation consequences, giving a unified structural picture for KPZ-class planar disordered systems.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Structural theorems rest on Condition 2.2, which is not implied by ergodicity or Condition 2.3; bounded diagonal-constant ergodic weights satisfy Condition 2.3(a)(ii) but violate Condition 2.2, so the 'mild hypothesis' does essential work.","rationale":"The reader's CONDITIONAL verdict is right, and the most load-bearing issue is Condition 2.2. Lemma 3.5 is the first structural result and every later statement about total order, closedness, coalescence, extension to all roots, and the Busemann process builds on it; if Condition 2.2 fails, the entire pointwise structure can fail, as Remark 3.6 shows. The diagonal-constant example proves Condition 2.2 is not redundant with the paper's other standing hypotheses: it is ergodic, bounded, class L, hence satisfies Condition 2.3(a)(ii), while S_{z,j,n}≡0. This is not an internal inconsistency, because the paper explicitly assumes Condition 2.2 and restricts theorems to Ω0; it is a scope limitation in the abstract's first sentence and in the interpretation of 'mild hypothesis.' I also weighed the limitations the authors themselves flag: Theorem 3.44 at β=∞ requires Condition 3.43, a zero-temperature strong-uniqueness input that is not proved, and the limit tilt must be extreme (Remarks 3.45 and 3.48). This independently justifies a CONDITIONAL rather than ACCEPT verdict. I did not find a concrete gap in the arguments conditional on the stated hypotheses: the total order, closedness, and strong uniqueness proofs are coherent, and the L1 continuity argument is sound once Condition 3.43 is granted. Therefore the reader's verdict should stand unchanged.","tokens_in":71156,"tokens_out":21675,"duration_ms":208549,"concrete_test":"Fix a bounded non-degenerate i.i.d. sequence (f_k), e.g. uniform on [0,1], and define the ergodic weight field ω_{(a,b)}=f_{a+b}. For each z and n, S_{z,1,n}=Σ_{i=0}^{n-1}(ω_{z+i e1}-ω_{z-e2+(i+1)e1})=0 almost surely, so P(lim_n S_{z,1,n}=-∞)=0; the same holds for j=2. Verify Condition 2.3(a)(ii): weights are bounded, and for fixed x, n^{-1}Σ_{0≤k≤εn}|ω_{x+k e_j}|→ε E|f_0| then ε→0 gives (2.9). This confirms that an environment satisfying all regularity hypotheses except Condition 2.2 is excluded, so the structural theorems cannot be extended to plain ergodic disorder without strengthening assumptions. A numerical check on finite boxes could additionally exhibit non-fully-supported extremal Gibbs states, matching the failure mechanism of Remark 3.6.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Lemma 3.5 is the load-bearing base of the paper: it classifies non-fully-supported Gibbs-DLR measures as convex combinations of the two coordinate rays, and it feeds the total order (Prop. 4.6), closedness (Prop. 4.18), extension to all roots (Prop. 3.21), the global total order (Thms. 3.24-3.25), and the coalescence criterion (Prop. 3.28), hence also strong existence and uniqueness of the Busemann process. Its proof needs the weight-difference walk S_{z,j,n} to drift to -∞ (Condition 2.2). Ergodicity alone does not give this, and neither does the regularity Condition 2.3. For instance, let (f_k) be bounded, non-degenerate i.i.d. and set ω_{(a,b)}=f_{a+b}. Then S_{z,1,n}=Σ_{i=0}^{n-1}(f_{s+2i}-f_{s+2i})=0 almost surely, so Condition 2.2 fails; yet the environment is ergodic, bounded, and satisfies the class-L condition (2.9), hence Condition 2.3(a)(ii). Remark 3.6 already constructs ergodic environments failing Condition 2.2 and exhibiting trapped geodesics. Thus Condition 2.2 is a substantive scope restriction: the structural theorems and the Busemann construction are not valid under plain ergodic disorder, despite the abstract's phrase 'additional mild hypothesis.' Separately, as the reader notes, Theorem 3.44 is conditional for β=∞ on the unproved Condition 3.43 and on extreme tilts (Remarks 3.45, 3.48), but the foundational concern is Condition 2.2.