Long-time asymptotics for Airy wanderer line ensembles

TL;DR

This paper studies the long-time behavior of Airy wanderer line ensembles, revealing their organization into slope-based groups and their convergence to Dyson Brownian motion or Airy line ensembles, depending on parameters.

Contribution

It provides a detailed characterization of the asymptotic fluctuations of Airy wanderer line ensembles and establishes new limit theorems for their behavior at the edges.

Findings

Ensembles organize into groups with common asymptotic slopes.

Each group converges to a Dyson Brownian motion after scaling.

When finitely many slopes are positive, upper curves follow parabolic trajectories, others remain flat.

Abstract

We investigate the long-time behavior of the Airy wanderer line ensembles, an infinite-parameter family of Brownian Gibbsian line ensembles arising as edge-scaling limits of inhomogeneous models in the Kardar--Parisi--Zhang universality class. These ensembles are governed by sequences of nonnegative parameters that encode the asymptotic slopes of the curves at positive and negative infinity. Our main results characterize the fluctuations around this leading-order behavior and establish functional limit theorems for the ensembles near both ends of the spatial axis. We show that, at a macroscopic level, an Airy wanderer line ensemble organizes into groups of finitely many curves sharing a common asymptotic slope. After appropriate centering and scaling, each such group converges to a Dyson Brownian motion whose dimension equals the size of the group. In the case where only finitely many slope parameters are positive, we further prove a curve separation phenomenon: the upper curves follow deterministic parabolic trajectories, while the remaining lower curves remain globally flat and converge to the classical Airy line ensemble.