Actions in the Airy line ensemble and convergence to the Airy sheet
Balint Virag, Xuan Wu
TL;DR
The paper addresses convergence to the Airy sheet, a fundamental component of the directed landscape in the KPZ class, by introducing action-based representations that encode distances to infinity in the Airy line ensemble. The authors develop a modular framework of $\mathcal{A}$-actions and action representations, proving that convergence to the Airy line ensemble implies convergence to the Airy sheet without relying on intricate analyses of pre-limiting Busemann functions. They establish general convergence theorems for last passage and polymer models, develop discrete-to-continuous limits, and apply the framework to Brownian LPP, O'Connell--Yor semidiscrete polymers, log-gamma polymers, and the KPZ equation, yielding unified, conceptually transparent proofs of Airy sheet convergence. The results provide a versatile toolkit for deriving the full scaling limit, the directed landscape, in broad KPZ-class settings. Overall, the paper offers a modular, broadly applicable method that bypasses case-by-case Busemann analyses and clarifies the link between Airy-line ensemble limits and Airy-sheet structure with significant implications for KPZ universality.
Abstract
Actions in the Airy line ensemble represent distances from an infinitely far object. We characterize the Airy sheet by S(x,.)=T^x(.,1), where T^x is the unique action in the Airy line ensemble satisfying a growth condition depending on x. This provides a new simple framework for establishing convergence to the Airy sheet. We present simple conceptual proofs of such results in the case of Brownian last passage, the O'Connell-Yor semidiscrete polymer, the log-gamma polymer and the KPZ equation.