Noise sensitivity of crossings for high temperature Ising model

Hugo Vanneuville; Vincent Tassion

REVIEW 4 major objections 5 minor 34 references

Noise sensitivity of crossings for high temperature Ising model

T0 review · 4 major / 5 minor · reviewed 2026-08-15 · deepseek-v4-flash

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Pith's one-line read The paper proves that crossing events in the subcritical Ising model are noise sensitive under Glauber dynamics, with joint probability tending to 1/4.

desk verdict 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. read the letter →

arxiv 2505.11457 v2 pith:F7ZSOVDY submitted 2025-05-16 math.PR math-phmath.MP

Vincent Tassion , Hugo Vanneuville This is my paper · ORCID

classification math.PRmath-phmath.MP MSC 60K3582B2082C20

keywords noisesensitivityIsingmodelGlauberdynamicstriangularlatticecrossingeventfour-armsubcriticaldifferentialinequalities

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

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.

What carries the argument

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.

What would settle it

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.

Watch

Extended reading notes

Core claim

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.

Load-bearing premise

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.

Editorial extensions

If this is right

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.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

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.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Dear —,

The 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.

What 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.

But 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.

There 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.

For 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.

Best,

Referee Report

4 major / 5 minor

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.

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 (4)

[§6.1.1, proof of Proposition 6.5] 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.
[§6.2, Lemma 6.15] 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.
[§6.1.1, proof of Lemma 6.9] 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.
[Appendix A.2.1, exponentially fast spatial mixing] 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.

minor comments (5)

[Definition 1.2] 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.
[§5.1, proof of Lemma 5.2] 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.
[Appendix A.2.1, exponentially fast spatial mixing] 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'.
[§6.1.1, proof of Lemma 6.8] 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.
[Appendix A.2.3] 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.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the noise-sensitivity theorem is derived from independent Ising/Glauber inputs and external percolation theorems; self-citations are methodological, not load-bearing.

full rationale

The derivation chain for Theorems 1.3 and 1.6 is not circular. The paper's new content is the proof that the pair (σ, σ_t) inherits finite-energy (Prop. 3.2), spatial mixing (Prop. 5.1), a Russo-type differential inequality (Prop. 4.8), and quasi-multiplicativity (Prop. 6.2) for the Ising Glauber dynamics; these are proved inside the paper from the static Ising properties (FKG, MON, SMP, BXP) and the generator formula. The conclusion 1/4 is obtained by integrating these differential inequalities, not by assuming noise sensitivity of crossings. The framework borrowed from [TV23] is a proof method, and the noise-sensitivity theorem for Bernoulli percolation from [TV23] is not used as an input; the Ising result is a genuinely new application. The use of [KT23] for the box-crossing property cites an independent published theorem whose stated hypotheses (FKG, invariance under lattice symmetries, and P(Cross)=1/2) are verified or asserted in the paper; it is not a restatement of the target result. The identity P(σ∈Cross_n)=1/2 in Definition 1.2 is an input assumption, not a fitted parameter or a renamed prediction; if it were false the theorem would fail, but that is a correctness concern rather than circularity. Likewise, the uniformity issue in the induction in Prop. 6.5 is a proof gap, not a circular reduction, since no equation in the paper is equivalent by construction to the noise-sensitivity conclusion. There are no fitted constants renamed as predictions, and no cited 'uniqueness theorem' is used to forbid alternatives. The central claim therefore has independent mathematical content, and the self-citations, while numerous, are not load-bearing in a circular sense.

Assumptions & free parameters 0 free parameters · 6 assumptions · 0 invented entities

No parameters are fitted to data. The constants c_FE and τ are proof devices, not model parameters. The listed axioms are the background results the central claim rests on; the most fragile is the asserted crossing probability 1/2 for off-critical β.

assumptions (6)

standard math Static Ising properties: FKG inequality, boundary monotonicity (MON), spatial Markov property (SMP).
Taken from [FV17] and used throughout Sections 2, 4, 5, and the appendix.
domain assumption Box-crossing property (BXP) for β ≤ β_c.
Proved in Appendix A.2.1, but for β < β_c it relies on the asserted identity P(Cross_n)=1/2 and on [KT23], [DT19].
domain assumption Exponentially fast spatial mixing for the static Ising model when β < β_c.
Stated in Appendix A.2.1; proof is left to the reader and based on FK coupling and sharpness of the phase transition.
domain assumption Dynamical mixing bound (5): for β < β_c, |P(σ,σ_t∈A) - μ_n(A)^2| ≤ e^{-ct}.
Quoted from [MOS94] under a strong mixing condition for boxes; the paper states the condition follows from FK coupling.
domain assumption Four-arm exponent strictly below 2: α_n/α_m ≥ c (m/n)^{2-c} for β < β_c.
Proved in Appendix A.2.3 using [KT23] and [Tas14]; it is essential for Proposition 7.1.
domain assumption Three-arm half-plane exponent at least 2 (Lemma A.3).
Proved in appendix from (BXP), (SMP), (MON).

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Cite this review

Pith. "Pith review of Noise sensitivity of crossings for high temperature Ising model." pith.science (2026). https://pith.science/paper/F7ZSOVDY

@misc{pith2026250511457,
  author       = {Pith},
  title        = {Pith review of: Noise sensitivity of crossings for high temperature Ising model},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/F7ZSOVDY}},
  note         = {Machine review of arXiv:2505.11457}
}

abstract

Consider the event that there is a $+$ crossing from left to right in a box for the Ising model on the triangular lattice. We show that this event is noise sensitive under Glauber dynamics $t \mapsto \sigma_t$ in the subcritical regime $\beta<\beta_c$. We rely on the non-spectral approach from our previous work [TV23]. An important aspect in this more general setup is the study of the pair $(\sigma_0,\sigma_t)$ and in particular the establishment of properties such as finite-energy and spatial mixing.

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Pith tools

Reviewed August 15, 2026 · model on record in the stance chip above.

[1] [1]

Ahlberg and R

D. Ahlberg and R. Baldasso. Noise sensitivity and V oronoi percolation. Electronic Journal of Probability , 23, 2018

work page 2018

[2] [2]

Ahlberg, E

D. Ahlberg, E. Broman, S. Griffiths, and R. Morris. Noise sensitivity in continuum percolation. Israel Journal of Mathematics , 201, 2014

work page 2014

[3] [3]

Ahlberg, S

D. Ahlberg, S. Griffiths, R. Morris, and V. Tassion. Quenched V oronoi percolation. Advances in Mathematics , 286, 2016

work page 2016

[4] [4]

Bakry, I

D. Bakry, I. Gentil, and M. Ledoux. Analysis and geometry of M arkov diffusion operators . Springer Science & Business Media, 2013

work page 2013

[5] [5]

Broman, C

E.I. Broman, C. Garban, and J.E. Steif. Exclusion sensitivity of B oolean functions. Probability theory and related fields , 155, 2013

work page 2013

[6] [6]

Benjamini, G

I. Benjamini, G. Kalai, and O. Schramm. Noise sensitivity of B oolean functions and applications to percolation. Publications math \'e matiques de l'IH \'E S , 90, 1999

work page 1999

[7] [7]

Broman and J.E

E. Broman and J.E. Steif. Dynamical stability of percolation for some interacting particle systems and -stability. Annals of Probability , 2005

work page 2005

[8] [8]

Duminil-Copin and I

H. Duminil-Copin and I. Manolescu. Planar random-cluster model: scaling relations. Forum of Mathematics, Pi , 2022

work page 2022