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Enhanced noise sensitivity, 2D directed polymers and Stochastic Heat Flow

T0 review · 3 major / 5 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read This paper proves that the critical 2D Stochastic Heat Flow is asymptotically independent of the white noise obtained as the scaling limit of the same disorder, through a new general criterion for noise sensitivity of functions of…

desk verdict Sharp noise-sensitivity criteria with a clean polymer application, but the advertised SHF independence is proven only for finitely valued disorder while the abstract says it unconditionally. read the letter →

arxiv 2507.10379 v2 pith:JBSYHW26 submitted 2025-07-14 math.PR

Francesco Caravenna , Anna Donadini This is my paper · ORCID
classification math.PR MSC 05D4082B4460H15
keywords noisesensitivityBKScriterioninfluenceshypercontractivitydirectedpolymerinrandomenvironmentstochasticheatflowwhiteenhanced
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

The paper generalizes the classical BKS noise-sensitivity criterion from binary input variables to a wide class of functions of general independent random variables: under a hypercontractivity condition (Assumption 2.2), vanishing summed squared influences of the variables forces the function to decorrelate under small random perturbations, with quantitative bounds that are optimal in the binary case. It introduces an enhanced form of noise sensitivity that upgrades decorrelation to full asymptotic independence, and proves this enhanced property for the partition functions of 2D directed polymers in the critical window, where their summed squared influences decay like $1/\log N$. As the main application, the paper shows that the critical 2D Stochastic Heat Flow, the universal scaling limit of these partition functions, is asymptotically independent of the white noise obtained as the scaling limit of the same disorder: the limiting object forgets the noise that produced it. The independence statement is proved for disorder variables that take finitely many values, the setting in which the enhanced noise-sensitivity machinery is established.

What carries the argument

The engine is the pair (Assumption 2.2, Theorem 4.5): a hypercontractivity condition on the conditional slices $\omega_i \mapsto f(\omega)$ yields the bound $\|f^{(\le d)}\|_2^2 / \mathrm{Var}[f] \le \eta_q^{-2d} (W[f]/\mathrm{Var}[f])^{1-2/q}$ on the low-degree part of the orthogonal chaos decomposition, which via the explicit covariance formula $\mathrm{Cov}[f(\omega^\varepsilon),f(\omega)] = \sum_d (1-\varepsilon)^d \|f^{(d)}\|_2^2$ turns small total influence into small covariance. The paper's new objects are the $L^1$ influence $\mathrm{Inf}^{(1)}_k[f] = E[|f - E_k[f]|]$ and its sum of squares $W[f]$, which generalize the classical influence to arbitrary distributions and functions; in the binary case they reduce to the classical quantities up to a factor $2p(1-p)$. The quantitative criteria are then upgraded to enhanced noise sensitivity (Theorem 2.15) for finitely valued i.i.d. variables, and the application to polymers reduces to the influence computation of Proposition 3.4, giving $W[f_N] = O(1/\log N)$. The modified Tribes function (Theorem 2.19) provides the matching lower bound establishing the optimality of the exponent.

What would settle it

Take Bernoulli disorder at criticality and estimate $\mathrm{Cov}[\phi(Z_N), \psi(\xi_N)]$ for bounded smooth $\phi,\psi$: Theorem 3.6 predicts convergence to zero for every such pair. If for some $\varepsilon>0$ and some $\phi,\psi$ the covariance is bounded away from zero as $N\to\infty$, the independence claim is false; a numerical check at $N$ of order $10^4$-$10^5$ with the $O(1/\log N)$ prediction for $W[f_N]$ would be a direct test.

