Brownian motion

Physics

Neuroscience

Mathematics

Random matrix

Real line

Biology

Politics

Statistical physics

Statistics

Mathematical analysis

Political science

Eigenvalues and eigenvectors

Line (geometry)

Geometry

Representation (politics)

Percolation (cognitive psychology)

Probabilistic logic

Quantum mechanics

The Annals of Probability

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.

Bulk properties of the Airy line ensemble