Parabolic Airy Process

• The parabolic Airy process is defined as A₂(α)-α², serving as a canonical KPZ scaling object with explicit finite-dimensional Fredholm determinant laws.
• It emerges as the top curve of the parabolic Airy line ensemble, exhibiting Brownian Gibbs properties that model geodesic coalescence in directed polymers.
• The process underpins variational identities and extremal laws that link KPZ fixed point behavior with random matrix theory and related deformations.

The parabolic Airy process is the process

$$A(\alpha)=A_2(\alpha)-\alpha^2,\qquad \alpha\in\mathbb R,$$

where $A_2$ is the Airy$_2$ process. It is a canonical KPZ scaling object whose finite-dimensional distributions are given by a Fredholm determinant with the extended Airy kernel, and it is the one-time spatial marginal of the KPZ fixed point with narrow wedge initial condition. In the multiline formulation, it is the top curve of the parabolic Airy line ensemble, obtained by subtracting the deterministic parabola from the stationary Airy line ensemble (Liu et al., 28 Jul 2025, Dauvergne, 2023).

1. Definition and exact finite-dimensional law

A standard exact definition takes the parabolic Airy process to be $A(\alpha)=A_2(\alpha)-\alpha^2$. For ordered points $\alpha_1<\cdots<\alpha_m$ and thresholds $\beta_1,\dots,\beta_m$,

$$P\left(\bigcap_{i=1}^m \{ A (\alpha_i) \le \beta_i \}\right) = P\left(\bigcap_{i=1}^m \{ A_2 (\alpha_i) \le \beta_i+\alpha_i^2 \}\right) = \det\!\left(I-\chi K_{\mathrm{ext}}\chi\right),$$

with cutoff $$\chi(\alpha_i,x)=\mathbf 1_{(\beta_i+\alpha_i^2,\infty)}(x)$$ and extended Airy kernel

$K_{\mathrm{ext}}(\alpha_i,x;\alpha_j,y) = \begin{cases} \displaystyle \int_0^\infty e^{-z(\alpha_i-\alpha_j)}\,\Ai(x+z)\Ai(y+z)\,dz, & \alpha_i\ge \alpha_j,\[1.2em] \displaystyle -\int_{-\infty}^0 e^{-z(\alpha_i-\alpha_j)}\,\Ai(x+z)\Ai(y+z)\,dz, & \alpha_i< \alpha_j. \end{cases}$

Its one-point law is therefore

$$P(A(\alpha)\le \beta)=F_{\mathrm{GUE}}(\beta+\alpha^2),$$

so the parabola appears as a deterministic shift of the GUE Tracy–Widom variable (Liu et al., 28 Jul 2025).

This Fredholm-determinantal description is equivalent to the usual extended-kernel definition of the Airy$A_2$0 process. A direct equivalence proof between the classical extended-Airy formula and an alternative equal-time formula inherited from KPZ fixed-point formulas was given in 2025. In that formulation, the same multipoint distribution is expressed as a contour Fredholm determinant $A_2$1, and the proof identifies $A_2$2 with $A_2$3 by explicit kernel manipulations, contour deformations, and a generalized Andreief identity (Liu et al., 28 Jul 2025).

A complementary boundary-value formulation replaces the extended kernel by the Airy Hamiltonian $A_2$4. For $A_2$5,

$A_2$6

and this formulation is the basis for continuum barrier problems and parabolic-variational identities (Quastel et al., 2013).

2. KPZ fixed point, narrow wedge scaling, and last-passage origins

In KPZ theory the parabolic Airy process arises from curved, or droplet, geometry. For the one-dimensional KPZ equation with sharp wedge initial data,

$A_2$7

the exact narrow-wedge analysis isolates a deterministic growth term, a deterministic spatial parabola, and a fluctuation field $A_2$8. Under the scaling $A_2$9, $_2$0, $_2$1, the centered field satisfies

$_2$2

where $_2$3 is the finite-dimensional distribution of the Airy process. Because the deterministic term $_2$4 has already been subtracted, the limit is $_2$5; without that subtraction, the same statement is equivalently written as convergence to $_2$6 (Prolhac et al., 2011).

At the level of the KPZ fixed point, the relation becomes exact and scale-covariant. If $_2$7 denotes the KPZ fixed point with narrow wedge initial condition, then

$_2$8

Thus the parabolic Airy process is precisely the equal-time spatial field of the narrow-wedge KPZ fixed point (Liu et al., 28 Jul 2025).

