On the multipoint distribution formulas of the parabolic Airy process

Aaron Ortiz; Zhipeng Liu

On the multipoint distribution formulas of the parabolic Airy process

T0 review · 0 major / 5 minor · reviewed 2026-08-15 · deepseek-v4-flash

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Pith's one-line read This paper proves that the equal-time multipoint distribution formula for the KPZ fixed point with narrow-wedge initial condition gives the same parabolic Airy process as the extended Airy kernel Fredholm determinant.

desk verdict A solid, honest equivalence proof that Liu's equal-time KPZ fixed point formula really is the parabolic Airy process; it deserves peer review, not a desk reject. read the letter →

arxiv 2507.20487 v1 pith:TME3C2V7 submitted 2025-07-28 math.PR

Zhipeng Liu , Aaron Ortiz This is my paper · ORCID

classification math.PR MSC 60K35

keywords parabolicAiryprocessKPZfixedpointmultipointdistributionFredholmdeterminantextendedkernelcontourintegralAndreiefidentitynarrowwedgeinitialcondition

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 establishes that two seemingly different formulas for the finite-dimensional distributions of the parabolic Airy process are actually the same: the equal-time multipoint formula obtained from the KPZ fixed point (Theorem 1.1) is shown, after contour-integral rewriting, to equal the original Fredholm determinant with the extended Airy kernel. A reader would care because the parabolic Airy process is conjectured to be a universal limit of models in the KPZ universality class with narrow-wedge data, and a multipoint formula is only usable if it is known to describe the same process as the defining kernel. The proof supplies a chain of explicit identities, evaluating auxiliary integrals by residues, reorganizing determinants through a generalized determinant identity (Lemma 1.2), and factoring the final kernel into upper-triangular pieces. It also yields several new contour-integral formulas for the process along the way.

What carries the argument

The machinery is the factorization of the kernel as $L=-A_1(\sum_{k=0}^{m-1}(-1)^k B^k)A_2$, where $B$ is strictly upper-triangular, meaning it vanishes on and below the diagonal; this makes $B^m=0$ and $\det(I+B)$ harmless inside the Fredholm determinant, so $\det(I+L)=\det(I+B-A)$, and the latter is the conjugated Airy kernel determinant. The generalized Andreief identity (Lemma 1.2) is the combinatorial tool that replaces determinants of single-variable integrals, $\det(\int_X A_i(x)B_j(x)\,d\mu(x))$, by integrals over products of the block determinants $\det[B_i(x_j)]_{i,j\in I_k}$, and it is what lets the paper reorganize the $m$-level contour integrals from the KPZ fixed point formula.

What would settle it

Compute both sides for the first non-trivial case, $m=2$, at specific ordered points such as $\alpha_1=0$, $\alpha_2=1$, $\beta_1=\beta_2=0$: evaluate the Fredholm determinant of $K$ in Proposition 2.1 by truncating its series expansion and evaluate the extended-Airy-kernel determinant in (1.1) by a separate numerical scheme; any discrepancy beyond numerical error would disprove the stated equality.

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Extended reading notes

Core claim

The central claim is that formula (1.10), the equal-time case of the KPZ fixed point multipoint distribution with narrow-wedge initial condition, coincides with (1.1), the extended Airy kernel Fredholm determinant that defines the parabolic Airy process, for every ordered $m$-tuple $\alpha_1<\cdots<\alpha_m$. The argument passes through an intermediate Fredholm determinant $\det(I+K)$ (Proposition 2.1) whose kernel $K$ is built from contour integrals of the functions $f_i(w)=e^{-w^3/3+\alpha_i w^2+\beta_i w}$. By evaluating the auxiliary integrations, conjugating the kernel, and using $\det(I+AB)=\det(I+BA)$, the determinant is reduced to the extended Airy kernel determinant, with the Gaussian factors in (3.16)--(3.17) providing the conjugation and the final identification.

Load-bearing premise

The paper relies, without reproving it, on the multipoint distribution formula for the KPZ fixed point with narrow-wedge initial condition (Theorem 1.1) and on the absolute convergence of several contour integrals cited to other works; if either gave way, the equality of the two formulas would not follow.

