  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.