Watch

Extended reading notes

Core claim

At the core is a quantitative noise-sensitivity bound: for any $f$ in $L^2$ of independent variables satisfying Assumption 2.2, $\mathrm{Cov}[f(\omega^\varepsilon),f(\omega)]/\mathrm{Var}[f] \le 4\,(W[f]/\mathrm{Var}[f])^{\gamma_{\varepsilon,q}}$, with $W[f]$ the sum of squared $L^1$ influences defined via the probabilistic gradient $\delta_k f = f - E_k[f]$ (Theorem 2.9). In the optimal hypercontractivity case the exponent becomes the sharp $\varepsilon/(2-\varepsilon)$ (Theorem 2.17), and a generalized Tribes construction shows this exponent cannot be improved (Theorem 2.19). Under the same Assumption, vanishing $W[f_N]$ implies classical noise sensitivity (Theorem 2.10); when the variables are i.i.d. and finitely valued, the same criterion implies enhanced noise sensitivity, i.e. asymptotic independence of $f_N(\omega^\varepsilon)$ and $f_N(\omega)$ (Theorem 2.15). For the 2D directed polymer partition functions in the critical window the paper proves $W[f_N] = O(1/\log N)$ (Theorem 3.2), and since the associated white-noise field is a degree-one chaos, the drift of the variance spectrum to infinity forces the joint limit to split into independent parts: the Stochastic Heat Flow and the white noise are independent (Theorem 3.6).

Load-bearing premise

The independence of the Stochastic Heat Flow from the white noise (Theorem 3.6) is proved only for disorder variables that take finitely many values, because the enhanced noise-sensitivity criterion (Theorem 2.15) is established only in that setting; without it, the paper shows decorrelation but not independence.

Editorial extensions

If this is right

  • Classical noise sensitivity of any sequence satisfying Assumption 2.2 follows from the general BKS criterion (Theorem 2.10), with explicit quantitative control from Theorem 2.9.
  • For finitely valued disorder, enhanced noise sensitivity holds for any functions with $W[f_N] \to 0$, so decorrelation upgrades to asymptotic independence (Theorem 2.15).
  • The directed polymer partition functions in the critical window satisfy $W[f_N] = O(1/\log N)$, hence are enhanced noise sensitive (Theorem 3.2).
  • The critical 2D Stochastic Heat Flow $Z^\vartheta$ and the white noise $\xi$ from the disorder are asymptotically independent; the SHF is not driven by that white noise in the limiting sense (Theorem 3.6).
  • The exponent $\varepsilon/(2-\varepsilon)$ in the refined BKS bound cannot be improved, as shown by the modified Tribes construction (Theorem 2.19).

Reading between the lines

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

  • The independence result is proved for finitely valued disorder; extending Theorem 2.15 to general distributions under Assumption 2.2 would remove this restriction and give independence for Gaussian or exponential-moment disorder, but that step is not taken in the paper.
  • The mechanism is generic: any sequence of $L^2$ disorder observables whose variance spectrum drifts to infinity will be asymptotically independent of the white-noise field, so the same scheme could apply to other critical disordered systems with logarithmic renormalization.
  • The $O(1/\log N)$ decay of $W[f_N]$ suggests a quantitative coupling bound on the joint law of $(Z_N, \xi_N)$, beyond the qualitative f.d.d. independence proved here.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 5 minor

Summary. The paper develops a quantitative BKS-type noise-sensitivity criterion for functions of independent, not necessarily binary, random variables under a hypercontractivity assumption (Assumption 2.2), with optimal exponents in the hypercontractive case, and proves an enhanced noise-sensitivity property yielding asymptotic independence for finitely valued variables. It then applies these results to the two-dimensional directed polymer partition functions in the critical window, proving that their normalized sum of squared L1 influences is O(1/log N) and hence that the partition functions are noise sensitive. Under the added assumption that the disorder variables are finitely valued, it derives enhanced noise sensitivity for the partition functions and, from that, the asymptotic independence of the critical 2D Stochastic Heat Flow (SHF) from the white noise arising from the same disorder, in the sense of joint convergence in finite-dimensional distributions.

