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This paper establishes a quantitative two-time spatial decorrelation bound for the flat KPZ fixed point: for every s, t > 0, the covariance between heights at times s and t and spatial distance x decays at least as fast as C exp(-c|x|^3), t

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2026-08-01 18:59 UTC pith:AKRUV4HX

load-bearing objection First quantitative two-time decorrelation for flat KPZ; proof is convincing, with one imported half-plane estimate that needs a line of justification. the 2 major comments →

arxiv 2607.17113 v1 pith:AKRUV4HX submitted 2026-07-19 math.PR

Two-time spatial decorrelation for the flat KPZ fixed point

Le Chen , Fei Pu This is my paper
classification math.PR MSC 60K3560G6082C22
keywords KPZ fixed pointAiry1 processdirected landscapespatial decorrelationalpha-mixingcentral limit theoremflat initial datadynamic susceptibility
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

The paper proves that the flat KPZ fixed point decorrelates in space across two different times at a cubic-exponential rate. This matters because the later height is not merely a second point of the same Airy1 profile; it is a nonlinear variational functional of the entire earlier profile. The proof works by splitting the covariance into a part where the earlier profile is seen only far from the origin, controlled by strong mixing of Airy1, and a rare event where the variational optimizer escapes a localization window, controlled by a directed-landscape estimate. As a consequence, centered spatial averages of the height field converge to a Gaussian process whose covariance is the space-integrated two-time correlation.

Core claim

The central claim is Theorem 1.1: for every s, t > 0 there are constants C, c > 0 such that |Cov(h(t,x), h(s,0))| ≤ C exp(-c|x|^3) for |x| ≥ 1. In scaling form, for 0 < s ≤ t and ρ = s/t, this is |Cov(h(t,x), h(s,0))| ≤ t^{2/3} C_ρ exp(-c_ρ |x|^3 / t^2). The paper identifies the decorrelation mechanism as the combination of cubic-exponential mixing of the Airy1 process with uniform localization of intermediate optimizers in the directed landscape. It does not identify the sharp constant or prove a matching lower bound.

What carries the argument

The engine is a two-part mechanism. First, Proposition 3.1 establishes that the Airy1 process, hence a fixed-time flat KPZ slice, has an alpha-mixing coefficient decaying like exp(-c d^3 / s^2); this comes from a half-plane separation argument in exponential last-passage percolation. Second, Propositions 4.1 and 4.3 show that the variational maximizer in the directed landscape representation h(t,x) = sup_y {h(s,y) + L(y,s;x,t)} stays within distance η|x| of x with probability at least 1 - C exp(-c|x|^3). The covariance is then split into a distant-profile term, handled by mixing, and a truncation-error term, handled by optimizer localization.

Load-bearing premise

The proof leans on a fast (cubic-exponential) decorrelation of the fixed-time Airy1 process across distant half-lines; the paper adapts a known bound to a reflected coordinate system without proving that reflected version, and if that decay were only polynomial, the main theorem would not follow.

What would settle it

In the exponential last-passage percolation model used in Section 3, compute the probability that the unrestricted geodesic crosses a separator rotated by reflection (ψ = -d q_N). If that probability does not decay like exp(-c d^3), Proposition 3.1 fails. Alternatively, evaluate Cov(h(t,x), h(s,0)) at fixed s, t for large x in the flat KPZ fixed point; decay slower than exp(-c|x|^3) would contradict Theorem 1.1.

Watch this falsifier. Get emailed when new claim-graph text bears on it.

