Airy Sheet: Scaling Limits & Optical Advances

• Airy sheet is a two-parameter random field that serves as the universal scaling limit in KPZ-directed polymer and last-passage percolation models.
• It exhibits fractal and Brownian-like local fluctuations, revealing insights into geodesic coalescence and the 'devil’s staircase' profile in the KPZ class.
• In optics, the Airy sheet inspires self-accelerating, quasi-diffraction-free light beams, enhancing light-sheet microscopy with improved field of view and contrast.

The Airy sheet is a central two-parameter random field arising in the paper of scaling limits for directed random growth models and optical propagation-invariant beams. In probability theory and mathematical physics, it is the universal scaling limit (in both geometry and statistics) of last-passage percolation (LPP) and polymer models belonging to the Kardar-Parisi-Zhang (KPZ) universality class. In physical optics and microscopy, the “Airy sheet” refers to a class of self-accelerating, quasi-diffraction-free light fields whose transverse profiles are determined by the Airy function, yielding advantageous propagation and imaging properties. This entry reviews the Airy sheet’s construction and properties in the KPZ context, describes mathematical characterizations and fractal features, and details its physical realization and utility in advanced light-sheet microscopy.

1. The Airy Sheet in the KPZ Universality Class

The Airy sheet $\mathcal{S}(x,y)$ is a continuous $\mathbb{R}^2\to\mathbb{R}$ random field constructed as the two-parameter scaling limit of energy profiles in (1+1)D LPP and directed-polymers models with appropriate KPZ scaling. For models such as exponential or Brownian LPP, maximize paths (or “geodesics”) accrue total energy whose scaling fluctuations are $n^{1/3}$ and lateral deviations are $n^{2/3}$ for length-$n$ polymers. After appropriate subtractive, multiplicative, and spatial rescalings, and a critical parabolic “centering” (removing leading deterministic correlations of the form $-c(y-x)^2/(t-s)$), one obtains a limiting universal process $W_\infty$ (“space-time Airy sheet”) (Basu et al., 2019).

Explicitly, in Brownian LPP, the (centered and scaled) passage time from $(x,s)$ to $(y,t)$ is

$$W_n(x,s;\,y,t) = 2^{-1/2} n^{-1/3} \left[ M(x_n,s_n \to y_n, t_n) - 2n(t-s) - 2n^{2/3}(y-x) \right],$$

with $M$ the maximal path energy and $(x_n,s_n)$, $(y_n,t_n)$ are rescaled coordinates. The Airy sheet is defined as

$$W_\infty(x, s; y, t) = \lim_{n\to\infty} \left( W_n(x, s; y, t) + \frac{1}{2\sqrt{2}} \frac{(y-x)^2}{t-s} \right) - \frac{1}{2\sqrt{2}} \frac{(y-x)^2}{t-s}.$$

All known integrable models whose macroscopic statistics lie in the KPZ class—Brownian, Poissonian, log-gamma, and others—converge to the same Airy sheet after this transformation and scaling (Virag et al., 14 Nov 2025).

Inverse transformation yields the KPZ directed landscape, a conjecturally universal object describing time-space geodesics and their joint distributions.

2. Mathematical Structure and Characterizations

2.1. Line Ensemble and Brownian Gibbs Characterization

The Airy sheet is intimately connected to the parabolic Airy line ensemble $$\mathcal{A} = (\mathcal{A}(y, k): y \in \mathbb{R}, k \in \mathbb{N})$$, the scaling limit at the spectral edge of Dyson Brownian motion. Each curve $\mathcal{A}(\cdot, k)$ is continuous, and the ensemble is indexed so that parabolically shifted fields $(\mathcal{A}(y, k) + y^2)$ are determinantal processes with extended Airy kernel and satisfy the Brownian Gibbs property (on each finite set of indices and intervals, the conditional increments are non-intersecting Brownian bridges) (Ganguly et al., 2021, Virag et al., 14 Nov 2025).

