{"id":"4bf81cbc-6468-433a-b836-088d22f85faf","arxiv_id":"2505.11457","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"high","formal_verification":"none","parameter_count":0,"one_line_summary":"For β < β_c, the probability that a + crossing holds at both times 0 and t under Glauber dynamics converges to 1/4, with a sharp transition at the sensitivity scale ε_n = 1/(n^2 α_n).","lead":"This paper proves that crossing events in the high-temperature Ising model on the triangular lattice become noise sensitive under Glauber dynamics: after any fixed positive time, a left-to-right + crossing at time 0 and at time t become asymptotically independent. The proof uses differential inequalities and requires new finite-energy and spatial-mixing properties for the two-time pair of configurations.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Separation of arms (Prop 6.5) has a sketchy induction with a base case not shown uniform in m; the stated bound via max over all k is not finite, so the uniform quasi-multiplicativity used in Theorem 1.6 lacks a complete proof.","rationale":"The reader's weakest assumption was the identity P(σ∈Cross_n)=1/2 asserted in Definition 1.2. I examined this carefully and found it is correct: for site percolation on the triangular lattice, the matching graph equals the original graph, so the lattice is self-dual; combined with the reflection (k,m)↦(m,k) and global spin flip, which are symmetries of µ_{2n}, this gives exactly one of {+ left-right crossing} and {− top-bottom crossing} and equal probabilities, hence P=1/2 for every β. Thus the reader's concern does not land. However, the paper does have a genuine gap in Proposition 6.5 (separation of arms). The induction step r_{m,n} ≤ C_δ + C''δ r_{m,n/4} requires a base case that is not written down; the stated bound via max over all k≥(4m)∨(1/δ) is mathematically non-finite because four-arm probabilities decay polynomially. The uniformity in m of the base-case constant is not established, yet Prop 6.2 (quasi-multiplicativity), Lemma 6.13, Lemma 6.15, and ultimately Theorem 1.6 all depend on scale-uniform constants. This is a serious but likely repairable omission, matching a conditional verdict with high correctness risk. My recommendation is to keep the reader's CONDITIONAL verdict, but for a different reason than the one identified by the reader.","tokens_in":37356,"tokens_out":53976,"duration_ms":502383,"concrete_test":"Write out the full induction in Proposition 6.5 with an explicit base case: for fixed m and n in [4m, C(m∨δ^{-1})], prove that r_{m,n} is bounded by a constant independent of m, using BXP and the fact that the annulus Λ_{4n}\\Λ_{m-1} has bounded aspect ratio. Check whether the lower bound on µ_{8k}(A^sep_4(m,4k)) for k≈m can be made uniform in m. If the bound diverges as m→∞, the uniform quasi-multiplicativity Proposition 6.2 fails and Theorem 1.6 is unsupported.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The paper's central quantitative claim (Theorem 1.6) rests on the quasi-multiplicativity of dynamical 4-arm probabilities (Prop 6.2), which in turn relies on the separation-of-arms property (Prop 6.5). In the proof of Prop 6.5, the authors derive r_{m,n} ≤ C_δ + C''δ r_{m,n/4} and then claim a direct induction gives r_{m,n} ≤ 2C_δ + max_{k≥(4m)∨(1/δ)} r_{m,k} ≤ 2C_δ + max_k 1/π^sep_{m,4k}(t) ≤ 2C_δ + max_k 1/µ_{8k}(A^sep_4(m,4k))^2, asserting the last maximum is bounded by the box-crossing property (BXP). Taken literally, this is false: for k→∞, µ_{8k}(A^sep_4(m,4k)) is essentially the four-arm probability α_{m,4k}, which decays polynomially, so 1/µ^2 is unbounded. The correct induction must restrict the base case to k in a bounded range where n/4^ℓ is comparable to m∨δ^{-1}; the bound there is uniform in m only if the probability of a separated four-arm event across an annulus of bounded aspect ratio is bounded below by a constant independent of m. This uniformity is not established in the text. If the constant in Prop 6.5 depended on m, the constants in Prop 6.2 would become scale-dependent, and the differential inequalities in Section 6.2 and the proof of Theorem 1.6 would collapse. The issue is concrete and repairable, but as written the proof is incomplete. Note that the reader's identified weakness (P(Cross_n)=1/2 in Definition 1.2) is actually valid: it follows from the matching-graph self-duality of the triangular lattice site percolation together with the reflection (k,m)↦(m,k) combined with spin flip, so that concern does not land.","agreement_with_reader":"disagree"},"referee_report":{"model":"deepseek-v4-flash","summary":"This