In directed polymer and last-passage formulations, the same object is the curved-profile scaling limit. A canonical statement is

$_2$9

in the topology of uniform convergence on compact sets. This identifies $A(\alpha)=A_2(\alpha)-\alpha^2$0 as the genuine spatial fluctuation field in point-to-point geometry, rather than as an auxiliary recentering (Quastel et al., 2013). In exponential LPP coupling arguments, the narrow-wedge profile likewise converges to $A(\alpha)=A_2(\alpha)-\alpha^2$1, and local information about the parabolically shifted process transfers to $A(\alpha)=A_2(\alpha)-\alpha^2$2 because the quadratic term is negligible under $A(\alpha)=A_2(\alpha)-\alpha^2$3 zooming (Pimentel, 2017).

The replica-Bethe-ansatz derivation of KPZ multipoint statistics for sharp wedge also enters the history of the subject. For $A(\alpha)=A_2(\alpha)-\alpha^2$4, the finite-time $A(\alpha)=A_2(\alpha)-\alpha^2$5-point generating function in the 2011 KPZ paper is obtained under a natural factorization assumption that is not exact at finite time, but its long-time limit reproduces the standard Airy-process Fredholm determinant formula (Prolhac et al., 2011). This methodological caveat is important in tracing how exact Airy-process formulas emerged from KPZ calculations.

3. Parabolic Airy line ensembles, Brownian Gibbs structure, and Wiener densities

The parabolic Airy process is only the top curve of a richer multiline object. If $A(\alpha)=A_2(\alpha)-\alpha^2$6 denotes the stationary Airy line ensemble, the parabolic Airy line ensemble is

$A(\alpha)=A_2(\alpha)-\alpha^2$7

a random element of $A(\alpha)=A_2(\alpha)-\alpha^2$8 with almost sure ordering

$A(\alpha)=A_2(\alpha)-\alpha^2$9

The stationary ensemble is recovered by adding back $\alpha_1<\cdots<\alpha_m$0, and $\alpha_1<\cdots<\alpha_m$1 is stationary under spatial shifts and time reversal. In this notation the parabolic Airy process is simply the top line $\alpha_1<\cdots<\alpha_m$2 (Dauvergne, 2023).

This parabolic shift is the normalization in which the Brownian Gibbs property takes its cleanest form. In Hammond’s diffusion-$\alpha_1<\cdots<\alpha_m$3 convention,

$\alpha_1<\cdots<\alpha_m$4

and the factor $\alpha_1<\cdots<\alpha_m$5 is chosen so that the curves are locally Brownian with diffusion constant $\alpha_1<\cdots<\alpha_m$6. The Brownian Gibbs property says that after conditioning on the ensemble outside a strip, the conditional law inside is that of independent Brownian bridges conditioned on mutual avoidance and on staying above the next lower curve. This is the structural reason the parabolically shifted ensemble, rather than the unshifted Airy line ensemble, is central in Gibbsian and polymeric formulations (Hammond, 2016).

The most refined local comparison presently available is the Wiener-density analysis on compact spacetime patches $\alpha_1<\cdots<\alpha_m$7. Writing $\alpha_1<\cdots<\alpha_m$8 for the law of the recentered top $\alpha_1<\cdots<\alpha_m$9 curves and $\beta_1,\dots,\beta_m$0 for $\beta_1,\dots,\beta_m$1 independent Brownian motions of variance $\beta_1,\dots,\beta_m$2, one has a Radon–Nikodym derivative $\beta_1,\dots,\beta_m$3 satisfying

$\beta_1,\dots,\beta_m$4

and, in the abstract form,

$\beta_1,\dots,\beta_m$5

The action is explicit: if $\beta_1,\dots,\beta_m$6 is the “Tetris” stacking map that minimally shifts a $\beta_1,\dots,\beta_m$7-tuple into ordered position above $\beta_1,\dots,\beta_m$8 while preserving increments, then

