{"id":"af48794d-1c89-4c92-be2a-5d8833fb439a","arxiv_id":"2507.10707","paper_version":2,"verdict":"ACCEPT","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"low","formal_verification":"none","parameter_count":0,"one_line_summary":"For disordered pinning models with non-degenerate disorder, conditioning the contact density between 0 and 1 yields largest gaps of order log n, so the pure-model big jump phenomenon disappears.","lead":"This paper proves that random disorder destroys the classic big jump mechanism in pinning models: conditioning the number of contacts never forces a single macroscopic gap, and the largest gap stays logarithmic in the system size. The proof uses a new quenched local central limit theorem and a smoothing inequality that extends known disorder smoothing results to much weaker disorder assumptions.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Internally coherent, but the O(log n) claim rests on unverified companion-preprint results in [34]; conditional acceptance is appropriate.","rationale":"I read the main proof carefully and found no manifest internal error: the change-of-measure argument, the characteristic-function bound in Lemma 3.5, the Fourier-inversion local CLT, and the smoothing inequality are coherent, and the paper gives independent supporting results (Theorem 1.5, soft conditioning, circular-DNA application). The central claim is therefore not implausible or post-hoc. The load-bearing risk is exactly the one the Reader flagged: the paper imports the localized-phase machinery of the companion preprint [34], and the specific estimates used here—two-replica exponential decay, positivity of μ, and the quenched CLT—are not reproved. Because Lemma 3.7 and Lemma 3.6(ii) sit directly beneath Theorem 1.7, a hidden error there would change the main conclusion. This does not justify rejection, but it does justify making acceptance conditional on independent verification of those companion statements, or on the authors supplying a self-contained version of the needed corollaries. The paper's own acknowledged limitation near contact density 1 does not affect the stated closed R⊂(0,1) theorems.","tokens_in":35527,"tokens_out":29346,"duration_ms":347564,"concrete_test":"Check the original [34, Lemma 3.2] against the form used here: verify that the exponential two-replica no-overlap bound E[sup_{h∈[h_s,∞)} E⊗²_{j,h}(∏_{k=1}^{j-1}(1-X_kX'_k))] ≤ G e^{-γj} holds for the unbounded interval [h_s,∞), or supply the monotonicity argument reducing the supremum to a compact set. Independently re-derive [34, Prop 1.7/Cor 2.5] showing μ(h)>0 and E[P_{n,h,·}[T_1=n]] ≤ C e^{-μ(h)n} under Assumptions 1.1–1.2; if either step requires extra hypotheses such as bounded disorder, Lemma 3.7 and Theorem 1.7 need revision.","verdict_should_be":"CONDITIONAL","load_bearing_attack":"Theorem 1.7 is proved by combining the change of measure to h_{n,l,ω}, the quenched local CLT (Theorem 1.9), and the gap estimate Lemma 3.7. The weakest load-bearing input is external: Lemma 3.7 needs [34, Prop 1.7/Cor 2.5] for the alternative free energy μ(h)>0 and the annealed return-probability decay that makes Λ_s := Σ_j e^{μ(h_s)j/2} P_{j,h_s,·}[T_1=j] integrable; Theorem 1.9 needs [34, Lemma 3.2] for the exponential two-replica no-overlap bound in Lemma 3.6(ii), and Theorem 3.4 for the quenched CLT. None of these is reproved or machine-checked here, and [34] is by the same authors. If μ(h)>0 fails for some disorder satisfying Assumptions 1.1–1.2, or if the two-replica decay constant is not uniform on the interval [h_s,∞) as used in Lemma 3.6(ii), then the polynomial decay in Lemma 3.7 and the local CLT would not follow, and with them the O(log n) bound. This is a verification risk at the exact point where the central claim is most exposed, not an internal inconsistency I can exhibit.