Editorial extensions

If this is right

The two known multipoint formulas for the parabolic Airy process, the defining extended Airy kernel determinant and the equal-time KPZ fixed point formula, can now be used interchangeably.
Proposition 2.1 gives a new contour-integral Fredholm determinant for the process, which is simpler than the original $D$-function expansion and may be better suited to asymptotic analysis.
The intermediate identities recorded as (2.21), Lemmas 2.3 and 2.4, (3.5), and (3.15) are new explicit formulas for the same multipoint distributions.
The generalized Andreief identity stands on its own as a tool for converting Fredholm determinants into block-determinant integrals, independent of the probabilistic context.

Reading between the lines

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

The same residue-and-factorization pattern should apply to the unequal-time KPZ fixed point formula, where time differences enter only through extra Gaussian factors in the $F_i$; proving that would give a direct, limit-free derivation of the alternative multitime formula discussed in the paper.
The strict upper-triangular structure of $B$ suggests a hierarchical decomposition of the $m$-point distribution: each added spatial point contributes a new layer conjugated by Gaussian kernels, which could make asymptotic regimes such as widely spaced $\alpha_i$ tractable from the formula alone.
Replacing the narrow-wedge initial condition by other initial data in the KPZ fixed point would likely require a different $F_1$, and the proof indicates exactly where the Gaussian-factor structure is indispensable.

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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

Short version: this is a careful, honest equivalence proof. The authors show that Liu's equal-time multipoint formula for the KPZ fixed point (Liu22a) reduces, after contour surgery, to the original extended Airy kernel determinant that defines the parabolic Airy process. It does not resolve a long-open question and it does not produce a new physical phenomenon, but it closes a real gap: the two formulas were believed to agree, and now the chain of simplifications is written down in detail.

The genuinely new pieces are the intermediate contour-integral formulas, especially the operator factorization through the nilpotent matrix B that turns the determinant into det(I+B−A), and the generalized Andreief identity in Lemma 1.2. The proof of that identity is self-contained and correct; the m=1 case is the classical Andreief identity. The reduction from the multi-contour mess to a Fredholm determinant is coherent, and the final conjugation to the extended Airy kernel checks out at the operator level.

The soft spots are real but minor. The paper takes Theorem 1.1 of Liu22a as a black box; if that theorem failed, everything downstream fails. That is a reasonable starting point, since the theorem is published and the authors state plainly that they are not reproving it, but readers should know the conclusion is conditional on it. The interchange of the n-sum with the z-contours in (2.20)–(2.21) is not proved here and is referred to LZ25. The contour deformations in Lemma 2.3 and Lemma 3.1 are plausible, and I did not find a pole-crossing problem, but a few are sketched rather than fully justified; a referee should ask for those details to be filled in. None of these look load-bearing.

The citation pattern is fine. Liu22a is the natural input, the dependence is flagged transparently, and the rest of the references are standard.

Who should read it: people working on integrable probability and the KPZ fixed point who need multipoint formulas or want to convert between the two known formulas. It is not a paper for a general audience. I would send it to a serious referee. My verdict matches the reader's: accept after minor revision, with the convergence details tightened.

Referee Report

0 major / 5 minor

Summary. The paper proves that the equal-time multipoint distribution formula for the KPZ fixed point with narrow-wedge initial condition obtained in [Liu22a], specialized to τ_i = 1 and α_1 < ... < α_m, coincides with the extended Airy kernel Fredholm determinant (1.1) that defines the parabolic Airy process. The proof evaluates the z-contour integrals in the [Liu22a] formula (Lemma 2.3), reorganizes the remaining u-integrals using a generalized Andreief identity (Lemma 2.4), obtains a new contour-integral Fredholm determinant (Proposition 2.1), and then factorizes that kernel through L2(R+) and a strictly upper-triangular nilpotent operator B to identify the result with the conjugated extended Airy kernel. The paper also records several intermediate formulas for the parabolic Airy process and proves a generalized Andreief identity that is independent of the process.