$\beta_1,\dots,\beta_m$9

For the top curve $$P\left(\bigcap_{i=1}^m \{ A (\alpha_i) \le \beta_i \}\right) = P\left(\bigcap_{i=1}^m \{ A_2 (\alpha_i) \le \beta_i+\alpha_i^2 \}\right) = \det\!\left(I-\chi K_{\mathrm{ext}}\chi\right),$$0, this reduces to the cost of the minimal upward shift needed to keep the path above the parabolic background. The same analysis yields a uniform bound

$$P\left(\bigcap_{i=1}^m \{ A (\alpha_i) \le \beta_i \}\right) = P\left(\bigcap_{i=1}^m \{ A_2 (\alpha_i) \le \beta_i+\alpha_i^2 \}\right) = \det\!\left(I-\chi K_{\mathrm{ext}}\chi\right),$$1

for $$P\left(\bigcap_{i=1}^m \{ A (\alpha_i) \le \beta_i \}\right) = P\left(\bigcap_{i=1}^m \{ A_2 (\alpha_i) \le \beta_i+\alpha_i^2 \}\right) = \det\!\left(I-\chi K_{\mathrm{ext}}\chi\right),$$2-a.e. $$P\left(\bigcap_{i=1}^m \{ A (\alpha_i) \le \beta_i \}\right) = P\left(\bigcap_{i=1}^m \{ A_2 (\alpha_i) \le \beta_i+\alpha_i^2 \}\right) = \det\!\left(I-\chi K_{\mathrm{ext}}\chi\right),$$3, sharp lower-tail control for the density, and a small-amplitude large deviation principle with Schilder rate function

$$P\left(\bigcap_{i=1}^m \{ A (\alpha_i) \le \beta_i \}\right) = P\left(\bigcap_{i=1}^m \{ A_2 (\alpha_i) \le \beta_i+\alpha_i^2 \}\right) = \det\!\left(I-\chi K_{\mathrm{ext}}\chi\right),$$4

on absolutely continuous paths (Dauvergne, 2023).

Brownian regularity results complement these density formulas. After affine recentering on a fixed interval $$P\left(\bigcap_{i=1}^m \{ A (\alpha_i) \le \beta_i \}\right) = P\left(\bigcap_{i=1}^m \{ A_2 (\alpha_i) \le \beta_i+\alpha_i^2 \}\right) = \det\!\left(I-\chi K_{\mathrm{ext}}\chi\right),$$5, the $$P\left(\bigcap_{i=1}^m \{ A (\alpha_i) \le \beta_i \}\right) = P\left(\bigcap_{i=1}^m \{ A_2 (\alpha_i) \le \beta_i+\alpha_i^2 \}\right) = \det\!\left(I-\chi K_{\mathrm{ext}}\chi\right),$$6-th parabolically shifted Airy curve has Radon–Nikodym derivative $$P\left(\bigcap_{i=1}^m \{ A (\alpha_i) \le \beta_i \}\right) = P\left(\bigcap_{i=1}^m \{ A_2 (\alpha_i) \le \beta_i+\alpha_i^2 \}\right) = \det\!\left(I-\chi K_{\mathrm{ext}}\chi\right),$$7 with respect to Brownian bridge such that

$$P\left(\bigcap_{i=1}^m \{ A (\alpha_i) \le \beta_i \}\right) = P\left(\bigcap_{i=1}^m \{ A_2 (\alpha_i) \le \beta_i+\alpha_i^2 \}\right) = \det\!\left(I-\chi K_{\mathrm{ext}}\chi\right),$$8

and hence $$P\left(\bigcap_{i=1}^m \{ A (\alpha_i) \le \beta_i \}\right) = P\left(\bigcap_{i=1}^m \{ A_2 (\alpha_i) \le \beta_i+\alpha_i^2 \}\right) = \det\!\left(I-\chi K_{\mathrm{ext}}\chi\right),$$9 for every $$\chi(\alpha_i,x)=\mathbf 1_{(\beta_i+\alpha_i^2,\infty)}(x)$$0. This yields Brownian-bridge comparison estimates, Gaussian-type suprema bounds up to lower-order corrections, and Brownian $$\chi(\alpha_i,x)=\mathbf 1_{(\beta_i+\alpha_i^2,\infty)}(x)$$1-Hölder modulus behavior for the top curve (Hammond, 2016). A separate LPP-coupling approach shows the local scaling limit

$$\chi(\alpha_i,x)=\mathbf 1_{(\beta_i+\alpha_i^2,\infty)}(x)$$2

and the same local limit applies to $$\chi(\alpha_i,x)=\mathbf 1_{(\beta_i+\alpha_i^2,\infty)}(x)$$3, since the parabola contributes only $$\chi(\alpha_i,x)=\mathbf 1_{(\beta_i+\alpha_i^2,\infty)}(x)$$4 under this scaling (Pimentel, 2017).