If this is right

  • Theorem 1.2: the centered spatial averages N^{-1/2} ∫_0^N (h(t,x) - E h(t,x)) dx converge in finite-dimensional distributions to a centered Gaussian process G with covariance χ(t,s) = ∫_R Cov(h(t,x), h(s,0)) dx.
  • The dynamic susceptibility χ is symmetric, positive semidefinite, and homogeneous of degree 4/3, so χ(λt, λs) = λ^{4/3} χ(t,s).
  • On the diagonal, χ(t,t) = 2^{4/3} t^{4/3} σ², where σ² is the space-integrated Airy1 covariance, recovering the one-time fluctuation scale.
  • The scaling form of the bound shows the natural KPZ transversal exponent: the decay rate is cubic in the scaled distance |x|/t^{2/3}.
  • Because the proof gives only an upper bound, a matching lower bound and the sharp constant for the two-time covariance remain open.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • Beyond the paper, the mixing-plus-localization scheme should extend to other initial data, such as narrow-wedge or two-sided Brownian initial profiles, provided a suitable fixed-time mixing estimate exists.
  • The coarse cubic-exponential bound likely hides a more precise exponent related to the 4/3 exponent in the Airy1 correlation tail; a sharp constant might be extracted by refining the alpha-mixing analysis rather than the optimizer localization.
  • A functional central limit theorem for the time process would require a tightness argument in the time variable; the homogeneity of χ suggests the limiting process is not fractional Brownian motion without an additional stationarity-increments check.
  • The same covariance splitting could be applied to multipoint two-time correlations, potentially giving joint Gaussian limits for spatial averages at several times and spatial shifts.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

2 major / 4 minor

Summary. The paper proves a quantitative two-time spatial decorrelation bound for the flat KPZ fixed point: for every s,t>0 there are C,c>0 such that |Cov(h(t,x),h(s,0))| ≤ C exp(-c|x|^3) for |x|≥1 (Theorem 1.1). The proof decomposes the variational formula for h(t,x) into a truncated observable H_{η|x|} and a truncation error. The first term is controlled by a cubic-exponential α-mixing estimate for the Airy_1 process (Proposition 3.1 and Lemma 3.2), and the second term by a uniform localization estimate for intermediate directed-landscape optimizers (Propositions 4.1 and 4.3). As an application, Theorem 1.2 establishes that normalized spatial averages N^{-1/2}∫_0^N (h(t,x)-E h(t,x)) dx converge in finite-dimensional distributions to a centered Gaussian process whose covariance is the space-integrated two-time correlation χ(t,s).

Significance. If correct, Theorem 1.1 is the first quantitative two-time spatial decorrelation estimate for the flat KPZ fixed point; its cubic exponent matches the fixed-time Airy_1 covariance exponent established in [3], while the two-time quantity is genuinely variational. The paper's mechanism — Airy_1 mixing plus directed-landscape localization — is clean and likely transferable. The auxiliary lemmas are proved in detail, the constants are explicit up to unspecified dependence on s,t, and Theorem 1.2 is a substantive consequence. The main risk is a single imported estimate used in a non-verbatim form, discussed below.

major comments (2)
  1. [Section 3, Proposition 3.1 (half-planes H±_{N,d}, Eq. (3.1))] This is the load-bearing import. The proof invokes [2, Remark 1] after reversing the sign of the separator, and justifies the flip only by the assertion that 'the transverse-excursion estimate ... is unchanged by interchanging the two coordinate axes.' That assertion is not proved or supplied with a precise reference. Interchanging coordinates maps the endpoint u_N(s) to u_N(-s), so the reduction to the printed estimate in [2] requires an additional argument (coordinate swap plus stationarity/reversibility of the Airy_1 limit, or a direct proof of the reflected separation bound). Since Proposition 3.1 is the sole source of the cubic-exponential α-mixing bound used in Proposition 3.3 and Theorem 1.1, the reflected separation estimate must be supplied. Without it, the distant-profile term in (3.6) would not be controlled at the claimed rate.
  2. [Section 6, proof of Theorem 1.2 (association step)] The proof that the sequence (X_j) is associated is only sketched: one needs to approximate each X_j by Riemann sums that are coordinatewise nondecreasing functions of finitely many field values, then pass to the limit. The argument is standard, but the paper does not write the approximation explicitly. This is not a fatal gap, but since the CLT relies on Newman's theorem for associated sequences, the approximation step should be made precise.
minor comments (4)
  1. [Section 3, after (3.1)] The phrase 'The remaining bounded range follows from the trivial bound α(d)≤1 after increasing C' is terse; it would be clearer to write α(d)≤1≤C_0 e^{-c d^3} for d≤d_0.
  2. [Section 3, display (3.7)] The notation E_{ω'}[|Cov_ω(...)|] is somewhat informal. Define Cov_ω explicitly as the covariance computed under P_ω for fixed ω'.
  3. [Section 4, proof of Proposition 4.1] The grid reduction is condensed. In particular, the monotonicity of leftmost/rightmost maximizers is cited to [7, Proposition 9.2], but the statement that moving to a northeast/southwest grid point changes m_{s,t} by at most one mesh step could be displayed as an equation for readability.
  4. [Section 5, proof of Theorem 1.1] The scaling-covariant form (1.4) is asserted without proof. A one-sentence verification using h(t,x) = t^{1/3} h(1, x/t^{2/3}) and ρ=s/t would make the paper self-contained.