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"This paper develops a structural theory of rooted Gibbs–DLR measures for the planar directed polymer with nearest-neighbor up-right paths, in a fixed realization of an ergodic random environment. The hypotheses are Condition 2.2 (a weight-difference random walk drifts to −∞), assumed throughout and used to prove that non-fully-supported Gibbs measures are trivial mixtures (Lemma 3.5), and Condition 2.3 (mixing and moment hypotheses of class-L type), used once the limiting free energy and Busemann cocycles enter. The main structural results are: extremal rooted Gibbs measures form a closed and totally ordered set (Theorem 3.8); extremality is characterized by the existence of a coalescing self-coupling (Theorem 3.16); every fully supported extremal measure generates a consistent, coalescing, globally totally ordered family of extremal measures indexed by all lattice sites (Proposition 3.21, Theorems 3.24–3.25, Proposition 3.28); and cocycle equality is equivalent to consistency and coalescence.","tokens_in":71509,"tokens_out":13154,"duration_ms":110891,"significance":"If the results hold, this is a significant advance for the mathematical theory of planar directed polymers. The proofs are fully rigorous, are built from explicit couplings (the tree coupling of Lemma 4.1, Goldstein's maximal-coupling criterion, inverse-CDF sampling via a measurable quantile function, and an adaptation of the Licea–Newman argument), and contain no fitted parameters or post-hoc numerical claims. The closedness of the extremal set (Theorem 3.8) is a genuinely new structural fact for which no a priori reason was known, and the strong existence and strong uniqueness of the Busemann process on the canonical space resolve questions left open in [46] for positive temperature. The joint L1 continuity of the Busemann cocycles is, to my knowledge, the first result of its kind, and the large-deviation corollaries give a novel quantitative bridge between positive and zero temperature. The manuscript is unusually candid: Remarks 3.6, 3.45, 3.48, and 5.4 state limitations, unproved inputs, and open problems explicitly, and the conditional theorems are stated with their hypotheses.","major_comments":[{"comment":"The abstract's phrase 'additional mild hypothesis' materially understates the role of Condition 2.2. The condition is not implied by ergodicity or by Condition 2.3(a): for a bounded, non-degenerate i.i.d. sequence (f_k), set ω_{(a,b)} = f_{a+b}. The environment is ergodic under the Z^2 shifts and satisfies Condition 2.3(a)(ii) (weights are bounded and the class-L estimate (2.9) holds trivially), yet S^1_{z,n} = Σ_{i=0}^{n-1}(f_{s+2i} − f_{s+2i}) = 0 almost surely for every z, so (2.6) fails. Condition 2.2 is load-bearing: the proof of Lemma 3.5 uses the almost-sure divergence of S_{v,2}^N to −∞ to rule out non-trivial coordinate rays, and Lemma 3.5 in turn feeds the total order (Prop. 4.6), closedness (Prop. 4.18), the extension to all roots (Prop. 3.21), the global total order (Thms. 3.24–3.25), and the coalescence criterion (Prop. 3.28), hence the entire Busemann construction. Remark 3.6 already exhibits ergodic environments failing Condition 2.2 with trapped zero-temperature geodesics and conjectures the same phenomenon at positive temperature, so the restriction is known to be substantive, not merely technical. I ask that the abstract and Introduction describe Condition 2.2 as a substantive non-trapping hypothesis, with an example such as the diagonal-constant field above to calibrate its scope, and state explicitly that the structural theorems are not asserted under plain ergodic disorder.","section":"Abstract; Condition 2.2; Lemma 3.5; Remark 3.6"},{"comment":"The headline L1 continuity theorem is not unconditional in its zero-temperature regime, as the abstract's wording suggests. At β = ∞, Theorem 3.44 requires Condition 3.43, which is precisely the unproved statement that zero-temperature shift-covariant recovering L1 cocycles are strongly unique without a finite-energy/coalescence hypothesis; Remark 3.45 concedes that removing it would require different methods planned for future work, with an i.i.d. alternative that only covers the β_n < ∞ cases. Remark 3.48 concedes an additional restriction: the condition (β_∞, h_∞) ∈ H^{ν_∞,ext} leaves a potential gap in the convergence of the Busemann process if cocycles with non-extremal tilts exist. Since Corollaries 3.49–3.53, including the quenched subsequential large deviation principles, inherit these hypotheses, the abstract's claim of a joint L1 continuity theorem