Bulk properties of the Airy line ensemble

Abstract

The Airy line ensemble is a central object in random matrix theory and last passage percolation defined by a determinantal formula. The goal of this paper is to provide a set of tools which allow for precise probabilistic analysis of the Airy line ensemble. The two main theorems are a representation in terms of independent Brownian bridges connecting a fine grid of points, and a modulus of continuity result for all lines. Along the way, we give tail bounds and moduli of continuity for nonintersecting Brownian ensembles, and a quick proof of tightness for Dyson's Brownian motion converging to the Airy line ensemble.

arXiv comment43 pages, 4 figures. The title has changed from the previous version (to `Bulk properties...' from `Basic properties...'). Journal version available at https://projecteuclid.org/journals/annals-of-probability/volume-49/issue-4

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• The directed landscape
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• Last Passage Percolation with a Defect Line and the Solution of the Slow Bond Problem
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Cited by in corpus (12)

• The directed landscape
• Uniform convergence to the Airy line ensemble
• Three-halves variation of geodesics in the directed landscape
• The heat and the landscape I
• Disjoint optimizers and the directed landscape
• Brownian regularity for the KPZ line ensemble
• Tightness of Bernoulli Gibbsian line ensembles
• Characterization of $H$-Brownian Gibbsian line ensembles
• Uniqueness, mixing, and optimal tails for Brownian line ensembles with geometric area tilt
• Short- and long-time path tightness of the continuum directed random polymer
• Edge scaling limit of Dyson Brownian motion at equilibrium for general $\beta \ge 1$
• A convergence framework for Airy${}_{\beta }$ line ensemble via pole evolution