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies disordered pinning models with i.i.d. charges satisfying a subexponential moment condition. Its main result (Theorem 1.7) is that, for P-a.e. disorder, conditioning the contact count to lie in nR for a closed R⊂(0,1) forces the maximal gap between contacts to be O(log n) with high probability, uniformly in the conditioning level; this strengthens previous o(n) bounds. The proof combines a new quenched local CLT for the contact number (Theorem 1.9), a smoothing inequality f(h_c+δ)≤cδ² (Theorem 1.5), and a change of measure to a canonical parameter h_{n,l,ω}; Theorem 1.8 states a mesoscopic homogeneity result. Section 4 treats soft conditioning and circular-DNA models. The paper is carefully written and the main derivations are explicit, but several localized-phase inputs are imported from the companion preprint [34].","tokens_in":35727,"tokens_out":19138,"duration_ms":214323,"significance":"If the results hold, this is a substantial advance in disordered pinning: it shows the big-jump regime is completely washed out under minimal integrability, not merely at the macroscopic o(n) scale, and it provides a local CLT and a smoothing inequality that are of independent interest. The proof is parameter-free in the sense that no fitted constants drive the conclusion; the O(log n) bound is derived from explicit estimates. I note that the central claim rests on the companion paper [34] (Sections 3.1 and Lemma 3.7), so the verification risk flagged in the stress-test note is real; however, the dependency is explicit and the imported statements are listed, and I do not see an internal inconsistency.","major_comments":[],"minor_comments":[{"comment":"Please add a sentence on the status of the companion paper [34] and specify exactly which results are used in the form stated here; the central theorems inherit the correctness of [34], so the reader should know whether it is published, accepted, or available only as a preprint.","section":"Section 3.1"},{"comment":"In the upper-tail Chernoff bound there is a sign typo: the display \"e^{−ρ(h)ζn+ϵζn/2}\" should read \"e^{−ρ(h)ζn−ϵζn/2}\" to match the subsequent line and the final estimate (3.20).","section":"Proof of Lemma 3.9"},{"comment":"The abstract states the O(log n) result without the restriction to closed subsets of (0,1); please make this restriction explicit in the abstract so that it matches Theorem 1.7 and the limitation discussion in Section 1.6.3.","section":"Abstract and Section 1.5"},{"comment":"The derivation of Theorem 1.8 from Theorem 3.8 is only a one-sentence reduction; please spell out the change-of-measure argument or give a precise reference to the analogous steps in the proof of Theorem 1.7, since Theorem 1.8 is a headline result.","section":"Section 3.4"}],"recommendation":"minor_revision","confidential_remarks":"The main theorems depend on the companion paper [34], which is by the same authors and is not reproved here. I recommend that the editor ensure the referees have access to [34] and that its results are independently verified before final acceptance; this is a verification risk rather than an internal flaw, but it should be resolved before publication."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Bottom line: this is a real strengthening of the known o(n) bound to O(log n) for the largest contact gap under contact-number conditioning, and it comes with a new quenched local CLT and a generalized smoothing inequality. The math is dense but coherent, and the central claim holds up as far as I can see. The soft spot is exactly where the stress-test note points: the proof leans on the companion preprint [34] for the localized-phase machinery (alternative free energy positivity, two-replica decorrelation, quenched CLT), and that preprint is by the same authors and not reproved here. I do not see a circular derivation—[34] is cited as established, not derived from this paper's results—but the O(log n) bound is load-bearing on results I have not independently verified. That is a verification risk, not an inconsistency I can exhibit.\n\nWhat is new: Theorem 1.7 improves the o(n) bound in [29] to O(log n) under weaker assumptions than bounded disorder, and Theorem 1.8 gives mesoscopic density uniformity down to scale ζ_n log n. Theorem 1.9 is a quenched local CLT with an explicit characteristic-function proof; Lemma 3.6 gives the needed mean/variance control. Theorem 1.5 extends the smoothing inequality to disorder with only exponential moments, a real generalization of [11,32]. The proof of the pure-model big-jump proposition in Appendix A is a clean local-CLT/tilt argument. No fitted parameters, no post-hoc selection.