2023/01/31 by Duncan Dauvergne, Dauvergne, Duncan · 1 citation
Mathematics · #60K35 #FOS: Mathematics #Geometry and complex manifolds #Probability (math.PR) #Random Matrices and Applications #Stochastic processes and statistical mechanics
paper · pdf · doi:10.48550/arxiv.2302.00097
openalex publication_date 2023/01/31 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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, …, k\ × [a, a + t], the law of the recentered ensemble \mathfrak A - \mathfrak A(a) has a density XK with respect to the law of k independent Brownian motions. We show that XK(f) = exp (-\textsfS(f) + o(\textsfS(f))) where \textsfS is an explicit, tractable, non-negative function of f. We use this formula to show that XK is bounded above by a K-dependent constant, give a sharp estimate on the size of the set where XK < ε as ε→ 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.