Significance. If the stated results hold, the paper makes several valuable contributions: it generalizes the Benjamini–Kalai–Schramm criterion beyond Boolean functions, gives an explicit refined exponent ε/(2−ε) with an optimality example via a modified Tribes function, and links noise sensitivity of polymer partition functions to a striking structural property of the critical SHF, its independence from the driving white noise. The proofs are largely self-contained and include the key influence computation for directed polymers and the chaos-decomposition arguments. The paper is honest about a central limitation: the enhanced noise-sensitivity theorem and the SHF independence theorem are proved only for finitely valued disorder, and this restriction is stated explicitly in Section 2.7 and in the statements of Theorems 2.15 and 3.6.

major comments (3)
  1. [Abstract and Section 3.3] The abstract states unconditionally that 'the Stochastic Heat Flow is independent of the white noise arising from the disorder', but Theorem 3.6 is proved only under the assumption that the disorder variables ω(n,z) take finitely many values. The proof of (3.12) uses enhanced noise sensitivity (3.6) for φ(Z_N), which relies on Theorem 2.15; Theorem 2.15 requires finitely valued variables, and Section 2.7 explicitly notes that Assumption 2.2 is not obviously stable under composition with smooth functions. Thus, as written, the paper establishes the independence theorem only in the finitely valued case; the abstract and the introduction should be amended to state this condition, or an argument covering general i.i.d. disorder with exponential moments must be supplied.
  2. [Section 5.5 and Theorem 2.15] The proof of Theorem 2.15 applies Theorem 2.10 to the composed function φ(f_N), which requires φ(f_N) to satisfy Assumption 2.2. This is justified for finitely valued variables by Example 2.3, because every L2 function of such variables lies in a common finite-dimensional space of functions. For non-finitely-valued variables, however, the argument breaks: for Gaussian or other continuous disorder, f_N is affine in each tiled variable ζ(n,z), but for a generic smooth φ, φ(f_N) is not affine in ζ(n,z), so condition (2.8) with V_i = span{1, ζ_i} fails. No alternative verification of (2.8) or of the hypercontractivity bound (2.9) is given for φ(f_N) in this setting. Consequently the enhanced noise-sensitivity criterion (2.26) and, through it, the enhanced polymer statement (3.6) are not established for continuous disorder.
  3. [Proof of Theorem 3.6, paragraph after (3.12)] The proof invokes Remark 2.11 to claim that the variance spectrum of φ(Z_N) drifts to infinity. Remark 2.11 states an equivalence under the assumption that Var[f_N] is bounded above and below away from zero; here only boundedness above (from boundedness of φ) is available. This does not invalidate the argument, because the needed vanishing of the low-degree chaos components follows directly from (3.6) together with (4.14): for fixed d, the truncated covariance is at least (1−ε)^d times the squared L2 norm of the projection onto chaos of degree ≤ d. The citation is therefore misleading and should be replaced by this direct argument.
minor comments (5)
  1. [Section 3.4] The heading 'Proof of Theorem 3.4' should read 'Proof of Proposition 3.4', since the result being proved is Proposition 3.4.
  2. [Remark 3.5] There is a typo: 'non ehnanced' should be 'non-enhanced'.
  3. [Section 3.1] The phrase 'phase trasition' in the penultimate paragraph should be 'phase transition'.
  4. [Section 2.4, Theorem 2.9] The theorem statement says γ_{ε,q} depends on the hypercontractivity constant η_q, but the displayed bound (2.16) uses γ_{ε,q} without explicitly indicating that the constant 4 is universal; this is clear from the proof but could be stated more cleanly.
  5. [Appendix C, Lemma C.5] In the proof of Lemma C.5, after equation (C.8) the text says 'by Fubini's theorem' in a context that also uses martingale convergence; adding a sentence explaining the limiting step for infinite T would improve readability.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: all predicted bounds follow from explicit influence/hypercontractivity estimates; the finite-valued-disorder restriction in Thm 3.6 is a scope limitation, not a circular step.