\n\nSoft spots, in proportion: (1) The reliance on [34] is heavy—Lemma 3.6(ii), Theorem 1.9, and Lemma 3.7 all import results from it. The paper is honest about this, but a referee should read [34] before signing off on Theorem 1.7. (2) The main theorem only covers closed subsets of (0,1); the paper itself notes it cannot handle densities approaching 1 without boundedness. That is a stated limitation, not a flaw. (3) Assumption 1.2 still requires exponential moments, so \"minimal integrability\" means minimal relative to prior work, not absolute.\n\nWho this is for: specialists in disordered pinning and disorder smoothing. It deserves a serious referee, and the companion preprint should be vetted in parallel. My recommendation: send it to review, with the understanding that acceptance hinges on [34] holding up.","headline":"Genuine O(log n) strengthening of the no-big-jump result, but acceptance should be conditional on the companion preprint [34] checking out.","tokens_in":36314,"tokens_out":2185,"would_cite":true,"duration_ms":24597,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["60K37","82B44","60K35","60F05"],"pacs":[],"model":"deepseek-v4-flash","headline":"Disordered pinning models with non-degenerate random charges have no big-jump regime: conditioning the contact number to any density in $(0,1)$ leaves the largest gap at most logarithmic in system size.","keywords":["disordered pinning models","big jump phenomenon","disorder smoothing","contact number constraint","quenched local central limit theorem","large deviations","renewal processes","contact density"],"falsifier":"Run a simulation of a disordered pinning model with i.i.d. charges of small non-zero variance and a heavy-tailed renewal law, condition on $L_n=l$ with $l/n$ fixed in $(0,1)$, and estimate $P_{n,h,\\omega}(M_n>c\\log n\\mid L_n=l)$ as $n$ grows; if for some fixed $c$ this probability fails to tend to zero, Theorem 1.7 is false. A cheaper internal check is to compute the two-replica overlap sum in Lemma 3.2: if $\\mathbb E[\\sup_{h\\in H}\\mathbb E^{\\otimes 2}_{n,h,\\omega}[\\prod_{k=1}^{n-1}(1-X_kX'_k)]]$ does not decay exponentially in $n$, the decorrelation input fails and the proof collapses.","tokens_in":35277,"feed_emoji":"📉","tokens_out":8975,"duration_ms":93878,"temperature":0.7,"pith_summary":"In a disordered pinning model, renewal epochs are contacts that pay an energy $h+\\omega_a$, with the charges $\\omega_a$ independent and identically distributed. In the pure (no-disorder) model with a first-order localization transition, conditioning the contact density to lie below the localized minimum forces one macroscopic gap between contacts: the big jump phenomenon. This paper proves that adding disorder with non-zero variance destroys that phenomenon: for typical disorder, whenever the contact number is conditioned to a density in a closed subinterval of $(0,1)$, the largest gap $M_n$ between contacts is at most $O(\\log n)$, with probability tending to $1$ uniformly in the conditioning level. The mechanism is that disorder smooths the transition, making the critical contact density $\\rho_c$ equal to zero, and a new quenched local central limit theorem controls the probability of each contact number. This matters because the big jump is the one-dimensional analogue of phase-separation droplets, so the result says quenched disorder erases that droplet mechanism entirely.","feed_headline":"Disorder shrinks the largest contact gap to O(log n)","feed_subtitle":"Disorder smooths the transition, so conditioned pinning models keep logarithmic contact gaps instead of a big jump.","key_machinery":"The central object is the maximal gap $M_n=\\max\\{T_1,\\dots,T_{L_n}\\}$ under the sharp conditioning $L_n=l$. The argument changes measure to the unique $h_{n,l,\\omega}$ with $E_{n,h_{n,l,\\omega},\\omega}[L_n]=l$, so the conditioned probability factors into a numerator $P_{n,h,\\omega}(M_n>c\\log n)$ and a denominator $P_{n,h,\\omega}(L_n=l)$. The numerator is controlled by Lemma 3.7, which uses the alternative free energy $\\mu(h)$, the exponential decay rate of the probability of a renewal at time $n$, to show $P_{n,h,\\omega}(M_n>c\\log