paper studies the Ising model on the triangular lattice at inverse temperature β < β_c, sampled in a finite rhombus Λ_{2n} with free boundary conditions and evolved by Glauber dynamics. The main results are that the left-to-right crossing event Cross_n is noise sensitive for every fixed time t > 0 (Theorem 1.3), and, more sharply, that P(σ, σ_t ∈ Cross_n) tends to 1/4 when t_n/ε_n → ∞ and to 1/2 when t_n/ε_n → 0, where ε_n = 1/(n^2 α_n) is the inverse of n^2 times the static four-arm probability (Theorem 1.6). The proof follows the non-spectral differential-inequality method of [TV23]: the authors establish finite-energy, spatial mixing, quasi-invariance by translation, and Russo-type differential formulas for the pair (σ, σ_t), then use percolation arguments to prove quasi-multiplicativity for dynamical four-arm probabilities and differential inequalities for π_n(t) and for the crossing probability.","tokens_in":37803,"tokens_out":18772,"duration_ms":182032,"significance":"If the proof gaps identified below are repaired, this is a substantial advance: it brings sharp noise-sensitivity theory beyond product measures via a non-Fourier method and identifies the correct sensitivity length for a dependent planar model at high temperature. The paper's structural contributions—finite-energy and spatial mixing for the law of (σ, σ_t), the dynamical FKG step, and the reduction of sharp noise sensitivity to a four-arm exponent—are reusable and clearly formulated. The results are quantitative, falsifiable statements with no fitted parameters, and the reliance on independent published theorems is appropriate rather than circular. I therefore view the paper as a good candidate for publication after a revision that completes the missing technical proofs.","major_comments":[{"comment":"The final displayed chain in the proof of Proposition 6.5 is not a valid induction. From the inequality r_{m,n} ≤ Cδ + C''δ r_{m,n/4}, the conclusion should be r_{m,n} ≤ 2Cδ + r_{m,k} for k = n/4^L in the base range, not a maximum over all k ≥ (4m)∨(1/δ). The subsequent bound r_{m,k} ≤ 1/π^sep_{m,4k}(t) ≤ 1/µ_{8k}(A^sep_4(m,4k))^2 does not make the maximum finite: for fixed m and k → ∞, A^sep_4(m,4k) is a four-arm event across an annulus of diverging modulus, whose probability is not bounded below uniformly in k, so the reciprocal is unbounded. The cited box-crossing property (BXP) only gives lower bounds for annuli of bounded aspect ratio and cannot justify boundedness over this infinite range. The proof must restrict the base case to k comparable to m (or δ^{-1}) and prove the needed uniform lower bound on π^sep_{m,4k}(t) for such k, independent of m. This gap propagates to Proposition 6.2, Lemma 6.13, and Theorem 1.6.","section":"§6.1.1, proof of Proposition 6.5"},{"comment":"The proof of Lemma 6.15 is not contained in the manuscript: after stating that the multiscale pivotal analysis yields (21), the authors write 'We leave the details to the reader' and give only a two-bullet description. This lemma is an essential ingredient of the first and second properties in §7.1, which in turn feed Proposition 7.1 and the proof of Theorem 1.6. In particular, the claimed bound on −π'_n(t) in terms of ∑ k π_k(t) must be derived in full, because the constants there control the exponential factor in the first property and hence the superquadratic decay. Please supply a complete proof or a precise reference to a published argument that contains this exact statement.","section":"§6.2, Lemma 6.15"},{"comment":"The transition from the well-separated event A^{δ,X}_4(m,n) to the separated event A^sep_4(m,4n) is dismissed with 'We omit the precise details of this construction, which is standard in separation arguments.' Since Lemma 6.9 is one of the three lemmas used to prove Proposition 6.5, and Proposition 6.5 is load-bearing for the paper, the corridor construction should be written out or a precise reference with the exact construction in this two-time dynamical setting should be given. The standard static construction does not automatically transfer, because the events are required simultaneously at times 0 and t and the pair (σ, σ_t) lacks the spatial Markov property.","section":"§6.1.1, proof of Lemma 6.9"},{"comment":"The proof of the exponentially fast spatial mixing statement is only sketched: it cites the FK–Ising coupling and [DT19, Proposition 16] and says 'We leave the details of the proof to the reader.' This statement is used to derive the box-crossing property for β < β_c, which is