full rationale

I found no circular step. The paper's chain is: (i) a general BKS-type bound (Thm 2.9) is derived from Assumption 2.2 by the Efron-Stein chaos decomposition (Prop. 4.2), hypercontractivity of the noise operator (Thm 4.5, obtained from the external [MOO10]), and interpolation; the exponent gamma_{epsilon,q} is explicit and no quantity is fitted from the data being predicted. (ii) The polymer application computes the sum of squared L1-influences directly as W[f_N] = O(1/log N) (Prop. 3.4, proof of Thm 3.2), so the noise sensitivity of the partition functions is a derived estimate, not an input. (iii) Thm 3.6 combines the resulting vanishing of the low-degree variance spectrum of phi(Z_N) with the fact that psi(xi_N) has chaos degree at most deg psi, using orthogonality (4.10); this is not equivalent to assuming the conclusion. The only load-bearing imported result is the convergence Z_N -> Z^theta from [CSZ23] (Thm 3.1); although that is a same-group citation, it is a previously published theorem, independent of the present paper's fits, and is corroborated by [Tsa24, GT25]. The genuine limitation--not circularity--is that enhanced noise sensitivity and hence Thm 3.6 are proved only for finitely valued disorder, as the paper itself notes in Sec. 2.7 and states in Thm 3.6; the abstract's unconditional wording overstates the proved scope. No fitted parameter is relabelled as a prediction, and no uniqueness claim is imported from the authors' prior work.

Assumptions & free parameters 3 free parameters · 6 assumptions · 0 invented entities

The central claims rest on Assumption 2.2 hypercontractivity (whose necessity is demonstrated by Example 2.21), the imported critical-window convergence theorem [CSZ23], standard hypercontractivity and local-CLT results, and the paper's own chaos-decomposition calculus. No parameters are fitted to data; the only hand-chosen parameters are ϑ and γ, which parametrize inherited scaling and the optimality example respectively, plus the abstract constants (q, M_q) from the standing assumption. No new entities are postulated.

free parameters (3)
  • Critical-window parameter ϑ ∈ R
    Parameter in the critical scaling (3.1), σ_N² = (1/R_N)(1 + (ϑ+o(1))/log N); inherited from [CSZ23], not fitted in this paper, indexes the limiting SHF Z^ϑ.
  • Tribes scaling exponent γ ∈ (0,1/2) = γ ∈ (0,1/2)
    Chosen by hand in (2.32), a_t = t^{1/2+γ+o(1)}, to make the Modified Tribes example have the right influence-to-covariance trade-off; affects only the logarithmic correction power in Theorem 2.19.
  • Abstract constants q and M_q
    Constants in Assumption 2.2 (2.9); existence for the polymer partition function is shown via the tilted variables ζ = e^{β_N ω} - λ(β_N) - 1 in Remark 3.3; they are assumptions, not fitted values.
assumptions (6)
  • domain assumption Assumption 2.2: for each i, the map ω_i ↦ f(ω) lies in a vector space V_i ⊂ L^2(E_i, μ_i) with hypercontractive bound ||g||_q ≤ M_q ||g||_2 for centred g ∈ V_i, uniformly over i (q > 2, M_q < ∞).
    Core structural input for Theorems 2.9, 2.10, 4.5 and hence for the polymer application; enters in Section 2.2 and is invoked throughout Section 5. The paper shows (Example 2.21) that a hypercontractivity-type condition is necessary for a general BKS criterion.
  • domain assumption Critical-window convergence of averaged partition functions to the SHF: Theorem 3.1 of [CSZ23], i.e., f.d.d. convergence Z_N(g,h) → Z^ϑ(g,h) under the scaling σ_N² = (1/R_N)(1 + (ϑ+o(1))/log N).
    Imported from Caravenna-Sun-Zygouras (Inventiones 233, 2023, peer-reviewed); defines the limit object Z^ϑ and is the entry point for both Theorem 3.2 and Theorem 3.6. Same-group authorship but externally established.
  • standard math Ensemble hypercontractivity theory of Mossel-O'Donnell-Oleszkiewicz: Propositions 3.11 and 3.16 on p2,q,η-hypercontractive ensembles and multilinear polynomial hypercontractivity.
    Used to prove Theorem 4.5 (general hypercontractivity of the noise operator). The paper extends the finite setting to infinite index sets via truncation and Fatou in Appendix C.
  • standard math Chaos decomposition (Efron-Stein/Hoeffding) for L^2 functions of independent variables, Proposition 4.2, and the covariance identity (4.14): Cov[f(ω^ε), f(ω)] = Σ_d (1-ε)^d ||f^{(d)}||².
    Proven in Appendix B; the backbone of the noise sensitivity criteria and of the orthogonality argument in Theorem 3.6.
  • standard math Local central limit and Gaussian estimates for the simple random walk and for binomial variables, used in Section 6.3, e.g., (2.35), (6.10)-(6.12).
    Standard estimates backing the Modified Tribes optimality computation; presented in compact form.
  • domain assumption The disorder has zero mean, unit variance, and finite exponential moments (Section 3.1).
    Standard polymer model assumptions; ensures the tilted variables have finite uniform moment bounds needed for Assumption 2.2.