Circularity Check

0 steps flagged

No significant circularity: the two-time decorrelation bound is derived from external Airy1 mixing and directed-landscape localization inputs, not from the target covariance.

full rationale

Theorem 1.1 is not circular. Its proof decomposes Cov(h(t,x), h(s,0)) via (2.5) into a truncated-profile term controlled by the fixed-time Airy1 alpha-mixing estimate (Proposition 3.1, imported from Basu–Bhattacharjee [2] and Basu–Busani–Ferrari [3]) and a truncation-error term controlled by optimizer localization (Propositions 4.1 and 4.3, built on the directed-landscape results of Dauvergne–Ortmann–Virág [7] and Dauvergne–Sarkar–Virág [8]). Neither input assumes the two-time covariance being proved. Theorem 1.2 follows from Theorem 1.1 plus spatial stationarity, association, and Newman's CLT [16]; the limiting covariance kernel chi(t,s) is defined as the space-integrated covariance and is computed, not fitted. The self-citations in the paper ([4], [6], [15], [18]) are auxiliary: [18] supplies Airy1 moment/tail bounds and the single-time CLT, [6] supplies association of the SHE field, and [15] supplies a standard Gaussian-vector lemma; none of these carry the central two-time decorrelation claim. The sign-reflected half-plane separation estimate from [2, Remark 1] used in Proposition 3.1 is asserted by symmetry rather than proved, so it is a verification risk, but it is an imported external premise, not a circular reduction. The paper's own Remark 1.3 also honestly lists open problems (matching lower bound, identification of chi, functional CLT), indicating the results are not presented as trivial consequences of earlier work.

Axiom & Free-Parameter Ledger

0 free parameters · 10 axioms · 0 invented entities

No free parameters are fitted to data; constants depend on s,t,eta but are existential bounds. The central claim is carried by prior KPZ/landscape results rather than by new postulated objects. The most delicate import is the sign-reflected separation estimate from preprint [2].

axioms (10)
  • domain assumption KPZ fixed point variational formula h(t,x)=sup_y {h0(y)+L(y,0;x,t)} (Nica–Quastel–Remenik [17, Cor. 4.2])
    Gives the flat representation (2.1) and, via metric composition, the two-time evolution (2.2) that the whole proof is built on.
  • domain assumption Directed landscape independent increments and metric composition ([7, Def. 10.1(II)–(III)])
    Used to decompose h(t,x) through time s and to condition on the future landscape increment independently of h(s,·).
  • domain assumption Spatial stationarity of the flat KPZ fixed point ([14, Thm 4.5(iv)])
    Used in Lemma 2.1, in the s>t symmetry argument, and in proving stationarity of the block variables X_j.
  • domain assumption Fixed-time identification h(t,x) ~ (2t)^{1/3} A1((2t)^{-2/3}x) ([14, (4.15)])
    Transfers Airy1 mixing to h(s,·) and supplies the s=t case via the Airy1 covariance decay of [3].
  • domain assumption Half-plane separation estimate for exponential LPP geodesics ([2, Remark 1]), in the sign-reflected side-specific form used here
    The engine behind the cubic-exponential alpha-mixing bound (3.1); the paper asserts the reflected orientation follows by coordinate symmetry but does not prove it.
  • domain assumption Tail bound for the maximizer of two independent parabolic Airy processes ([7, Lemma 9.5])
    Gives the fixed endpoint-pair optimizer localization (4.4) after the skew-stationarity reduction in Proposition 4.1.
  • domain assumption Airy sheet bound S(q,0)+q^2 ≤ C+c0 log^{2/3}(2+|q|) with tail bound on C ([8, Lemma 5.3])
    Used to localize the initial-time maximizer Z_x in Proposition 4.3.
  • domain assumption Association of the SHE field and preservation of association under weak limits ([6, Thm A.4], [10, (P5)]); convergence of KPZ equation to the KPZ fixed point ([22, Thm 1.8])
    Needed to show the block variables X_j are associated and to apply Newman's CLT in Theorem 1.2.
  • standard math Davydov's alpha-mixing covariance inequality ([9])
    Converts the alpha-mixing coefficient of Proposition 3.1 into the L4 covariance bound in Lemma 3.2.
  • standard math Newman's CLT for associated stationary sequences ([16, Thm 2]) and a lemma identifying Gaussian vectors from nonnegative linear combinations ([15, Lemma A.1])
    These are the limit-theorem machinery for Theorem 1.2.