n)$ decays faster than any power of $n$. The denominator is controlled by the quenched local central limit theorem of Theorem 1.9, proved by bounding the characteristic function of $L_n$ through a sparse observable $J_{n,h,\\omega}$ built on even sites between odd contacts and using exponential two-replica decorrelation. The disorder-smoothing Theorem 1.5 enters by guaranteeing $\\rho_c=0$, so the inverse map of the contact density covers all of $(0,1)$ and the tilted parameter $h_{n,l,\\omega}$ stays in a compact localized set uniformly in $l$.","core_discovery":"The paper's central claim is Theorem 1.7: under Assumptions 1.1 and 1.2 with $\\int \\omega_0^2\\,d\\mathbb P>0$, for $\\mathbb P$-a.e. disorder realization $\\omega$, for every $h\\in\\mathbb R$ and every closed $R\\subset(0,1)$ there is a constant $c>0$ independent of $\\omega$ such that $\\lim_{n\\to\\infty}\\sup_{l\\in nR\\cap\\mathbb N}P_{n,h,\\omega}(M_n>c\\log n\\mid L_n=l)=0$. In words, conditioning the contact density to be any fixed fraction of the volume leaves the configuration localized at the microscopic scale: no single inter-contact gap can be macroscopic. Theorem 1.8 refines this to mesoscopic windows, showing the local contact density is uniformly close to the conditioned density on every interval longer than $\\zeta_n\\log n$ with $\\zeta_n\\to\\infty$. The structural reason is Corollary 1.6: disorder forces the rate function $I_h$ for the contact density to be strictly convex on $(0,1)$, eliminating the affine stretch that in the pure model is realized by exactly one large gap. The proof also establishes a quenched local central limit theorem for $L_n$ in the localized phase.","pith_inferences":["A natural test of sharpness is to simulate the conditional law for bounded i.i.d. charges and see whether the gap distribution is actually $\\Theta(\\log n)$, not $o(\\log n)$; the paper proves an upper bound and does not claim tightness.","The local-CLT route is likely transferable to contact densities approaching $1$, where there is almost no room for a large gap; the paper notes that bounded disorder should suffice to push the argument into that range.","The disorder-smoothing result strengthens the analogy with random-field Ising systems: a random field that rounds a first-order transition should destroy macroscopic phase-separation droplets in any model whose free energy is sufficiently regular, an analogy the paper draws but does not prove.","For the generalized Poland–Scheraga model built on two-dimensional renewals, it remains open whether disorder washes out the big jump; a numerical check of the conditional gap profile in that model would be a direct extension of Theorem 1.7."],"forward_implications":["For typical disorder, every closed subinterval conditioning on contact density gives uniform logarithmic gaps: the pure model's big-jump mechanism has no disordered analogue at any $h$.","The largest gap in the conditioned model has the same $O(\\log n)$ scale as the largest gap in the unconditioned localized phase, so conditioning does not create a new phase.","The quenched local central limit theorem supplies a uniform Gaussian approximation of the contact-number mass function in the localized phase, a tool usable beyond conditioning.","Soft conditioning inherits the bound: $P_{n,h,\\omega}(M_n>c\\log n\\mid L_n\\ge rn)\\to0$, and for $h>h_c$ also under $L_n\\le rn$; circular-DNA models with nonlocal potentials acquire $M_n\\le c\\log n$ when the contact density sits in a localized window.","The rate function for the contact density loses its affine stretch and becomes strictly convex and Gevrey-3 on $(0,1)$, merging the two large-deviation regimes of the pure model into one."],"supporting_citations":[{"why":"Supplies the localized-phase estimates—strict convexity and Gevrey-3 regularity of the free energy, exponential two-replica decorrelation, concentration of the free energy, and the quenched CLT—that the main theorems import.","marker":"[34]"},{"why":"Provides the smoothing inequality for tilted disorder used in the proof of Theorem 1.5.","marker":"[11]"},{"why":"Establishes the bounded-disorder smoothing bound that Theorem 1.5 generalizes to minimal integrability conditions.","marker":"[32]"},{"why":"Proves