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Pith. "Pith review of Enhanced noise sensitivity, 2D directed polymers and Stochastic Heat Flow." pith.science (2026). https://pith.science/paper/JBSYHW26

@misc{pith2026250710379,
  author       = {Pith},
  title        = {Pith review of: Enhanced noise sensitivity, 2D directed polymers and Stochastic Heat Flow},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/JBSYHW26}},
  note         = {Machine review of arXiv:2507.10379}
}
read the original abstract

We investigate noise sensitivity beyond the classical setting of binary random variables, extending the celebrated result by Benjamini, Kalai, and Schramm to a wide class of functions of general random variables. Our approach yields improved bounds with optimal rates. We also consider an enhanced form of noise sensitivity which yields asymptotic independence, rather than mere decorrelation. We apply these result to establish enhanced noise sensitivity for the partition functions of 2D directed polymers, in the critical regime where they converge to the critical 2D Stochastic Heat Flow. As a consequence, we prove that the Stochastic Heat Flow is independent of the white noise arising from the disorder.

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Reference graph

Works this paper leans on

52 extracted references · 37 canonical work pages · cited by 1 Pith paper

  1. [1]

    Ahlberg, E

    D. Ahlberg, E. Broman, S. Griffiths, R. Morris. Noise sensitivity in continuum percolation. Israel J. Math. 201 (2014), 847--899

    work page 2014

  2. [2]

    Ahlberg, M

    D. Ahlberg, M. Hillairet, E. Toropova. Noise sensitivity in Last-passage percolation. Preprint (2025)

    work page 2025

  3. [3]

    Alberts, K

    T. Alberts, K. Khanin, J. Quastel. The intermediate disorder regime for directed polymers in dimension 1+1 . Ann. Probab. 42 (2014), 1212--1256

    work page 2014

  4. [4]

    Ben-Or, N

    M. Ben-Or, N. Linial. Collective coin flipping, robust voting schemes and minima of Banzhaf values. 26th Annual Symposium on Foundations of Computer Science (sfcs 1985) . IEEE. 1985, 408–416

    work page 1985

  5. [5]

    Ben-Or, N

    M. Ben-Or, N. Linial. Collective coin flipping. Randomness and Computation (S. Micali, ed.), 1987

    work page 1987

  6. [6]

    M. Benaïm. R. Rossignol. Exponential concentration for first passage percolation through modified Poincaré inequalities. Ann. Inst. H. Poincaré 44 (2008), 544--573

    work page 2008

  7. [7]

    Benjamini, G

    I. Benjamini, G. Kalai, O. Schramm. Noise sensitivity of boolean functions and applications to percolation. Inst. Hautes Études Sci. Publ. Math. 90 (1999), 5--43

    work page 1999

  8. [8]

    Berger, F

    Q. Berger, F. Caravenna, N. Turchi. Strong disorder for Stochastic Heat Flow and 2D Directed Polymers. Preprint (2025), arXiv:2508.02478

    arXiv 2025

Show all 52 references
  1. [9]

    Bertini, N

    L. Bertini, N. Cancrini. The stochastic heat equation: F eynman- K ac formula and intermittence. J. Statist. Phys. , 78(5-6) (1995), 1377--1401

    1995

  2. [10]

    Bhattacharjee, G

    C. Bhattacharjee, G. Peccati, D. Yogeshwaran. Spectra of Poisson functionals and applications in continuum percolation. Preprint (2024), arXiv:2407.13502

    2024 arXiv

  3. [11]

    A. Bonami. Étude des coefficients de Fourier des fonctions de L^p(G) . Ann. Inst. Fourier 20 (1970), 335--402

    1970

  4. [12]