pith-pipeline@v1.3.0-alltime-deepseek · 16250 in / 28854 out tokens · 239759 ms · 2026-08-01T18:59:56.241866+00:00 · methodology

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read the original abstract

We establish quantitative two-time spatial decorrelation for the Kardar--Parisi--Zhang fixed point with flat initial data. For every $s,t>0$,there exist constants $C,c>0$ such that \[ \big|{\rm Cov}(\mathfrak{h}(t,x),\mathfrak{h}(s,0))\big| \le C\exp\{-c|x|^3\},\qquad |x|\ge1. \] Unlike the fixed-time covariance, which is governed directly by the Airy$_1$ process, the two-time covariance involves the nonlinear variational evolution of the entire earlier height profile. Our proof combines cubic-exponential mixing of the Airy$_1$ process with a uniform localization estimate for intermediate optimizers in the directed landscape. As a consequence, the centered spatial averages, normalized by $N^{1/2}$, converge in finite-dimensional distributions to a centered Gaussian process whose covariance is the space-integrated two-time correlation of the flat KPZ fixed point.

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Reference graph

Works this paper leans on

22 extracted references · 4 canonical work pages

  1. [1]

    and Hegde, M

    Aggarwal, A., Corwin, I. and Hegde, M. (2024). KPZ fixed point convergence of the ASEP and stochastic six-vertex models. Preprint arXiv:2412.18117

    Pith/arXiv arXiv 2024

  2. [2]

    and Bhattacharjee, S

    Basu, R. and Bhattacharjee, S. (2024). Limit theorems for extrema of Airy processes. Preprint arXiv:2406.11826

    Pith/arXiv arXiv 2024

  3. [3]

    and Ferrari, P

    Basu, R., Busani, O. and Ferrari, P. L. (2023). On the exponent governing the correlation decay of the Airy 1 process.Comm. Math. Phys.398, no. 3, 1171–1211. MR4561801; doi:10.1007/s00220-022-04544-1. TWO-TIME SPATIAL DECORRELATION 23

    work page doi:10.1007/s00220-022-04544-1 2023

  4. [4]

    and Pu, F

    Bhattacharjee, S. and Pu, F. (2026). Macroscopic Hausdorff dimension of the level sets of the Airy processes.Electron. J. Probab.31, Paper No. 36, 21 pp. Preprint arXiv:2501.00772

    Pith/arXiv arXiv 2026

  5. [5]

    and Jim´ enez, J

    Chen, L. and Jim´ enez, J. J. (2026). Spatial covariance of KPZ from flat initial profile. Preprint arXiv:2603.14174

    arXiv 2026

  6. [6]

    and Pu, F

    Chen, L., Khoshnevisan, D., Nualart, D. and Pu, F. (2023). Central limit theorems for spatial averages of the stochastic heat equation via Malliavin– Stein’s method.Stoch. Partial Differ. Equ. Anal. Comput.11, no. 1, 122–176. MR4563698; doi:10.1007/s40072-021-00224-8

    work page doi:10.1007/s40072-021-00224-8 2023

  7. [7]

    and Vir´ ag, B

    Dauvergne, D., Ortmann, J. and Vir´ ag, B. (2022). The di- rected landscape.Acta Math.229, no. 2, 201–285. MR4554223; doi:10.4310/acta.2022.v229.n2.a1

    doi:10.4310/acta.2022.v229.n2.a1 2022

  8. [8]

    and Vir´ ag, B

    Dauvergne, D., Sarkar, S. and Vir´ ag, B. (2022). Three-halves variation of geodesics in the directed landscape.Ann. Probab.50, no. 5, 1947–1985. MR4474505; doi:10.1214/22-aop1574; arXiv:2010.12994