a central input for Sections 5 and 6. Please either include a complete proof or state the result as a theorem with a precise reference that proves it in the exact form needed here.","section":"Appendix A.2.1, exponentially fast spatial mixing"}],"minor_comments":[{"comment":"The identity µ_{2n}(Cross_n) = 1/2 is correct, but the one-line justification 'by symmetry of the rhombus and of the measure µ_{2n} and by self-duality' is too terse for a claim used in Appendix A.2.1; the argument uses spin-flip symmetry, the reflection swapping the two pairs of sides of the rhombus, and planar duality for site configurations on the triangular lattice.","section":"Definition 1.2"},{"comment":"In the displayed computation, 'where is the two last equalities' should read 'where in the two last equalities', and the abbreviation 'indep.' should be spelled out.","section":"§5.1, proof of Lemma 5.2"},{"comment":"In the statement of the exponentially fast spatial mixing result, 'Let W be the set of vertices at (Euclidean) distance less than k from W' should say 'from V'.","section":"Appendix A.2.1, exponentially fast spatial mixing"},{"comment":"The proof refers to 'the exponent of the 3-arm event is (at least) 2'; please make the sign and terminology consistent with Lemma A.3, which proves the upper bound µ_{2n}(A^+_3(m,n)) ≤ C(m/n)^2.","section":"§6.1.1, proof of Lemma 6.8"},{"comment":"The three 'one can prove' observations in the proof of (1) should either be written out or accompanied by precise references, since the conclusion α_n/α_m ≥ c(m/n)^{2−c} is used in Remark 1.5 and in the proof of Theorem 1.6.","section":"Appendix A.2.3"}],"recommendation":"major_revision","confidential_remarks":null},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Dear —,\n\nThe headline is that the paper's central claim is not supported: the asserted identity P(σ∈Cross_n)=1/2 for β<β_c is almost certainly false, and without it Theorem 1.3 collapses. This is not a minor gap; it is the hinge of the whole argument.\n\nWhat is genuinely good: the idea of transplanting the [TV23] differential-inequality machinery to a dependent spin system is attractive, and the finite-energy/spatial-mixing analysis of the pair (σ,σ_t) is nontrivial and might be reusable. The writing is careful and the authors flag several technical difficulties honestly.\n\nBut the identity is load-bearing. For β<β_c, the Ising model is in the disordered phase: correlations decay exponentially and the probability of a + crossing of a large box decays to zero. The exact value 1/2 would require a self-duality that the off-critical Ising model simply does not have (the model is dual to a different temperature, not to itself). The proof of BXP in Appendix A.2.1 relies on this identity via [KT23]; if it fails, the box-crossing property fails, and the percolation arguments in Section 6 have no foundation. The reader's concern lands. The stress-test note's claim that the identity follows from matching-graph self-duality is, I think, wrong: that argument works for independent site percolation at p=1/2, not for the Ising measure at positive β.\n\nThere is a secondary issue, correctly identified by the stress-test: the induction in Proposition 6.5 has an unbounded max over k that is not finite, so the uniform quasi-multiplicativity is not proved as written. That one looks repairable, but it is another unresolved point.\n\nFor whom would this be useful? If the identity were removed and the theorem corrected (say, to a statement about the noise sensitivity of near-critical percolation observables, or about the FK representation), the methods could be of interest. As it stands, the paper is not ready for a serious referee. My recommendation is to desk reject and ask the authors to re-examine the foundation.\n\nBest,","headline":"The paper's main theorem depends on a false crossing probability identity; as written it cannot be right, though the pair-process machinery has some value.","tokens_in":38319,"tokens_out":18968,"would_cite":false,"duration_ms":182136,"reading_group":"no","serious_thinker":"yes","would_accept_peer_review":false},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["60K35","82B20","82C20"],"pacs":[],"model":"deepseek-v4-flash","headline":"The paper proves that crossing events in the subcritical Ising model are noise sensitive under Glauber dynamics, with joint probability tending to 1/4.","keywords":["noise sensitivity","Ising model","Glauber dynamics","triangular lattice","crossing event","four-arm event","subcritical","differential inequalities"],"falsifier":"Estimate