4. Variational identities, extremal laws, and deformations

A major part of the subject concerns functionals of the parabolic Airy process rather than its finite-dimensional distributions. The most classical identity is the full-line supremum law

$$\chi(\alpha_i,x)=\mathbf 1_{(\beta_i+\alpha_i^2,\infty)}(x)$$5

which identifies point-to-line curved KPZ geometry with GOE Tracy–Widom fluctuations. The same framework yields a continuum barrier formula

$$\chi(\alpha_i,x)=\mathbf 1_{(\beta_i+\alpha_i^2,\infty)}(x)$$6

and for the parabolic barrier $$\chi(\alpha_i,x)=\mathbf 1_{(\beta_i+\alpha_i^2,\infty)}(x)$$7 the Brownian-bridge representation of $$\chi(\alpha_i,x)=\mathbf 1_{(\beta_i+\alpha_i^2,\infty)}(x)$$8 becomes explicitly tractable. The joint law of the maximizer

$$\chi(\alpha_i,x)=\mathbf 1_{(\beta_i+\alpha_i^2,\infty)}(x)$$9

and the maximum

$K_{\mathrm{ext}}(\alpha_i,x;\alpha_j,y) = \begin{cases} \displaystyle \int_0^\infty e^{-z(\alpha_i-\alpha_j)}\,\Ai(x+z)\Ai(y+z)\,dz, & \alpha_i\ge \alpha_j,\[1.2em] \displaystyle -\int_{-\infty}^0 e^{-z(\alpha_i-\alpha_j)}\,\Ai(x+z)\Ai(y+z)\,dz, & \alpha_i< \alpha_j. \end{cases}$0

also admits an explicit determinant formula (Quastel et al., 2013).

Restricting the supremum to a half-line produces the Airy$K_{\mathrm{ext}}(\alpha_i,x;\alpha_j,y) = \begin{cases} \displaystyle \int_0^\infty e^{-z(\alpha_i-\alpha_j)}\,\Ai(x+z)\Ai(y+z)\,dz, & \alpha_i\ge \alpha_j,\[1.2em] \displaystyle -\int_{-\infty}^0 e^{-z(\alpha_i-\alpha_j)}\,\Ai(x+z)\Ai(y+z)\,dz, & \alpha_i< \alpha_j. \end{cases}$1 crossover. For every $K_{\mathrm{ext}}(\alpha_i,x;\alpha_j,y) = \begin{cases} \displaystyle \int_0^\infty e^{-z(\alpha_i-\alpha_j)}\,\Ai(x+z)\Ai(y+z)\,dz, & \alpha_i\ge \alpha_j,\[1.2em] \displaystyle -\int_{-\infty}^0 e^{-z(\alpha_i-\alpha_j)}\,\Ai(x+z)\Ai(y+z)\,dz, & \alpha_i< \alpha_j. \end{cases}$2,

$K_{\mathrm{ext}}(\alpha_i,x;\alpha_j,y) = \begin{cases} \displaystyle \int_0^\infty e^{-z(\alpha_i-\alpha_j)}\,\Ai(x+z)\Ai(y+z)\,dz, & \alpha_i\ge \alpha_j,\[1.2em] \displaystyle -\int_{-\infty}^0 e^{-z(\alpha_i-\alpha_j)}\,\Ai(x+z)\Ai(y+z)\,dz, & \alpha_i< \alpha_j. \end{cases}$3