and of zero-temperature LDPs overstates what is currently proved. I request that Theorem 3.44 and the abstract explicitly distinguish the unconditional positive-temperature statement from the β = ∞ statement conditional on Condition 3.43, and that the extreme-tilt restriction of Remark 3.48 be reflected in the statement of the theorem or in a prominent remark attached to it.","section":"Theorem 3.44; Condition 3.43; Remarks 3.45 and 3.48; Corollaries 3.49–3.53"}],"minor_comments":[{"comment":"Duplicate word: 'would require either restricting to i.i.d. weights throughout throughout or extending one of our results from [50]'; delete the second 'throughout'.","section":"§1.2"},{"comment":"The phrase 'c` adl` ag functions' has corrupted accents; it should read 'càdlàg functions'.","section":"Remark 3.47"},{"comment":"The statement 'Assume that Condition 2.3 holds' should specify that Condition 2.3(a) is meant for β < ∞ and Condition 2.3(b) for β = ∞, since Condition 2.3 is parametrized by β and the two parts are mutually exclusive alternatives.","section":"Lemma 3.30"},{"comment":"The notation H^{β,ν} is introduced in (3.27) but the superscript order (β before ν) is not used consistently in the following displayed equations; a brief remark or consistent ordering would improve readability.","section":"§3.6, Eq. (3.27)"}],"recommendation":"major_revision","confidential_remarks":"The manuscript is honest in its body about limitations (Remarks 3.6, 3.45, 3.48, 5.4), and I found no internal inconsistency in the conditional theorems I checked; the structural coupling arguments in Sections 4–5 are coherent, and the paper clearly states which ingredients come from external published results ([38], [46], [50]). The two major issues concern the distance between the abstract's claims and the theorem statements: Condition 2.2 is called 'mild' although natural ergodic models such as diagonal-constant bounded weights violate it, and the zero-temperature L1 continuity theorem depends on an unproved Condition 3.43. Both are fixable by restating the scope precisely; neither requires new mathematics for the body of the paper. The self-citation pattern is appropriate given the cited works are the direct ingredients, and I see no novelty-disclosure concerns. The paper fits the journal's scope well and, after the requested revision, would be a strong contribution."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"This is the strongest structural result to date for rooted Gibbs-DLR measures in planar directed polymers. The positive-temperature structure theory is genuinely new: closedness and total order of extremal rooted measures (Thm 3.8), coalescence characterization of extremality (Thm 3.16), and canonical extension of a single extremal measure to a globally consistent, totally ordered family indexed by all lattice sites (Prop 3.21). Building on that, Thm 3.38 upgrades the earlier weak existence of the Busemann process to strong existence and strong uniqueness on the canonical weight space. The proofs are careful and coherent; the tree-coupling and inverse-CDF machinery is well matched to the problem, and Goldstein's coupling theorem is used where it belongs.\n\nThe soft spots are real but disclosed. The first is Condition 2.2. It is load-bearing: the classification of non-fully-supported Gibbs measures (Lemma 3.5), the total order, closedness, and coalescence all rest on it. It is not implied by ergodicity or by Condition 2.3. The stress-test example is a fair hit: bounded diagonal-constant ergodic weights satisfy Condition 2.3(a)(ii) but make the difference walk S_{z,j,n} identically zero, so Condition 2.2 fails. The paper does flag this restriction clearly in Section 2.2.1 and even supplies trapped-geodesic examples in Remark 3.6, but the abstract's phrase \"additional mild hypothesis\" undersells how much it prunes the universe of admissible environments.\n\nThe second soft spot is Theorem 3.44. The L1 continuity claim is not unconditional: for β=∞ it invokes Condition 3.43, an assumed strong strong uniqueness input for zero-temperature cocycles without finite-energy or coalescence hypotheses, and the limit tilt is required to be extreme (Remarks 3.45, 3.48). These limitations are explicitly acknowledged, and the i.i.d. case is unconditional, which is the flagship regime for KPZ-class models. Still, the abstract's opening sentence \"L1 continuity ... joint in inverse temperature, tilt, and environment\" overstates what is proven in general.