Significance. If the main theorem holds, the paper closes a gap between two independent published formulas for the same object and provides a direct algebraic verification that the equal-time [Liu22a] formula indeed reproduces the original definition of the parabolic Airy process. The proof is largely self-contained modulo Theorem 1.1, which is a published result used explicitly as an input, and the algebraic reductions in Lemmas 2.3, 2.4, and Section 3 are coherent. The generalized Andreief identity is proved in full, and the paper honestly flags its dependence on convergence details from [Liu22a, LZ25]. No ad hoc parameters are introduced, and the final claim is falsifiable through the explicit equality of two contour-integral formulas.

minor comments (5)

[Section 3, after Lemma 3.2] The equality of det(I+B-A) with the conjugated extended Airy determinant (1.1) is asserted in a single sentence ("after a conjugation and a simple notation change"), but this is the main conclusion of the paper. Please spell out the conjugation, for example by setting λ = x - β_i - α_i^2 and θ = y - β_j - α_j^2, and show explicitly how the (i,j) entries of I+B-A combine to reproduce the piecewise definition of K_ext, in particular how the strictly upper-triangular B corrects the i<j entries of A. As written, a reader cannot verify the orientation switch.
[Section 2.1, after (2.5)] The claimed Hadamard-inequality proof of absolute convergence of the series (2.5) is not immediate, because for i = 1 the kernel K(1,z;1,u) decays only like |u|^{-1} along Γ_L rather than super-exponentially in the second variable. Please either give a trace-class or Hilbert-Schmidt factorization before using the series expansion, or define det(I+K) via the absolutely convergent expression obtained from (2.21) and from the cited estimates in [Liu22a, LZ25], and correct the sentence accordingly.
[Lemma 2.3(i), around (2.25)] The derivation of (2.25) suppresses the cancellation between the prefactors (1-z_i^{-1})^{n_{i+1}} and the surviving contour coefficients (-z_i/(1-z_i))^{n_{i+1}} coming from the Γout_{i,R} integrals. Please display this cancellation explicitly; as printed, (2.25) appears to omit the R-contour coefficients, and the reader must reconstruct the analyticity argument by hand.
[Sections 2.1 and 2.3] The Fredholm determinant in (2.5) is written on the contour Γ_L defined in (2.4), while the proof of Proposition 2.1 in (2.50) uses the contour Γ1_L from Definition 2.2. Please state explicitly that Γ_L may be taken to be Γ1_L, or explain that the determinant is invariant under admissible deformations of the left contour.
[Lemma 1.2] In the statement of Lemma 1.2, "µ be a measure on Γ" should presumably read "measure on X"; the symbol Γ has not been defined at that point.

Circularity Check

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No significant circularity: the paper is an independent equivalence proof between the [Liu22a] equal-time KPZ fixed point formula and the defining extended Airy kernel formula.

full rationale

The paper's central claim is the equality of two independently sourced formulas for the multipoint distribution of the parabolic Airy process: (1.10), inherited from Theorem 1.1 of [Liu22a] for the KPZ fixed point with narrow wedge initial condition, and (1.1), the defining Fredholm determinant with the extended Airy kernel. The derivation does not assume (1.1) while proving the new determinant formula: Proposition 2.1 is obtained from (1.10) by evaluating the z-contour integrals (Lemma 2.3), reorganizing the u-integrals through a generalized Andreief identity (Lemma 2.4), and identifying the resulting series with det(I+K). Section 3 then factors K through the operator B and identifies the conjugated kernel with the extended Airy kernel via direct contour computations and the known identity [Oko02, Lemma 2.6]. No parameter is fitted to the target distribution, and no step defines the input in terms of the claimed output. Theorem 1.1 from [Liu22a] is used as a black box, and convergence details are deferred to [Liu22a, LZ25], but these are technical supports from externally derived TASEP/KPZ results, not ingredients that force the conclusion by construction. Self-citation by the first author is present but not load-bearing in a circular sense, since [Liu22a] itself does not rely on the equivalence proved here.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

The paper does not introduce fitted parameters or new physical entities. It relies on published theorems for the KPZ fixed point, standard Airy-function identities, and contour-analysis principles from the integrable probability literature.

