Half-space Airy line ensembles

TL;DR

This work constructs half-space analogues of the Airy line ensemble as a one-parameter family of infinite line ensembles on $[0, \infty)$, formed as weak limits of Pfaffian Schur processes (half-space geometric LPP). Away from the origin the ensembles exhibit Pfaffian correlation with a crossover kernel $K^{\mathrm{cross}}$, while near the boundary the top curves are described by reverse Brownian motions with alternating drifts, governed by the boundary parameter $\varpi$ through the half-space Brownian Gibbs property. The authors establish tightness via half-space interlacing Gibbs structure and KMT-type couplings, prove kernel convergence to a limiting Pfaffian kernel, and show the limiting object has infinitely many atoms at each fixed time, yielding subsequential finite-dimensional convergence to the half-space Airy line ensemble. The results sharpen the understanding of boundary effects in KPZ half-space universality and provide a rigorous, solvable path to the half-space Airy line ensemble with precise Pfaffian structure and Gibbs properties. They also connect discrete Pfaffian Schur processes to continuous half-space KPZ limits, enabling detailed asymptotics through steepest-descent analysis of the crossover kernels.

Abstract

We construct a one-parameter family of infinite line ensembles on $[0, \infty)$ that are natural half-space analogues of the Airy line ensemble. Away from the origin these ensembles are locally described by avoiding Brownian bridges, and near the origin they are described by a sequence of avoiding reverse Brownian motions with alternating drifts, that depend on the parameter of the model. In addition, the restrictions of our ensembles to finitely many vertical lines form Pfaffian point processes with the crossover kernels obtained by Baik-Barraquand-Corwin-Suidan (Ann. Probab., 46(6), 3015-3089, 2018)

Figures (4)

• Figure 1: The left side depicts the parabolic Airy line ensemble $\mathcal{L}^{\mathrm{pAiry}}$. The Brownian Gibbs property implies that the conditional law of the dashed lines is that of independent Brownian bridges connecting $(a,x_i)$ to $(b,y_i)$ that are conditioned to avoid each other and the thick curve. The right side depicts the line ensemble $\mathcal{L} = \{\mathcal{L}_i\}_{i \geq 1}$, where $\mathcal{L}_i(t) = 2^{-1/2} \cdot \mathcal{A}^{\mathrm{hs}}_i(t) - 2^{-1/2} \cdot t^2$, and $\mathcal{A}^{\mathrm{hs}}$ is the half-space Airy line ensemble. The half-space Brownian Gibbs property implies that the conditional law of the dashed lines is that of two independent reverse Brownian motions -- one started from $(b,y_1)$ with drift $-\sqrt{2} \varpi$, and the other started from $(b,y_2)$ with drift $\sqrt{2} \varpi$, which are conditioned to avoid each other and the thick curve.
• Figure 2: The figure depicts the top three curves in $\mathfrak{L}$, satisfying (\ref{['S1E1']}) and (\ref{['S1E2']}).
• Figure 3: The figure depicts $f(\cdot |A,B,y, \delta)$, $f(\cdot |-A,B,y, \delta)$ from (\ref{['eq:defining the corridor']}) for $A > 0$.
• Figure 4: The figure depicts the contours $\gamma^+_N(\mathsf{a}^q_3) = \gamma^+_N(\mathsf{a}^q_3, 0) \cup \gamma^+_N(\mathsf{a}^q_3,1)$ and $\gamma^-_N(-\mathsf{a}^q_2) = \gamma^-_N(-\mathsf{a}^q_2, 0) \cup \gamma^-_N(-\mathsf{a}^q_2,1)$ from Lemma \ref{['lem:PrelimitKernel']}.

Theorems & Definitions (103)

• Definition 1.1
• Definition 1.2
• Remark 1.3
• Theorem 1.4
• Remark 1.5
• Remark 1.6
• Remark 1.7
• Remark 1.8
• Definition 2.1
• Remark 2.2