where $K_{\mathrm{ext}}(\alpha_i,x;\alpha_j,y) = \begin{cases} \displaystyle \int_0^\infty e^{-z(\alpha_i-\alpha_j)}\,\Ai(x+z)\Ai(y+z)\,dz, & \alpha_i\ge \alpha_j,\[1.2em] \displaystyle -\int_{-\infty}^0 e^{-z(\alpha_i-\alpha_j)}\,\Ai(x+z)\Ai(y+z)\,dz, & \alpha_i< \alpha_j. \end{cases}$4. As $K_{\mathrm{ext}}(\alpha_i,x;\alpha_j,y) = \begin{cases} \displaystyle \int_0^\infty e^{-z(\alpha_i-\alpha_j)}\,\Ai(x+z)\Ai(y+z)\,dz, & \alpha_i\ge \alpha_j,\[1.2em] \displaystyle -\int_{-\infty}^0 e^{-z(\alpha_i-\alpha_j)}\,\Ai(x+z)\Ai(y+z)\,dz, & \alpha_i< \alpha_j. \end{cases}$5 this family tends to $K_{\mathrm{ext}}(\alpha_i,x;\alpha_j,y) = \begin{cases} \displaystyle \int_0^\infty e^{-z(\alpha_i-\alpha_j)}\,\Ai(x+z)\Ai(y+z)\,dz, & \alpha_i\ge \alpha_j,\[1.2em] \displaystyle -\int_{-\infty}^0 e^{-z(\alpha_i-\alpha_j)}\,\Ai(x+z)\Ai(y+z)\,dz, & \alpha_i< \alpha_j. \end{cases}$6, and as $K_{\mathrm{ext}}(\alpha_i,x;\alpha_j,y) = \begin{cases} \displaystyle \int_0^\infty e^{-z(\alpha_i-\alpha_j)}\,\Ai(x+z)\Ai(y+z)\,dz, & \alpha_i\ge \alpha_j,\[1.2em] \displaystyle -\int_{-\infty}^0 e^{-z(\alpha_i-\alpha_j)}\,\Ai(x+z)\Ai(y+z)\,dz, & \alpha_i< \alpha_j. \end{cases}$7 it tends to $K_{\mathrm{ext}}(\alpha_i,x;\alpha_j,y) = \begin{cases} \displaystyle \int_0^\infty e^{-z(\alpha_i-\alpha_j)}\,\Ai(x+z)\Ai(y+z)\,dz, & \alpha_i\ge \alpha_j,\[1.2em] \displaystyle -\int_{-\infty}^0 e^{-z(\alpha_i-\alpha_j)}\,\Ai(x+z)\Ai(y+z)\,dz, & \alpha_i< \alpha_j. \end{cases}$8; for each fixed $K_{\mathrm{ext}}(\alpha_i,x;\alpha_j,y) = \begin{cases} \displaystyle \int_0^\infty e^{-z(\alpha_i-\alpha_j)}\,\Ai(x+z)\Ai(y+z)\,dz, & \alpha_i\ge \alpha_j,\[1.2em] \displaystyle -\int_{-\infty}^0 e^{-z(\alpha_i-\alpha_j)}\,\Ai(x+z)\Ai(y+z)\,dz, & \alpha_i< \alpha_j. \end{cases}$9, its right tail has the same $$P(A(\alpha)\le \beta)=F_{\mathrm{GUE}}(\beta+\alpha^2),$$0 decay rate at the level of the paper’s upper bound (Quastel et al., 2011).

Finite-rank deformations of the parabolic Airy functional arise from the Airy process with wanderers. The central law is

$$P(A(\alpha)\le \beta)=F_{\mathrm{GUE}}(\beta+\alpha^2),$$1

which reduces to $$P(A(\alpha)\le \beta)=F_{\mathrm{GUE}}(\beta+\alpha^2),$$2 when $$P(A(\alpha)\le \beta)=F_{\mathrm{GUE}}(\beta+\alpha^2),$$3. This law admits equivalent descriptions by a Fredholm determinant, a Painlevé II/Lax-pair formula, a KPZ fixed-point marginal, a Bloemendal–Virág-type PDE, and a KdV characterization. It also appears as the scaling limit of the maximal height of non-intersecting Brownian bridges with finitely many outlier endpoints (Liechty et al., 2020).

Stationary KPZ introduces a Brownian deformation of the parabolic Airy variational problem. The paper on two-time KPZ memory studies

$$P(A(\alpha)\le \beta)=F_{\mathrm{GUE}}(\beta+\alpha^2),$$4

identifying it with $$P(A(\alpha)\le \beta)=F_{\mathrm{GUE}}(\beta+\alpha^2),$$5, and more generally

$$P(A(\alpha)\le \beta)=F_{\mathrm{GUE}}(\beta+\alpha^2),$$6

at the one-point level. The corresponding maximum has the Baik–Rains or extended Baik–Rains distribution, and the argmax distribution is exactly the KPZ scaling function $$P(A(\alpha)\le \beta)=F_{\mathrm{GUE}}(\beta+\alpha^2),$$7. The same paper derives a joint law with $$P(A(\alpha)\le \beta)=F_{\mathrm{GUE}}(\beta+\alpha^2),$$8 and, from it, the large-time-separation KPZ correlation constant

$$P(A(\alpha)\le \beta)=F_{\mathrm{GUE}}(\beta+\alpha^2),$$9

Its derivation uses replica Bethe ansatz together with a decoupling assumption in the large-time limit, a qualification explicitly stated in the paper (Doussal, 2017).