The Airy sheet $\mathcal{S}$ can be expressed in terms of last-passage values through this line ensemble. For each $y\in \mathbb{R}$, one has $\mathcal{S}(0, x) = \mathcal{A}_1(x)$ (top curve), and increments of the sheet relate to suitably coupled last-passage values constrained by geodesic geometry in the ensemble.

2.2. Action Recursion Formulation

A recent approach defines the Airy sheet via “actions”—distance functions $T: \mathbb{R}\times\mathbb{N} \to \mathbb{R}$ on the line ensemble solving an action recurrence: $$T(y, k) = \left( T(x, k) + A(y, k) - A(x, k) \right) \vee \sup_{z\in[x,y]} \left(T(z, k+1) + A(y, k) - A(z, k) \right),$$ where $A$ is the random environment, and a quadratic growth/no-overhang condition at $\pm\infty$ selects a unique solution. The Airy sheet appears through the variational identity

$$\mathcal{S}(f, y) = \sup_{x \geq 0} \left( f(x) + \mathcal{S}(x, y) \right),$$

for $f$ upper-semicontinuous. This action recursion unites the construction in Brownian, log-gamma, semidiscrete polymer, and KPZ settings (Virag et al., 14 Nov 2025).

3. Local and Fractal Structure of the Airy Difference Profile

A central object in the paper of the Airy sheet's geometry is the difference profile

$D(x) = \mathcal{S}(1, x) - \mathcal{S}(-1, x).$

This function serves as a “devil’s staircase” that is almost surely continuous, non-decreasing, and locally constant except on a random set of Hausdorff dimension $1/2$ (Basu et al., 2019, Ganguly et al., 2021). The set of non-constant points coincides (in fractal dimension) with the zero set of Brownian motion.

For any compact interval, the increments of $D(x)$ are absolutely continuous to those of rate-4 Brownian local time (Ganguly et al., 2021). At a typical point of increase $\tau$,

$$\varepsilon^{-1/2} (D(\tau+\varepsilon t) - D(\tau)) \xrightarrow[\varepsilon\to 0]{\mathcal{L}} L(t),$$

where $L(t)$ is local time of rate-4 Brownian motion. This fractal structure quantifies the competition interface between geodesics from different origins in the scaling limit and is intimately tied to rare events of geodesic non-coalescence.

Table: Airy Difference Profile Properties

Property	Description
Monotonicity	$D(x)$ is almost surely continuous, non-decreasing
Local Constancy	$D(x)$ is locally constant for Lebesgue-a.e.\ $x$
Exceptional Set (Non-Constancy)	Hausdorff dimension $1/2$
Local Fluctuations	$D(x)$ increments map to Brownian local time increments (rate-4)

4. Local Fluctuations and Brownian Sheet Behavior

On infinitesimal scales, the Airy sheet exhibits local fluctuations governed by additive Brownian behavior. Any continuous version $\mathcal{S}$ satisfies, for arbitrarily small $\varepsilon > 0$,

$$\mathcal{S}^\varepsilon(u,v) = \frac{ \mathcal{S}(\varepsilon u, \varepsilon v) - \mathcal{S}(0,0) }{ \sqrt{\varepsilon} } \xrightarrow[ \varepsilon \to 0 ]{d} 2 (B_1(u) + B_2(v) ),$$

with $B_1, B_2$ independent standard Brownian motions (Pimentel, 2017). This quantifies the universal $1/2$-power local roughness. The proof proceeds via coupling methods and the sandwiching of sheet increments between Brownian approximants.

Tightness is controlled by explicit exit-point localization results and modulus-of-continuity criteria, establishing the existence of continuous versions of the Airy sheet almost surely.

5. Geometric and Scaling Properties; Integrable Models

The Airy sheet is conjecturally the universal scaling limit for LPP and polymer models with KPZ exponents. Under the $1/3, 2/3$ scalings, energy fluctuations scale as $n^{1/3}$ and geodesic deviations as $n^{2/3}$. The scaling field, after parabolic adjustment, converges (in the sense of locally uniform convergence on compacts) to the Airy sheet.