    Pith/arXiv arXiv 2022

  9. [9]

    Davydov, J. A. (1968). The convergence of distributions which are generated by stationary random processes.Teor. Verojatnost. i Primenen.13, 730–737. MR243586

    1968

  10. [10]

    D., Proschan, F

    Esary, J. D., Proschan, F. and Walkup, D. W. (1967). Association of random variables, with applications.Ann. Math. Statist.38, 1466–1474. MR217826; doi:10.1214/aoms/1177698701

    arXiv 1967

  11. [11]

    and Pu, F

    Gu, Y. and Pu, F. (2025). Spatial decorrelation of KPZ from narrow wedge. Preprint arXiv:2506.23065

    Pith/arXiv arXiv 2025

  12. [12]

    and Rahman, M

    Johansson, K. and Rahman, M. (2021). Multitime distribution in dis- crete polynuclear growth.Comm. Pure Appl. Math.74, no. 12, 2561–2627. MR4373163; doi:10.1002/cpa.21980; arXiv:1906.01053

    Pith/arXiv arXiv 2021

  13. [13]

    Liu, Z. (2022). Multipoint distribution of TASEP.Ann. Probab.50, no. 4, 1255–1321. doi:10.1214/21-AOP1557; arXiv:1907.09876

    Pith/arXiv arXiv 2022

  14. [14]

    and Remenik, D

    Matetski, K., Quastel, J. and Remenik, D. (2021). The KPZ fixed point.Acta Math.227, no. 1, 115–203. MR4346267; doi:10.4310/acta.2021.v227.n1.a3

    doi:10.4310/acta.2021.v227.n1.a3 2021

  15. [15]

    and Pu, F

    Mueller, C. and Pu, F. (2025). Spatial fluctuation for stochastic heat equation with H¨ older coefficients. Preprint arXiv:2510.00807

    arXiv 2025

  16. [16]

    Newman, C. M. (1980). Normal fluctuations and the FKG inequalities. Comm. Math. Phys.74, no. 2, 119–128. MR576267; doi:10.1007/BF01197754

    work page doi:10.1007/bf01197754 1980

  17. [17]

    and Remenik, D

    Nica, M., Quastel, J. and Remenik, D. (2020). One-sided reflected Brownian motions and the KPZ fixed point.Forum Math. Sigma8, Paper No. e63, 16 pp. MR4190063; doi:10.1017/fms.2020.56

    work page doi:10.1017/fms.2020.56 2020

  18. [18]

    Pu, F. (2025). Ergodicity, CLT and asymptotic maximum of the Airy 1 pro- cess.Bernoulli31, 2624–2648. Preprint arXiv:2311.11217

    Pith/arXiv arXiv 2025

  19. [19]

    Vir´ ag, B. (2020). The heat and the landscape I. Preprint arXiv:2008.07241

    Pith/arXiv arXiv 2020

  20. [20]

    (1991).Probability with Martingales

    Williams, D. (1991).Probability with Martingales. Cambridge Mathemat- ical Textbooks. Cambridge University Press, Cambridge, xvi+251 pp. MR1155402; doi:10.1017/CBO9780511813658

    doi:10.1017/cbo9780511813658 1991

  21. [21]

    Widom, H. (2004). On asymptotics for the Airy process.J. Statist. Phys.115, no. 3–4, 1129–1134. MR2040024; doi:10.1023/B:JOSS.0000022384.58696.61; arXiv:math/0308157

    arXiv 2004

  22. [22]

    Wu, X. (2023). The KPZ equation and the directed landscape. Preprint arXiv:2301.00547. 24 LE CHEN AND FEI PU (Le Chen)Department of Mathematics and Statistics, Auburn Uni- versity, Auburn, AL 36849, USA.,Email: lzc0090@auburn.edu (Fei Pu)Laboratory of Mathematics and Complex Systems, School of Mathematical Sciences, Beijing Normal University, 100875, Beij...

    Pith/arXiv arXiv 2023