the earlier $o(n)$ no-macroscopic-big-jump result that Theorem 1.7 strengthens to $O(\\log n)$.","marker":"[29]"},{"why":"Supplies the quenched large deviation principle for the contact density used for Proposition 1.2 and soft conditioning.","marker":"[39]"},{"why":"Characterizes the localized phase and the logarithmic order of the largest gap, the scale Theorem 1.7 recovers under conditioning.","marker":"[31]"},{"why":"Inspires the characteristic-function strategy for the local CLT in Theorem 1.9.","marker":"[10]"},{"why":"Provides the free-energy framework and explicit solvability of the pure model used in Proposition 1.4.","marker":"[27]"}],"fun_headline_variants":["Disorder blocks big jump, forces log gaps","Conditioned pinning stays localized with disorder","Disorder kills macroscopic gaps in pinning models","Disorder forces log gaps under density conditioning"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is that the localized-phase estimates of the companion preprint [34] are correct as stated, because Theorem 1.5, Lemma 3.6, and Lemma 3.7 import them without independent proof, and the main theorems inherit any error there.","fun_headline_variants_meta":{"raw":{"variants":["Disorder blocks big jump, forces log gaps","Conditioned pinning stays localized with disorder","Disorder kills macroscopic gaps in pinning models","Disorder forces log gaps under density conditioning"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000506,"raw_usage":{"total_tokens":2549,"prompt_tokens":1108,"completion_tokens":1441,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":724,"completion_tokens_details":{"reasoning_tokens":1384}},"tokens_in":724,"tokens_out":1441,"duration_ms":11919,"temperature":1.0,"reasoning_tokens":1384,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-06T17:28:28.686452+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Run a simulation of a disordered pinning model with i.i.d. charges of small non-zero variance and a heavy-tailed renewal law, condition on $L_n=l$ with $l/n$ fixed in $(0,1)$, and estimate $P_{n,h,\\omega}(M_n>c\\log n\\mid L_n=l)$ as $n$ grows; if for some fixed $c$ this probability fails to tend to zero, Theorem 1.7 is false. A cheaper internal check is to compute the two-replica overlap sum in Lemma 3.2: if $\\mathbb E[\\sup_{h\\in H}\\mathbb E^{\\otimes 2}_{n,h,\\omega}[\\prod_{k=1}^{n-1}(1-X_kX'_k)]]$ does not decay exponentially in $n$, the decorrelation input fails and the proof collapses.","supporting_citations":[{"cited_title":"Concentration and fluctuation phenomena in the localized phase of the pinning model","cited_arxiv_id":"2405.16991","evidence_quote":"Supplies the localized-phase estimates—strict convexity and Gevrey-3 regularity of the free energy, exponential two-replica decorrelation, concentration of the free energy, and the quenched CLT—that the main theorems import."},{"cited_title":"Caravenna and F","cited_arxiv_id":null,"evidence_quote":"Provides the smoothing inequality for tilted disorder used in the proof of Theorem 1.5."},{"cited_title":"Giacomin and F.L","cited_arxiv_id":null,"evidence_quote":"Establishes the bounded-disorder smoothing bound that Theorem 1.5 generalizes to minimal integrability conditions."},{"cited_title":"Giacomin and B","cited_arxiv_id":null,"evidence_quote":"Proves the earlier $o(n)$ no-macroscopic-big-jump result that Theorem 1.7 strengthens to $O(\\log n)$."},{"cited_title":"den Hollander and M","cited_arxiv_id":null,"evidence_quote":"Supplies the quenched large deviation principle for the contact density used for Proposition 1.2 and soft conditioning."},{"cited_title":"Giacomin and F.L","cited_arxiv_id":null,"evidence_quote":"Characterizes the localized phase and the logarithmic order of the largest gap, the scale Theorem 1.7 recovers under conditioning."},{"cited_title":"Campanino, D","cited_arxiv_id":null,"evidence_quote":"Inspires the characteristic-function strategy for the local CLT in Theorem 1.9."},{"cited_title":"Giacomin, Random Polymer Models (Imperial College Press, World Scientific, 2007)","cited_arxiv_id":null,"evidence_quote":"Provides the free-energy framework and explicit solvability of the pure model used in Proposition 1.4."}],"review_version":1}