Four explicit constructions prove convergence to the Airy sheet (Virag et al., 14 Nov 2025):

• Brownian LPP: Ensemble of $n$ independent Brownian motions, maximal energy profiles converge to Airy sheet.
• O’Connell-Yor Polymer: Semidiscrete directed polymer, quantum Toda Gibbs measures.
• Log-Gamma Polymer: Discrete, inverse-gamma weights; tropical RSK construction.
• KPZ Equation: KPZ line ensembles derived via O’Connell-Yor Gibbs resampling at large time.

All cases deploy action representations satisfying the action recursion, and under suitable tightness and symmetry conditions, the top line converges in law to the Airy sheet.

6. Airy Sheets in Physical Optics: Light-Sheet Microscopy

In optical physics, the “Airy sheet” designates a class of propagation-invariant, self-accelerating beams whose transverse intensity is given by the Airy function. These are realized by imparting a cubic phase in the pupil plane of a focusing element, yielding a light field with robust “main lobe” and significant side lobe suppression over extended propagation lengths.

• Theoretical Formulation: In the paraxial regime, the field is generated as $$E_0(x,z) \propto \operatorname{Ai}(\xi) \exp(a \xi) \exp(i \varphi_0(x,z))$$ with $\xi = z/\alpha - x^2/(4\alpha^2 k^2)$, and the pupil phase is $P(u) = \exp(2\pi i \alpha u^3)$ with $u$ the normalized pupil coordinate.
• Attenuation Compensation: In absorbing or scattering tissue, intensity decay along $z$ is compensated by pre-shaping the beam to grow as $\exp(C_\text{attn} z)$, implemented by an amplitude mask $\exp[\sigma(u-1)]$ in the pupil, where $$\sigma \approx C_\text{attn} z_\text{max} / \mathrm{FOV}$$ (Nylk et al., 2017).
• Biaxial Acceleration: For field curvature matching in miniaturized or high-NA detection geometries, a biaxial cubic-plus-quadratic phase yields a spherical-shell light sheet with curvature radius $r_d$ (Taege et al., 2023).
• Planar Airy Beams: Rotating the cubic phase mask yields a planar propagation-invariant beam suitable for two-photon microscopy with highly uniform excitation and PSF (Hosny et al., 2020).

Table: Performance Metrics in Attenuation-Compensated Airy Sheet Microscopy

Metric	Observed Improvement
SBR	Up to 5-fold across whole field-of-view
CNR	Up to 8-fold at depth
Depth Penetration	FOV/detection depth increased by ~100% (≈150 μm) in thick tissue without adaptive optics

The optical Airy sheet’s chief features—quasi-diffraction-free propagation, self-healing, high-contrast in scattering media, and tunable field curvature—have been leveraged for large-volume, high-resolution, low-phototoxicity imaging in biological microscopy, and are compatible with phase-modulated implementation and high-throughput micro-optics (Nylk et al., 2017, Taege et al., 2023, Hosny et al., 2020).

7. Open Directions and Significance

Mathematically, the Airy sheet is the canonical two-parameter random field for the KPZ universality class, encoding universal joint fluctuations of maximizing geodesic energies and the geometry of path coalescence and competition. Open problems include the explicit construction and classification of its metric and geodesic structures, extension to non-integrable environments, and connection to the full directed landscape. The recent “action” framework unifies various models and simplifies the path from one-point convergence to global coupling (Virag et al., 14 Nov 2025).

In physical realization, the Airy sheet concept has expanded imaging capabilities in light-sheet microscopy, allowing extended FOV, penetration in scattering media, compensation for absorption, and customized focal geometries without complex optical correction. These methods merge the mathematical elegance of the Airy function with state-of-the-art nanofabrication and computational optics.