the crossing probability by high-precision Monte Carlo for the free-boundary Ising model on a large double box at a fixed subcritical temperature: if the estimates deviate from one half by an amount that does not vanish as the box grows, the identity behind the box-crossing property fails and the 1/4 limit is not the correct answer. A second check would compare the same probability under plus, minus, and free boundary conditions, since the paper's proof requires the limiting crossing probability to be insensitive to boundary choices.","tokens_in":37163,"feed_emoji":"🔄","tokens_out":8069,"duration_ms":83119,"temperature":0.7,"pith_summary":"At inverse temperature below the critical value, the Ising model on the triangular lattice has exponentially decaying spin correlations, yet its percolation observables are critical: the probability of a left-to-right crossing by plus spins in a rhombus is one half. The paper proves that this crossing event is noise sensitive under Glauber dynamics: for any fixed positive time, the crossing events at time 0 and time t become asymptotically independent, so their joint probability tends to 1/4. It also gives a sharp quantitative version: the amount of noise needed to destroy the crossing information is governed by the four-arm probability, with a characteristic time of order the reciprocal of n squared times that probability. If true, this shows that crossing information in a genuinely interacting high-temperature spin system is fragile under local resampling, extending the noise-sensitivity phenomenon beyond independent Bernoulli percolation.","feed_headline":"Ising crossings erased by Glauber noise","feed_subtitle":"At any fixed time, left-right crossing events at time 0 and time t become independent: joint probability falls to 1/4.","key_machinery":"The load-bearing object is the dynamical four-arm probability, the chance that four alternating-sign paths connect the origin to the boundary at both time 0 and time t under the coupled evolution. Its static counterpart defines the characteristic time. The argument proceeds through three mechanisms: a finite-energy property for the pair, proved by bounding the cost of local modifications through estimates on the random set of updated sites; spatial mixing and quasi-translation invariance for the pair, obtained by intersecting with a decoupling event on which the updated sites form small clusters; and a Russo-type differential inequality showing that the time derivative of the crossing probability is comparable to a sum of pivotal-event probabilities. Together with a quasi-multiplicativity property for the dynamical four-arm probabilities, these ingredients yield superquadratic decay of the four-arm ratio above the sensitivity length, which converts the differential inequality into the sharp 1/4 versus 1/2 threshold.","core_discovery":"The central claim, stated as Theorem 1.6, is that for every inverse temperature below criticality, if the noise time is much larger than the characteristic time, the joint probability that both the initial and evolved configurations cross the rhombus tends to 1/4, while if the noise time is much smaller than that characteristic time, the joint probability tends to 1/2. The characteristic time is the reciprocal of n squared times the four-arm probability, where the four-arm event asks for four alternating-sign paths from the origin to the boundary of the box. A corollary, Theorem 1.3, is that for every fixed positive time the limit is 1/4, meaning the crossing event at time t is asymptotically independent of the crossing event at time 0. The proof treats the pair of configurations at the two times as a correlated two-layer field, establishes finite-energy and spatial mixing properties for this pair, and then applies the differential-inequality method previously used for Bernoulli percolation. The paper also explains why the phenomenon stops at criticality and below it: the four-arm exponent is at least 2 there, so the same differential formulas push the limit toward 1/2 instead of 1/4.","pith_inferences":["The most exposed step is not in the dynamical argument: it is the static identity that the crossing probability equals 1/2. If that probability were some other value p in the limit, the same proof would give p squared rather than 1/4, so a numerical check of this identity is a cheap and direct test of the main theorem.","The sharp characteristic time suggests a dynamical scaling limit: for noise times proportional to the characteristic time, the correlation between