assumptions (5)

domain assumption Theorem 1.1 from [Liu22a]: the multipoint distribution formula for the KPZ fixed point with narrow wedge initial condition.
Used as the starting point for the equal-time formula (1.10); not reproved in this paper.
domain assumption The one-time marginal of the KPZ fixed point with narrow wedge at time one is the parabolic Airy process, equations (1.7) and (1.8), from [MQR21].
Bridges the KPZ fixed point formula to the parabolic Airy process.
standard math Okounkov's Airy integral identity (3.18), from [Oko02].
Used to evaluate B(i,lambda;j,theta) in Lemma 3.2.
standard math Fredholm determinant cyclicity and multiplicativity for absolutely convergent series expansions, used around (3.5) and (3.15).
Allows passing between det(I+K), det(I+L), and det(I+B-A).
domain assumption Absolute convergence and contour deformation principles for the chosen Gamma contours.
Justifies interchanging sums and integrals and deforming contours; details partly deferred to [Liu22a, LZ25].

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Pith. "Pith review of On the multipoint distribution formulas of the parabolic Airy process." pith.science (2026). https://pith.science/paper/TME3C2V7

@misc{pith2026250720487,
  author       = {Pith},
  title        = {Pith review of: On the multipoint distribution formulas of the parabolic Airy process},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/TME3C2V7}},
  note         = {Machine review of arXiv:2507.20487}
}

abstract

The parabolic Airy process is the Airy$_2$ process minus a parabola, initially defined by its finite-dimensional distributions, which are given by a Fredholm determinant formula with the extended Airy kernel. This process is also the one-time spatial marginal of the KPZ fixed point with the narrow wedge initial condition. There are two formulas for the space-time multipoint distribution of the KPZ fixed point with the narrow wedge initial condition obtained by arXiv:1906.01053 and arXiv:1907.09876. Especially, the equal-time case of arXiv:1907.09876 gives a different formula of the multipoint distribution of the parabolic Airy process. In this paper, we present a direct proof that this formula matches the one with the extended Airy kernel. Some byproducts in the proof include several new formulas for the parabolic Airy process, and a generalization of the Andreief's identity.

Figures

Figures reproduced from arXiv: 2507.20487 by the authors.

**Figure 2.** Illustration of the Γ-contours when m = 3. The two contours Γ1,L and Γ1,R are thickened. These notations are consistent with (2.1) and (2.2) when τ1 = · · · = τm = 1. We also introduce the notation of Cauchy determinant C(W; W˜ ) := det 1 wi − w˜j n i,j=1 = (−1)n(n−1)/2 Q 1≤i<j≤n (wj − wi)( ˜wj − w˜i) Qn i=1 Qn j=1(wj − w˜i) (2.16) for any n ≥ 1, and any vectors W = (w1, . . . , wn), W˜ = ( ˜w1, . . . , w˜n) ∈ C … view at source ↗

**Figure 3.** Deformation of the contours: Initially, for each 2 [PITH_FULL_IMAGE:figures/full_fig_p015_3.png] view at source ↗

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Pith tools

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

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Scaling limit of the colored ASEP and stochastic six-vertex models, 2024

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work page

[4] [4]

Macdonald processes

Alexei Borodin and Ivan Corwin. Macdonald processes. Probab. Theory Related Fields , 158(1-2):225--400, 2014

work page 2014

[5] [5]

Stochastic six-vertex model

Alexei Borodin, Ivan Corwin, and Vadim Gorin. Stochastic six-vertex model. Duke Math. J. , 165(3):563 -- 624, 2016

work page 2016

[6] [6]

On the distribution of the length of the longest increasing subsequence of random permutations

Jinho Baik, Percy Deift, and Kurt Johansson. On the distribution of the length of the longest increasing subsequence of random permutations. J. Amer. Math. Soc. , 12(4):1119--1178, 1999

1999

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The directed landscape

Duncan Dauvergne, Janosch Ortmann, and B \'a lint Vir \'a g. The directed landscape . Acta Mathematica , 229(2):201 -- 285, 2022

work page 2022

[8] [8]

Characterization of the directed landscape from the KPZ fixed point

Duncan Dauvergne and Lingfu Zhang. Characterization of the directed landscape from the KPZ fixed point. arXiv:2412.13032 , 2024

arXiv 2024