    Bourgain

    J. Bourgain. Walsh subspaces of L^p product spaces. Séminaire d’Analyse fonctionnelle (Polytechnique) (1979-80), exp.\ n. IV, 1--14

    1979

  5. [13]

    Caravenna, F

    F. Caravenna, F. Cottini, M. Rossi. Quasi-critical fluctuations for 2d directed polymers. Ann. Appl. Probab. 35(4) (2025), 2604--2643

    2025

  6. [14]

    Caravenna, R

    F. Caravenna, R. Sun, N. Zygouras. Universality in marginally relevant disordered systems. Ann. Appl. Probab. , 27(5) (2017), 3050--3112

    2017

  7. [15]

    Caravenna, R

    F. Caravenna, R. Sun, N. Zygouras. On the moments of the (2+1)-dimensional directed polymer and stochastic heat equation in the critical window. Commun. Math. Phys. 372 (2019), 385-440

    2019

  8. [16]

    Caravenna, R

    F. Caravenna, R. Sun, N. Zygouras. The two-dimensional KPZ equation in the entire subcritical regime. Ann. Prob. 48 (2020), 1086-1127

    2020

  9. [17]

    Caravenna, R

    F. Caravenna, R. Sun, N. Zygouras. The critical 2d Stochastic Heat Flow. Invent. math. 233 (2023), 325–460

    2023

  10. [18]

    Caravenna, R

    F. Caravenna, R. Sun, N. Zygouras. The critical 2d stochastic heat flow is not a G aussian multiplicative chaos. Ann. Probab. (2023), 51(6):2265--2300

    2023

  11. [19]

    Caravenna, R

    F. Caravenna, R. Sun, N. Zygouras. The critical 2d stochastic heat flow and related models. Preprint (2024), arXiv:2412.10311

    2024 arXiv

  12. [20]

    Caravenna, R

    F. Caravenna, R. Sun, N. Zygouras. Singularity and regularity of the critical 2D Stochastic Heat Flow. Preprint (2025), arXiv:2504.06128

    2025 arXiv

  13. [21]

    Y.-T. Chen. Delta-Bose gas from the viewpoint of the two-dimensional stochastic heat equation. Ann. Probab. 52 (2024), 127--187

    2024

  14. [22]

    Y.-T. Chen. Martingale problem of the two-dimensional stochastic heat equation at criticality. Preprint (2025), arXiv:2504.21791

    2025

  15. [23]

    F. Comets. Directed polymers in random environments. \'Ecole d'\'Et\'e de Probabilit\'es de Saint-Flour XLVI -- 2016. Springer, 2017

    2016

  16. [24]

    Clark, B

    J. Clark, B. Mian. Continuum polymer measures corresponding to the critical 2d stochastic heat flow. Preprint (2024), arXiv:2409.01510

    2024 arXiv

  17. [25]

    Clark, L .- C

    J. Clark, L .- C . Tsai. Conditional GMC within the stochastic heat flow. Preprint (2025), arXiv:2507.16056

    2025 arXiv

  18. [26]

    Eldan, R

    R. Eldan, R. Gross. Concentration on the Boolean hypercube via pathwise stochastic analysis. Invent. Math. 230 (2022), 935--994

    2022

  19. [27]

    Falik, A

    D. Falik, A. Samorodnitsky. Edge-isoperimetric inequalities and influences. Combin. Probab. Comput. 16 (2007), 693--712

    2007

  20. [28]

    Ganguly, K

    S. Ganguly, K. Nam. Sharp moment and upper tail asymptotics for the critical 2d Stochastic Heat Flow. Preprint (2025), arXiv:2507.22029

    2025 arXiv

  21. [29]

    Garban, J

    C. Garban, J. E. Steiff. Noise Sensitivity of Boolean Functions and Percolation. Cambridge University Press (2014)

    2014

  22. [30]

    Y. Gu, T. Komorowski. Noise sensitivity for stochastic heat and Schrödinger equation. Preprint (2025), arXiv:2502.04587

    2025 arXiv

  23. [31]

