Actions in the Airy line ensemble and convergence to the Airy sheet

TL;DR

This paper characterizes the Airy sheet through actions in the Airy line ensemble, offering a new framework for proving convergence in models like Brownian last passage and the KPZ equation.

Contribution

It introduces a novel characterization of the Airy sheet via actions, simplifying convergence proofs across multiple models.

Findings

Unified framework for convergence to the Airy sheet

Simplified proofs for Brownian last passage and KPZ models

New insights into the structure of the Airy line ensemble

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.