Wiener densities for the Airy line ensemble

TL;DR

This paper derives explicit formulas for the densities of the Airy line ensemble, providing bounds, large deviation principles, and tail estimates that enhance understanding of its probabilistic structure and applications in KPZ universality.

Contribution

It introduces an explicit, tractable formula for the densities of the Airy line ensemble and derives several key probabilistic estimates and bounds.

Findings

Density of the Airy line ensemble is approximately exponential in a non-negative function.

Established a large deviation principle for the Airy line ensemble.

Provided sharp tail bounds and density estimates considering line positions.

Abstract

The parabolic Airy line ensemble $\mathfrak A$ is a central limit object in the KPZ universality class and related areas. On any compact set $K = \{1, \dots, k\} \times [a, a + t]$, the law of the recentered ensemble $\mathfrak A - \mathfrak A(a)$ has a density $X_K$ with respect to the law of $k$ independent Brownian motions. We show that $$ X_K(f) = \exp \left(-\textsf{S}(f) + o(\textsf{S}(f))\right) $$ where $\textsf{S}$ is an explicit, tractable, non-negative function of $f$. We use this formula to show that $X_K$ is bounded above by a $K$-dependent constant, give a sharp estimate on the size of the set where $X_K < \epsilon$ as $\epsilon \to 0$, and prove a large deviation principle for $\mathfrak A$. We also give density estimates that take into account the relative positions of the Airy lines, and prove sharp two-point tail bounds that are stronger than those for Brownian motion. These estimates are a key input in the classification of geodesic networks in the directed landscape. The paper is essentially self-contained, requiring only tail bounds on the Airy point process and the Brownian Gibbs property as inputs.