Paper

Strong Characterization for the Airy Line Ensemble

Huang, Jiaoyang, Aggarwal, Amol

2023

Overview

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.

• Publisher: Cornell University Library, arXiv.org
• Subject: Parabolas