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.
Categories: math.PR cond-mat.stat-mech math-ph math.MP