\n\nWho gets value: anyone working on directed polymers, Busemann functions, or disordered Gibbs-DLR theory. The structural core is solid and will be cited regardless of the continuity caveats. The L1 theorem needs careful reading of conditions before use.\n\nRecommendation: this deserves a serious referee. Send it to a top probability journal, but ask the authors to state the scope of Condition 2.2 honestly in the introduction and to align the abstract's claims with the theorem hypotheses.","headline":"Major structural advance in planar directed polymers; the core theorems look right, but the print's most public claims outrun the actual hypotheses (Condition 2.2 is doing real work; the L1 continuity theorem is conditional at β=∞).","tokens_in":72045,"tokens_out":2704,"would_cite":true,"duration_ms":28259,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["60K35","60K37","82B44","60F10"],"pacs":[],"model":"deepseek-v4-flash","headline":"In planar directed polymers whose disorder satisfies a mild tail-divergence condition, the extremal rooted Gibbs-DLR measures form a closed, totally ordered set, and this order makes the tilt-indexed Busemann process canonical and…","keywords":["directed polymer","Gibbs-DLR measures","Busemann process","planar random environment","coalescence","total order","large deviations","zero-temperature limit"],"falsifier":"Build the positive-temperature analogue of the trapped-ray example described in Remark 3.6: an ergodic environment, as in the inhomogeneous corner growth model of the paper's [28] reference, where a labeled infinite path is trapped on a row or column once it meets a random barrier. If such a fully supported, non-trivial Gibbs-DLR measure exists whose mass is concentrated on trapped rays, then Lemma 3.5's conclusion fails, Condition 2.2 is violated, and the total-order and closedness theorems cannot hold in that environment — the single example would delineate exactly where the structural theory stops. A more direct test of strong uniqueness: in an i.i.d. environment at fixed positive $\\beta$, look for two distinct extremal rooted Gibbs measures with the same tilt cocycle $B^{\\beta,h}$; Theorem 3.32 says they must agree almost surely, so finding any such pair would falsify the uniqueness claim outright.","tokens_in":70936,"feed_emoji":"🧵","tokens_out":17877,"duration_ms":136704,"temperature":0.7,"pith_summary":"The paper studies the infinite-volume Gibbs states ('Gibbs-DLR measures') of the directed polymer on the square lattice $\\mathbb{Z}^2$ with ergodic random weights, and asks how those states are organized by the disorder. Its central claim is that, under a mild hypothesis on the environment — that the one-dimensional random walks built from differences of weights on adjacent rows and columns drift to $-\\infty$ almost surely — the extremal (irreducible) rooted Gibbs states form a closed and totally ordered collection under stochastic dominance. From that order the paper derives a complete structural picture: a fully supported state is extremal exactly when the path measures it induces admit a coalescing coupling; every extremal state canonically generates a consistent, totally ordered family of extremal states rooted at every lattice site; and the associated Busemann process exists canonically on the weight space and is strongly unique. This yields an $L^1$ continuity theorem for the Busemann cocycles jointly in inverse temperature, tilt, and environment, with corollaries giving stability in probability of the generated Gibbs measures under bounded weight perturbations and, in the zero-temperature limit, quenched large deviation principles whose rate functions vanish precisely on infinite geodesics. Why it matters: the same planar total order that organizes semi-infinite geodesics in zero-temperature growth models also organizes the entire positive-temperature Gibbs state space, making the Busemann process a genuine function of the disorder rather than a construction dependent on an extended probability space.","feed_headline":"One total order organizes all extremal polymer Gibbs states","feed_subtitle":"Extreme states