Strong Characterization for the Airy Line Ensemble

TL;DR

This work provides a sharp, quantitative characterization of the parabolic Airy line ensemble via a Brownian Gibbs framework. By establishing a sequence of on-scale estimates, global laws, curvature approximations, and Airy-statistics, the authors demonstrate that any Brownian Gibbsian line ensemble whose top curve tracks a parabola must coincide (up to an affine shift) with the parabolic Airy line ensemble. In the translation-invariant parabola-shift case, this yields the Okounkov–Sheffeld and Corwin–Hammond predictions that the edge limit is the Airy line ensemble. The analysis hinges on gap monotonicity, Dyson Brownian motion couplings, and boundary-removal couplings to transfer edge-boundary information into universal Airy statistics, including exact Airy gaps and the Airy line ensemble structure. The results reinforce the KPZ-edge universality by providing a strong, axiomatic route to Airy statistics under a parabolic decay condition, with potential implications for edge limits in non-integrable KPZ models.

Abstract

In this paper we show that a Brownian Gibbsian line ensemble whose top curve approximates a parabola must be given by the parabolic Airy line ensemble. More specifically, we prove that if $\boldsymbol{\mathcal{L}} = (\mathcal{L}_1, \mathcal{L}_2, \ldots )$ is a line ensemble satisfying the Brownian Gibbs property, such that for any $\varepsilon > 0$ there exists a constant $\mathfrak{K} (\varepsilon) > 0$ with $$\mathbb{P} \Big[ \big| \mathcal{L}_1 (t) + 2^{-1/2} t^2 \big| \le \varepsilon t^2 + \mathfrak{K} (\varepsilon) \Big] \ge 1 - \varepsilon, \qquad \text{for all $t \in \mathbb{R}$},$$ then $\boldsymbol{\mathcal{L}}$ is the parabolic Airy line ensemble, up to an independent affine shift. Specializing this result to the case when $\boldsymbol{\mathcal{L}} (t) + 2^{-1/2} t^2$ is translation-invariant confirms a prediction of Okounkov and Sheffield from 2006 and Corwin-Hammond from 2014.

Figures (45)

• Figure 1.1: Depicted above is an example of Brownian Gibbsian line ensemble, where the red curves can be resampled in the shaded region.
• Figure 1.2:
• Figure 1.3: Shown to the left is a depiction for height monotonicity. Shown to the right is a depiction for gap monotonicity
• Figure 1.4: Shown above are the three scenarios discussed in \ref{['Location0']}, where the black curves are of $\bm{\mathcal{L}}$; the red ones are the watermelons we eventually compare them to; and the orange one is the parabola that $\mathcal{L}_1$ should be close to, by \ref{['l1t']}. On the left, $\mathcal{L}_j$ cannot be too low at time $t_0$ (even after pushing some curves in $\bm{\mathcal{L}}$ down to form the red watermelon). The curve $\mathcal{L}_1$ fails to approximate the orange parabola on the middle (where it is too high, even after pushing some curves in $\bm{\mathcal{L}}$ down to form the red watermelon) and on the right (where it is too low, even after pushing some curves in $\bm{\mathcal{L}}$ up to form the red watermelon).
• Figure 1.5: Depicted above is the boundary removal coupling.

Theorems & Definitions (364)

• Definition 2.1
• Definition 2.2
• Definition 2.3
• Definition 2.4
• Lemma 2.5: PLE
• Lemma 2.6: ELE
• Definition 2.7
• Theorem 2.9
• Remark 2.10
• Corollary 2.11