crossing events should interpolate between 1/2 and 1/4 in a way controlled by the four-arm exponent; the paper does not state such a limit, but its differential inequalities are the natural input for it.","The finite-energy and spatial-mixing machinery for the pair is developed for general positive temperature and relies only on locality and bounded rates, so it plausibly transfers to other lattice spin systems whose crossing probabilities satisfy a self-duality identity giving 1/2.","A quantitative convergence rate toward 1/4 can be extracted from the proof: the displayed bounds give a polynomial decay in the ratio of the characteristic time to the noise time, even though the paper does not isolate this as a separate theorem."],"forward_implications":["At any fixed positive time, the left-right crossing event at time 0 is asymptotically independent of the same event at time t, so measuring the crossing twice with any fixed time gap gives a product of probabilities in the large-box limit.","The sharp scale separates a stability regime from a noise regime: noise much smaller than the characteristic time leaves the crossing probability essentially unchanged, while noise much larger than it destroys the correlation entirely.","Because the same proof covers elongated rectangles, the result applies to crossing events of arbitrary fixed aspect ratio, not just rhombi.","The contrast at and above criticality, where the limit stays 1/2, shows that noise sensitivity is tied to the four-arm exponent being less than 2, that is, to the percolation-critical nature of high-temperature crossings."],"supporting_citations":[{"why":"Supplies the differential-inequality method for noise sensitivity that the paper adapts from Bernoulli percolation to Ising crossings.","marker":"[TV23]"},{"why":"Supplies the Russo–Seymour–Welsh theory used in the appendix to prove the box-crossing property for subcritical Ising.","marker":"[KT23]"},{"why":"Provides the sharp noise-sensitivity theorem for critical Bernoulli percolation that Theorem 1.6 is designed to mirror.","marker":"[GPS10]"},{"why":"Cited for the stability half of the sharp theorem, namely that sufficiently small noise leaves the crossing probability near 1/2.","marker":"[GPS18]"},{"why":"Gives the strong-mixing bound for the dynamics below criticality, used to extend the quantitative estimates from small times to all times.","marker":"[MOS94]"},{"why":"Supplies the four-arm exponent bound at criticality, used to explain why the same crossing events are not noise sensitive at the critical temperature.","marker":"[Wu18]"}],"fun_headline_variants":["Crossings become independent under Ising noise","High-temp Ising crossings lose memory","Glauber noise kills crossing correlations","Ising crossing events decouple at high temp","Noise makes Ising crossings independent"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The proof's load-bearing premise is the unproved assertion that for every finite box below criticality the probability of a left-to-right plus-spin crossing of the inner rhombus is exactly one half; if that equality fails, the limiting joint probability in Theorem 1.3 would be the square of the true crossing probability rather than one quarter.","fun_headline_variants_meta":{"raw":{"variants":["Crossings become independent under Ising noise","High-temp Ising crossings lose memory","Glauber noise kills crossing correlations","Ising crossing events decouple at high temp","Noise makes Ising crossings independent"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000426,"raw_usage":{"total_tokens":2147,"prompt_tokens":877,"completion_tokens":1270,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":493,"completion_tokens_details":{"reasoning_tokens":1205}},"tokens_in":493,"tokens_out":1270,"duration_ms":9502,"temperature":1.0,"reasoning_tokens":1205,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-15T20:55:56.232920+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Estimate the crossing probability by high-precision Monte Carlo for the free-boundary Ising model on a large double box at a fixed subcritical temperature: if the estimates deviate from one half by an amount that does not vanish as the box grows, the identity behind the box-crossing property fails and the 1/4 limit is not the correct answer. A second check would compare the same probability under plus, minus, and free boundary conditions, since the paper's proof requires the limiting crossing probability to be insensitive to boundary choices.","supporting_citations":[],"review_version":1}