coalesce; the Busemann cocycle is unique; zero-temperature limits obey large deviation principles.","key_machinery":"The central objects are rooted Gibbs-DLR measures: probability laws on semi-infinite up-right paths from a site $x$ that satisfy the polymer Gibbs property $\\Pi(x_{k:n})=Q^{\\beta,\\omega}_{x,x_n}(x_{k:n})\\Pi(x_n)$ for every finite segment, together with their extreme points $\\operatorname{ext}\\mathrm{DLR}_x^{\\beta,\\omega}$. The load-bearing mechanism is the closed total order of Theorem 3.8 on the event $\\Omega_0$: it rests on Lemma 3.5 (any non-fully-supported Gibbs measure is a mixture of the two trivial coordinate rays), on a maximal-coupling criterion (two path measures agree on the tail $\\sigma$-algebra if and only if they admit a coalescing coupling), and on a random directed spanning tree whose root-to-point paths realize all point-to-point polymer measures simultaneously — this tree lets any coupling of Gibbs measures be approximated by concatenation couplings that preserve the Gibbs property after conditioning on joint tail events. Closure plus total order makes the extreme state space a random compact totally ordered space, so a shift-covariant inverse-CDF sampling scheme can select canonical representatives; coalescence forces the selection to depend only on the weights, giving strong existence, and the total order gives strong uniqueness. The Busemann cocycle $A(y,z)=\\lim_n \\beta^{-1}\\log(Z_{y,X_n}/Z_{z,X_n})$ is the additive field that carries states between roots: its recovery condition $\\sum_{i=1}^2 e^{\\beta\\omega_y-\\beta A(y,y+e_i)}=1$ defines the transition probabilities and, at zero temperature, the large-deviation rate functions.","core_discovery":"On the full-probability event $\\Omega_0$ where every row- and column-difference random walk $S_{z,j,n}$ diverges to $-\\infty$ (Condition 2.2), the paper proves that for each root $x$ and inverse temperature $\\beta \\in (0,\\infty)$ the set $\\operatorname{ext}\\mathrm{DLR}_x^{\\beta,\\omega}$ of extremal rooted Gibbs-DLR measures is closed in the weak topology and totally ordered by stochastic monotonicity (Theorem 3.8), with the two trivial coordinate-ray measures as the only non-fully-supported extremes. Extremality is characterized by coalescence: a fully supported rooted Gibbs measure is extreme if and only if the family of descendant measures it induces admits a coalescing coupling (Theorem 3.16). Each fully supported extreme state generates, through a backward-martingale limit, a globally consistent and totally ordered family of extreme states rooted at every lattice site, all sharing one $\\beta$-recovering Busemann cocycle $A^\\Pi$ (Proposition 3.21). Using the closed total order to view the extreme states as a random compact totally ordered space, the authors construct a shift-covariant quantile selection and prove strong existence and strong uniqueness: the tilt-indexed Busemann process $(B^{\\beta,h}_{\\pm})$ defined by monotone limits is a shift-covariant, $\\beta$-recovering $L^1$ cocycle on the canonical weight space, and any other process with the same tilts agrees almost surely (Theorem 3.38).","pith_inferences":["My inference: the coalescence characterization of extremality and the canonical extension of a rooted state to the whole lattice do not use planarity, and the paper says so for those steps; the genuinely planar input is the proof that shift-covariantly selected states are extreme, so transferring that one selection argument to higher dimensions would carry the whole program to random walks in rand","My inference: the closed totally ordered state space suggests a positive-temperature analogue of the zero-temperature 'no three geodesics' picture — at most two extremal states per asymptotic direction — and the paper's Remark 3.41 isolates the precise missing input (at most two distinct extreme states from the origin with a given direction); if established, it would imply that the Busemann proces","My inference for computational practice: the $L^1$ continuity of the cocycles in the environment is exactly the stability property a finite-volume sampling scheme needs, since it implies that small bounded perturbations of the disorder change infinite-volume Gibbs probabilities by a small amount in probability rather than amplifying