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Long-time asymptotics for Airy wanderer line ensembles

Alexander Clay, Evgeni Dimitrov, Rundong Ding, Alex Fu

TL;DR

This work analyzes long-time fluctuations of the Airy wanderer line ensembles, infinite-parameter Brownian Gibbsian line ensembles that arise as edge-scaling limits in KPZ universality with inhomogeneities. The authors show that curves cluster into groups by shared asymptotic slope and, after centering by $t^2$ and rescaling, each group converges to Dyson Brownian motion of the corresponding dimension; symmetry yields analogous results at the opposite end of the axis. When only finitely many slopes are positive, a curve-separation phenomenon occurs: the leading curves follow parabolic trajectories while the tail collapses to the standard Airy line ensemble, mirroring KPZ-edge universality with spiked data. The proofs fuse determinantal structures, Brownian Gibbs resampling, kernel asymptotics under contour deformations, and monotone couplings to extend results across all parameter choices, yielding a comprehensive edge-limit theory for inhomogeneous KPZ-type models.

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.

Long-time asymptotics for Airy wanderer line ensembles

TL;DR

This work analyzes long-time fluctuations of the Airy wanderer line ensembles, infinite-parameter Brownian Gibbsian line ensembles that arise as edge-scaling limits in KPZ universality with inhomogeneities. The authors show that curves cluster into groups by shared asymptotic slope and, after centering by and rescaling, each group converges to Dyson Brownian motion of the corresponding dimension; symmetry yields analogous results at the opposite end of the axis. When only finitely many slopes are positive, a curve-separation phenomenon occurs: the leading curves follow parabolic trajectories while the tail collapses to the standard Airy line ensemble, mirroring KPZ-edge universality with spiked data. The proofs fuse determinantal structures, Brownian Gibbs resampling, kernel asymptotics under contour deformations, and monotone couplings to extend results across all parameter choices, yielding a comprehensive edge-limit theory for inhomogeneous KPZ-type models.

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.
Paper Structure (26 sections, 27 theorems, 315 equations, 2 figures)

This paper contains 26 sections, 27 theorems, 315 equations, 2 figures.

Table of Contents

  1. Introduction and main results
  2. Preface
  3. The Airy wanderer line ensembles
  4. Properties of the Airy wanderer line ensembles
  5. Main results
  6. Key ideas and paper outline
  7. Kernel convergence
  8. Dyson Brownian motion limit
  9. Airy line ensemble limit
  10. Determinantal structure of Dyson Brownian motion
  11. Finite-dimensional convergence
  12. Upper tail estimates for the Dyson Brownian motion limit
  13. Upper tails for the Airy line ensemble limit
  14. Finite-dimensional convergence to Dyson Brownian motion
  15. Finite-dimensional convergence to the Airy line ensemble
  16. ...and 11 more sections

Key Result

Proposition 1.5

Assume the same notation as in Definition Def.DLP and fix $(a,b,c) \in \mathcal{P}_{\mathsf{fin}}$. Then, there exists a unique line ensemble $\mathcal{A}^{a,b,c} = \{\mathcal{A}_i^{a,b,c}\}_{i \geq 1}$ on $\mathbb{R}$ such that the following all hold. Firstly, the ensemble is non-intersecting, mean For each $m \in \mathbb{N}$, and $s_1, \dots, s_m \in \mathbb{R}$ with $s_1 < s_2 < \cdots < s_m$ w

Figures (2)

  • Figure 1: The left side depicts the curves $\{\mathcal{A}^{a,b,c}_i(t) - t^2\}_{i \geq 1}$ and the right depicts the curves $\{\mathcal{A}^{a,b,c}_i(t)\}_{i \geq 1}$ when $\mathsf{m}_1^a = 1$, $\mathsf{m}_2^a = 3$, $\mathsf{m}_1^b = 3$, $\mathsf{m}_2^b = 2$. Here, $J_a = \mathsf{m}_1^a + \mathsf{m}_2^a = 4$.
  • Figure 2: The figure depicts the contours $\Gamma_{\alpha + t_1}^+, \Gamma_{\beta + t_2}^-$ when they have two intersection points, denoted by $u_-$ and $u_+$. The contour $\gamma$ is the segment from $u_-$ to $u_+$.

Theorems & Definitions (65)

  • Definition 1.1
  • Definition 1.2
  • Definition 1.3
  • Definition 1.4
  • Proposition 1.5
  • Remark 1.6
  • Remark 1.7
  • Proposition 1.8
  • Remark 1.9
  • Proposition 1.10
  • ...and 55 more