    Y. Gu, J. Quastel, L .- C . Tsai. Moments of the 2 D SHE at criticality. Probab. Math. Phys. , 2(1) (2021), 179--219

    2021

  24. [32]

    Gu, L.-C

    Y. Gu, L.-C. Tsai. Stochastic heat flow is a black noise. Preprint (2025), arXiv:2506.16484

    2025 arXiv

  25. [33]

    Himwich, S

    Z. Himwich, S. Parekh. The directed landscape is a black noise. Preprint (2024), arXiv:2404.16801

    2024 arXiv

  26. [34]

    Ivanisvili, H

    P. Ivanisvili, H. Zhang. On the Eldan-Gross inequality. Preprint (2024), arXiv:2407.17864

    2024

  27. [35]

    S. Janson. Gaussian Hilbert spaces. Cambridge Tracts in Mathematics, Vol. 129. Cambridge University Press , Cambridge (1997)

    1997

  28. [36]

    S. Junk. Local limit theorem for directed polymers beyond the L^2 -phase. Preprint (2023), arXiv:2307.05097, to appear in J. Eur. Math. Soc

    2023 arXiv

  29. [37]

    S. Junk, H. Lacoin. Strong disorder and very strong disorder are equivalent for directed polymers Preprint (2024), arXiv:2402.02562

    2024

  30. [38]

    J. Kahn, G. Kalai, N. Linial. The influence of variables on Boolean functions. Proc. 29th Symp. on Foundations of Comp. Sci. (1988), 68--80

    1988

  31. [39]

    H. Lacoin. The localization transition for the directed polymer in a random environment is smooth. Preprint (2025), arXiv:2505.13382

    2025 arXiv

  32. [40]

    Keller, G

    N. Keller, G. Kindler. Quantitative relation between noise sensitivity and influences. Combinatorica 33 (2013), 45--71

    2013

  33. [41]

    G. Last, G. Peccati, D. Yogeshwaran. Phase transitions and noise sensitivity on the Poisson space via stopping sets and decision trees. Random Structures Algorithms 63 (2023), 457--511

    2023

  34. [42]

    Z. Liu, N. Zygouras. On the moments of the mass of shrinking balls under the Critical 2d Stochastic Heat Flow. Preprint (2024), arXiv:2410.14601

    2024

  35. [43]

    Mossel, R

    E. Mossel, R. O'Donnell, K. Oleszkiewicz. Noise stability of functions with low influences: Invariance and optimality. Ann. Math 171 (2010), 295--341

    2010

  36. [44]

    Nakashima

    M. Nakashima. Martingale measure associated with the critical 2d stochastic heat flow. Preprint (2025), arXiv:2503.20171

    2025 arXiv

  37. [45]

    Nakashima

    M. Nakashima. An upper bound of the lower tail of the mass of balls under the critical 2d stochastic heat flow. Preprint (2025), arXiv:2507.18080

    2025 arXiv

  38. [46]

    E. Nelson. The free Markov field. J. Funct. Anal. 12 (1973), 211--277

    1973

  39. [47]

    O'Donnell

    R. O'Donnell. Analysis of Boolean functions. Cambridge University Press (2014)

    2014

  40. [48]

    Rosenthal

    G. Rosenthal. Ramon van Handel’s Remarks on the Discrete Cube. Notes from a series of lectures (2020). https://www.cs.toronto.edu/ rosenthal/RvH_discrete_cube.pdf

    2020

  41. [49]

    Rossignol

    R. Rossignol. Threshold for monotone symmetric properties through a logarithmic Sobolev inequality. Ann. Probab. 35 (2006), 1707--1725

    2006

  42. [50]

    Talagrand

    M. Talagrand. On Russo's approximate zero-one law. Ann. Probab. 22 (1994), 1576--1587

    1994

  43. [51]

    L.- C . Tsai. Stochastic heat flow by moments, Preprint (2024), arXiv:2410.14657

    2024

  44. [52]

    Zygouras

    N. Zygouras. Directed polymers in a random environment: a review of the phase transitions. Stochastic Process. Appl. 177 (2024): Paper No. 104431, 34

    2024

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