exponentially.","My inference: whether left- or right-isolated extreme states exist is open even in the exactly solvable log-gamma polymer; if they do, Proposition 5.3(e) places them inside an explicit countable, shift-covariant set built from dyadic path-interval infima and suprema, giving a concrete search target in models where the full extremal set might be computable."],"forward_implications":["Every probability space that supports the weight field also supports a canonical realization of the tilt-indexed Busemann process, and any two such realizations agree almost surely: the cocycles are bona fide functions of the disorder, not of an extended probability space.","Any shift-covariant $\\beta$-recovering $L^1$ cocycle defined on an extended space is an ergodic mixture of the canonical tilt-indexed cocycles, with the law of its tilt vector as the mixing measure and no mass on discontinuity tilts (Theorem 3.42).","The Busemann cocycles and the extremal Gibbs measures they generate are jointly $L^1$-continuous, and continuous in probability, in inverse temperature, tilt, and environment; in directions where the limit shape is differentiable, bounded perturbations of the weights move the cocycles by $O(\\epsilon)$.","As $\\beta\\to 8$, along deterministic subsequences the positive-temperature rooted Gibbs measures satisfy quenched large deviation principles on path space, with rate functions built from the zero-temperature Busemann cocycle that vanish precisely on the infinite geodesics generated by that cocycle.","Extremality equals coalescence, so the phase structure of the polymer is read off from whether infinite paths eventually merge: two extreme states with the same Busemann cocycle are consistent and coalesce, while distinct ones are strictly stochastically ordered everywhere."],"supporting_citations":[{"why":"Supplies the cocycle characterization of rooted Gibbs measures, the backward martingale (Lemma 3.20), the tree coupling of finite-volume measures, and the weak-existence input for the planar Busemann process that the new theorems extend.","marker":"[46]"},{"why":"Provides the maximal-coupling theorem (tail sigma-algebra equality is equivalent to existence of a coalescing coupling) that yields the characterization of extremality and the family-wide coalescing couplings.","marker":"[36]"},{"why":"Supplies weak existence of generalized Busemann cocycles and Gibbs measures for random walks in random potentials on extended spaces, the input that Proposition 3.33 and Theorem 3.31 build on for strong existence.","marker":"[38]"},{"why":"Provides the template for strong uniqueness without shape hypotheses — sampling by a shift-covariant quantile/inverse-CDF construction — which Section 5 adapts to positive temperature.","marker":"[1]"},{"why":"The authors' zero-temperature analogue: strong existence and uniqueness of the tilt-indexed Busemann process in the corner growth model, whose methods, Theorem B.1, and Theorem 3.4 serve as the beta equals infinity input and proof pattern.","marker":"[50]"},{"why":"Furnishes the lack-of-space coalescence argument, adapted here to show that shift-covariantly selected Gibbs measures are extremal in planar polymers (Lemma 3.35).","marker":"[53]"},{"why":"The weak-strong convergence theorem used in the proof of Theorem 3.44 to bootstrap tightness of the Busemann cocycles to in-probability and L1 convergence.","marker":"[42]"},{"why":"Provides the quenched shape theorem for the limiting free energy (Lemma 3.1) that anchors the cocycle mean/vector convex analysis in Lemma 3.30.","marker":"[44]"}],"fun_headline_variants":["Extremal polymer Gibbs states form one totally ordered set","Coalescence decides which polymer Gibbs measures are extreme","Unique shift-covariant Busemann cocycle for every polymer extreme","Planar directed polymers: extreme states coalesce into a total order","Total order and coalescence pin down extremal polymer Gibbs measures"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is Condition 2.2: for every site, the one-dimensional random walk formed by taking differences of the weights on adjacent rows (and the analogous walk for columns) must converge to $-\\infty$ almost surely. If such a difference walk instead fails to diverge, non-trivial trapped rays can exist — Remark 3.6 describes an ergodic