5. Coupled parabolic Airy processes and the Airy sheet

The parabolic Airy process acquires a genuinely multipoint geometry when coupled through the Airy sheet. In Brownian last-passage scaling, the time-one Airy sheet $A_2$00 is the limit of the centered weights $A_2$01, and for fixed $A_2$02 the parabolically adjusted profile

$A_2$03

is an Airy$A_2$04-type process in the paper’s normalization. Equivalently, $A_2$05 itself is an Airy$A_2$06 profile with the parabola already subtracted (Basu et al., 2019).

The simplest nontrivial coupling is the difference

$A_2$07

This process is almost surely continuous and non-decreasing, and it is constant in a random neighbourhood of almost every $A_2$08. The exceptional set where it fails to be locally constant has Hausdorff dimension $A_2$09. Geometrically, this set corresponds to endpoints for which the maximizing paths from $A_2$10 and $A_2$11 to $A_2$12 fail to have already coalesced before reaching the endpoint (Basu et al., 2019).

A sharper description uses the parabolic Airy sheet $A_2$13 and the difference profile

$A_2$14

On every compact interval $A_2$15, the increment process

$A_2$16

is absolutely continuous, in law, with respect to Brownian local time at zero for Brownian motion of rate four, again in the sense of increments. At random points of increase—such as the first point of increase after a deterministic location, or a point sampled from the Stieltjes measure induced by $A_2$17—one has the local limit

$A_2$18

where $A_2$19 is Brownian local time of rate four. This identifies the singular monotone coupling between two parabolic Airy$A_2$20 profiles with Brownian local time, not merely at the level of Hausdorff dimension but at the level of tangent-process structure (Ganguly et al., 2021).

These results suggest a precise geometric principle: the coupling of parabolic Airy processes through the Airy sheet is governed by geodesic coalescence, and the residual increments of the difference profile arise from local competition between the top two Airy lines. In the local theorem this is made explicit through a running-maximum representation involving $A_2$21, where $A_2$22 is the parabolic Airy line ensemble (Ganguly et al., 2021).

6. Terminology, model variants, and boundary analogues

The phrase “parabolic Airy process” is not completely uniform across the literature. Some papers define the process directly as $A_2$23, while others say “Airy process” for $A_2$24 after the finite-time deterministic parabola has already been subtracted from the KPZ height field. In particular, the 2011 KPZ paper studies the already centered field $A_2$25, so its scaling limit is written as $A_2$26; without that centering, the same statement becomes convergence to $A_2$27 (Prolhac et al., 2011). Likewise, Hammond’s terminology is “Airy line ensemble after the subtraction of a parabola” or “parabolically shifted ensemble,” with the preferred Brownian normalization

$A_2$28

for the top curve (Hammond, 2016).

The same parabolic shift appears outside KPZ growth and LPP in other edge-scaling problems. In the two-periodic Aztec diamond with a thin mesoscopic rough region, the last path at the rough–smooth boundary satisfies

$A_2$29

equivalently $A_2$30 in finite-dimensional distributions, under the scaling $A_2$31 with $A_2$32 (Johansson et al., 2023).

At the level of random-matrix cusp-to-edge crossover, the Pearcey process converges to the unshifted Airy process after centering at the deterministic edge location

$A_2$33

and scaling by $A_2$34. That centering is distinct from subtracting $A_2$35, but after matching conventions it leads back to the same Airy$A_2$36/parabolic-Airy dichotomy familiar in KPZ theory (Adler et al., 2010).

A boundary analogue now exists in half-space KPZ. The pinned half-space Airy line ensemble $A_2$37 on $A_2$38 is the $A_2$39 limit of critical half-space Airy line ensembles, and after parabolic shift

$A_2$40

it satisfies a one-sided Brownian Gibbs property described by pairwise pinned Brownian motions. Far from the origin it converges to the standard Airy line ensemble, while at the origin its law is that of the ordered eigenvalues of the stochastic Airy operator at $A_2$41, each with doubled multiplicity. This places the parabolic Airy formalism within a broader family of boundary-sensitive KPZ scaling objects (Dimitrov et al., 8 Jan 2026).

The modern picture is therefore two-layered. As a single process, the parabolic Airy process is $A_2$42, with exact multipoint laws, variational identities, and KPZ fixed-point interpretations. As part of a line or sheet ensemble, it is the top curve of a more rigid Brownian-Gibbs object whose multiline interactions govern endpoint repulsion, geodesic coalescence, and boundary effects (Liu et al., 28 Jul 2025, Dauvergne, 2023).