Structural properties of the Airy wanderer line ensembles

Abstract

The Airy wanderer line ensembles are infinite-parameter generalizations of the classical Airy line ensemble that arise naturally as scaling limits of inhomogeneous (spiked) models in the Kardar-Parisi-Zhang universality class. In this paper, we establish several structural properties of these ensembles. Our results show their laws depend continuously on the parameters, which encode the asymptotic slopes of the ensemble's curves near positive and negative infinity. We further prove that these ensembles admit multiple monotone couplings with respect to their parameters. Finally, we show that the Airy wanderer line ensembles are extreme points in the space of all Brownian Gibbsian line ensembles on the real line.

Figures (4)

• Figure 1: 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 $u_+$.
• Figure 2: The figure depicts the contours $\Gamma^-_0$, $\Gamma^-_{u_1}$, $\Gamma_{u_2}^-$, $\Gamma_{v_1}^+$, $\Gamma_{v_2}^+$. The contours $\Gamma_{u_2}^-$ and $\Gamma_{v_2}^+$ intersect at two points, and $\Gamma_{v_2}^{+,0}$ is the portion of $\Gamma_{v_2}^+$ between them (drawn in dashed black). If $z_2 \in \Gamma_{v_2}^{+,0}$, then deforming $\Gamma_0^-$ to $\Gamma_{u_2}^-$ crosses $z_2$; otherwise, it does not.
• Figure 3: The figure depicts the contours $\Gamma_{u_2}^{-,0}$ and $\Gamma_{v_2}^{+,0}$. The points in $\{1/a_i\}_{m = 0}^{J_a}$ enclosed by the two contours are precisely $1/a_{q+1}, 1/a_{q+2}, \dots, 1/a_p$.
• Figure 4: The left figure depicts the top $N$ curves in $\mathcal{L}^{a,b,c}$ (in black) on $[-N,N]$ and $\hat{\mathcal{L}}^{\delta}$ (in dashed gray) on $[-\lambda_N^2 N, \lambda_N^2 N]$, for $J_a^+ = L = R = 1$ and $J_b^+ = 2$. The two dashed black parabolas are $g_N$ and $g_N + N^{-1/6}$. The right figure depicts the curve endpoints $\mathcal{L}_i^{a,b,c}(\pm N)$ and $\hat{\mathcal{L}}^{\delta,N}_i(\pm N)$ for $i = 1, \dots, N-1$, as well as the bottom curves $\mathcal{L}_N^{a,b,c}$ and $\hat{\mathcal{L}}^{\delta,N}_N$ on $[-N,N]$. The figure also depicts the locations of $x_i^N$ and $y_i^N$, which, for $i = 2, \dots, N-1$, are all squeezed between $g_N(\pm N)$ and $g_N(\pm N) + N^{-1/6}$. Distances in the figure have been exaggerated for clarity.

Theorems & Definitions (48)

• Definition 1.1
• Remark 1.2
• Definition 1.3
• Definition 1.4
• Definition 1.5
• Proposition 1.6
• Proposition 1.7
• Remark 1.8
• Remark 1.9
• Proposition 1.10