environment of this kind — and then the classification of Gibbs states as trivial mixtures collapses, taking the total order, the closedness, and the canonical Busemann process with it.","fun_headline_variants_meta":{"raw":{"variants":["Extremal polymer Gibbs states form one totally ordered set","Coalescence decides which polymer Gibbs measures are extreme","Unique shift-covariant Busemann cocycle for every polymer extreme","Planar directed polymers: extreme states coalesce into a total order","Total order and coalescence pin down extremal polymer Gibbs measures"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000417,"raw_usage":{"total_tokens":2231,"prompt_tokens":1107,"completion_tokens":1124,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":723,"completion_tokens_details":{"reasoning_tokens":1038}},"tokens_in":723,"tokens_out":1124,"duration_ms":8636,"temperature":1.0,"reasoning_tokens":1038,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-15T14:22:35.834884+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Build the positive-temperature analogue of the trapped-ray example described in Remark 3.6: an ergodic environment, as in the inhomogeneous corner growth model of the paper's [28] reference, where a labeled infinite path is trapped on a row or column once it meets a random barrier. If such a fully supported, non-trivial Gibbs-DLR measure exists whose mass is concentrated on trapped rays, then Lemma 3.5's conclusion fails, Condition 2.2 is violated, and the total-order and closedness theorems cannot hold in that environment — the single example would delineate exactly where the structural theory stops. A more direct test of strong uniqueness: in an i.i.d. environment at fixed positive $\\beta$, look for two distinct extremal rooted Gibbs measures with the same tilt cocycle $B^{\\beta,h}$; Theorem 3.32 says they must agree almost surely, so finding any such pair would falsify the uniqueness claim outright.","supporting_citations":[{"cited_title":"Busemann functions and Gibbs measures in directed polymer models onZ 2.Ann","cited_arxiv_id":null,"evidence_quote":"Supplies the cocycle characterization of rooted Gibbs measures, the backward martingale (Lemma 3.20), the tree coupling of finite-volume measures, and the weak-existence input for the planar Busemann process that the new theorems extend."},{"cited_title":"Maximal coupling.Z","cited_arxiv_id":null,"evidence_quote":"Provides the maximal-coupling theorem (tail sigma-algebra equality is equivalent to existence of a coalescing coupling) that yields the characterization of extremality and the family-wide coalescing couplings."},{"cited_title":"Existence of generalized Busemann functions and Gibbs measures for random walks in random potentials.Trans","cited_arxiv_id":null,"evidence_quote":"Supplies weak existence of generalized Busemann cocycles and Gibbs measures for random walks in random potentials on extended spaces, the input that Proposition 3.33 and Theorem 3.31 build on for strong existence."},{"cited_title":"Strong existence and uniqueness of the tilt-indexed Busemann process in the planar corner growth model","cited_arxiv_id":null,"evidence_quote":"The authors' zero-temperature analogue: strong existence and uniqueness of the tilt-indexed Busemann process in the corner growth model, whose methods, Theorem B.1, and Theorem 3.4 serve as the beta equals infinity input and proof pattern."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Furnishes the lack-of-space coalescence argument, adapted here to show that shift-covariantly selected Gibbs measures are extremal in planar polymers (Lemma 3.35)."},{"cited_title":"Existence of weak solutions for stochastic differential equations with driving semimartingales.Stochastics, 4(4):317–337, 1980/81","cited_arxiv_id":null,"evidence_quote":"The weak-strong convergence theorem used in the proof of Theorem 3.44 to bootstrap tightness of the Busemann cocycles to in-probability and L1 convergence."},{"cited_title":"A shape theorem and a variational formula for the quenched Lyapunov exponent of random walk in a random potential","cited_arxiv_id":null,"evidence_quote":"Provides the quenched shape theorem for the limiting free energy (Lemma 3.1) that anchors the cocycle mean/vector convex analysis in Lemma 3.30."}],"review_version":2}