Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

The pinned half-space Airy line ensemble

Evgeni Dimitrov, Christian Serio, Zongrui Yang

TL;DR

The work constructs a universal pinned half-space Airy line ensemble $\\mathcal{A}^{\\mathrm{hs};\\infty}$ as the $\\varpi\to\infty$ limit of the critical half-space ensembles, establishing a Pfaffian correlation structure with an explicit kernel and a parabolic Brownian Gibbs property after a shift. It shows that far from the boundary the ensemble recovers the full-space Airy line ensemble, while at the origin its law matches the ordered eigenvalues (with doubled multiplicity) of the stochastic Airy operator with $\\beta=4$, linking half-space KPZ universality to SAO spectra and Gaussian Symplectic Ensemble edge limits. The analysis combines a reformulated kernel $K^{\\varpi}$ with asymptotic kernel limits, tightness arguments, and a half-space Brownian Gibbs framework (via the DY25 program) to control both bulk and boundary behavior. The results solidify the pinned half-space Airy line ensemble as the canonical universal scaling limit for supercritical half-space KPZ models and reveal deep connections to Pfaffian processes, Airy universality, and SAO edge statistics with $\\beta=4$.

Abstract

Half-space models in the Kardar-Parisi-Zhang (KPZ) universality class exhibit rich boundary phenomena that alter the asymptotic behavior familiar from their full-space counterparts. A distinguishing feature of these systems is the presence of a boundary parameter that governs a transition between subcritical, critical, and supercritical regimes, characterized by different scaling exponents and fluctuation statistics. In this paper we construct the pinned half-space Airy line ensemble $\mathcal{A}^{\mathrm{hs}; \infty}$ on $[0,\infty)$ -- a natural half-space analogue of the Airy line ensemble -- expected to arise as the universal scaling limit of supercritical half-space KPZ models. The ensemble $\mathcal{A}^{\mathrm{hs}; \infty}$ is obtained as the weak limit of the critical half-space Airy line ensembles $\mathcal{A}^{\mathrm{hs}; \varpi}$ introduced in arXiv:2505.01798 as the boundary parameter $\varpi$ tends to infinity. We show that $\mathcal{A}^{\mathrm{hs}; \infty}$ has a Pfaffian point process structure with an explicit correlation kernel and that, after a parabolic shift, it satisfies a one-sided Brownian Gibbs property described by pairwise pinned Brownian motions. Far from the origin, $\mathcal{A}^{\mathrm{hs}; \infty}$ converges to the standard Airy line ensemble, while at the origin its distribution coincides with that of the ordered eigenvalues (with doubled multiplicity) of the stochastic Airy operator with $β= 4$.

The pinned half-space Airy line ensemble

TL;DR

The work constructs a universal pinned half-space Airy line ensemble as the limit of the critical half-space ensembles, establishing a Pfaffian correlation structure with an explicit kernel and a parabolic Brownian Gibbs property after a shift. It shows that far from the boundary the ensemble recovers the full-space Airy line ensemble, while at the origin its law matches the ordered eigenvalues (with doubled multiplicity) of the stochastic Airy operator with , linking half-space KPZ universality to SAO spectra and Gaussian Symplectic Ensemble edge limits. The analysis combines a reformulated kernel with asymptotic kernel limits, tightness arguments, and a half-space Brownian Gibbs framework (via the DY25 program) to control both bulk and boundary behavior. The results solidify the pinned half-space Airy line ensemble as the canonical universal scaling limit for supercritical half-space KPZ models and reveal deep connections to Pfaffian processes, Airy universality, and SAO edge statistics with .

Abstract

Half-space models in the Kardar-Parisi-Zhang (KPZ) universality class exhibit rich boundary phenomena that alter the asymptotic behavior familiar from their full-space counterparts. A distinguishing feature of these systems is the presence of a boundary parameter that governs a transition between subcritical, critical, and supercritical regimes, characterized by different scaling exponents and fluctuation statistics. In this paper we construct the pinned half-space Airy line ensemble on -- a natural half-space analogue of the Airy line ensemble -- expected to arise as the universal scaling limit of supercritical half-space KPZ models. The ensemble is obtained as the weak limit of the critical half-space Airy line ensembles introduced in arXiv:2505.01798 as the boundary parameter tends to infinity. We show that has a Pfaffian point process structure with an explicit correlation kernel and that, after a parabolic shift, it satisfies a one-sided Brownian Gibbs property described by pairwise pinned Brownian motions. Far from the origin, converges to the standard Airy line ensemble, while at the origin its distribution coincides with that of the ordered eigenvalues (with doubled multiplicity) of the stochastic Airy operator with .
Paper Structure (29 sections, 32 theorems, 322 equations, 1 figure)

This paper contains 29 sections, 32 theorems, 322 equations, 1 figure.

Table of Contents

  1. Introduction and main results
  2. The full-space and critical half-space Airy line ensembles
  3. Main results
  4. Ideas behind the proofs and paper outline
  5. Kernel asymptotics
  6. Alternative kernel formula
  7. Kernel asymptotics for $\varpi \rightarrow \infty$
  8. Kernel asymptotics for $T \rightarrow \infty$
  9. Finite-dimensional convergence
  10. Tightness from above
  11. Point process convergence
  12. Finite-dimensional convergence of $\mathcal{A}^{\mathrm{hs}; \varpi}$
  13. Finite-dimensional convergence of $X^{\mathsf{S}}$
  14. Gibbsian line ensembles
  15. Definitions and notation for line ensembles
  16. ...and 14 more sections

Key Result

Proposition 1.10

Fix $\varpi \in \mathbb{R}$. There exists a line ensemble $\mathcal{A}^{\mathrm{hs}; \varpi} = \{\mathcal{A}^{\mathrm{hs}; \varpi}_i\}_{i \geq 1}$ on $[0, \infty)$ that satisfies the following properties. The ensemble is non-intersecting in the sense of Definition Def.Ordered. In addition, for any f is a Pfaffian point process with correlation kernel $K^{\mathrm{hs}; \varpi}$ as in Definition Def.

Figures (1)

  • Figure 1: The figure depicts $\mathcal{A}^{\mathrm{hs}; \varpi}$ near the origin for $\varpi \in (-\infty, 0)$ (left), $\varpi \in (0, \infty)$ (middle) and $\varpi = \infty$ (right). At the origin, the curves $\mathcal{A}^{\mathrm{hs}; \varpi}_{2i-1}$ and $\mathcal{A}^{\mathrm{hs}; \varpi}_{2i}$ repel more for $\varpi \in (-\infty,0)$, repel less for $\varpi \in (0, \infty)$, and collide for $\varpi = \infty$.

Theorems & Definitions (93)

  • Definition 1.1
  • Definition 1.2
  • Definition 1.3
  • Definition 1.4
  • Remark 1.5
  • Remark 1.6
  • Definition 1.7
  • Remark 1.8
  • Remark 1.9
  • Proposition 1.10
  • ...and 83 more