Airy wanderer line ensembles

TL;DR

This work develops a multi-parameter generalization of the extended Airy kernel, introducing the Airy wanderer kernel $K_{a,b,c}$ built from two convergent parameter sequences and two nonnegative shifts. It proves the existence of a determinantal point process on $\mathbb{R}^2$ with this kernel and, under the finite-minus-parameter condition ($c^-=0$ and finite $J_a^-,J_b^-$), lifts it to a line ensemble on $\mathbb{R}$ with the Brownian Gibbs property, extending the wanderer ensemble framework. The approach hinges on a careful scaling and limiting procedure for Schur processes, via geometric last passage percolation with defects, yielding a double-contour kernel that converges to $K_{a,b,c}$. The paper also develops upper-tail and tightness estimates to establish finite-dimensional convergence and constructs the corresponding Wanderer line ensembles, providing a robust probabilistic framework for these generalized integrable models within the KPZ universality class. Together, these results broaden the class of exactly solvable line ensembles and connect multi-parameter deformations of the Airy kernel to concrete probabilistic objects with Gibbsian structure.

Abstract

In (J. Stat. Phys. 132, 275-290, 2008) Borodin and Péché introduced a generalization of the extended Airy kernel based on two infinite sets of parameters. For an arbitrary choice of parameters we construct determinantal point processes on $\mathbb{R}^2$ for these generalized kernels. In addition, for a subset of the parameter space we show that the point processes can be lifted to line ensembles on $\mathbb{R}$, which satisfy the Brownian Gibbs property. Our ensembles generalize the wanderer line ensembles introduced by Corwin and Hammond in (Invent. Math. 195, 441-508, 2014).

Figures (8)

• Figure 1: The figure depicts the contours $\Gamma_{\alpha}^+$ and $\Gamma_{\beta}^-$ shifted by $t_1$ and $t_2$, respectively. The figure also shows the relative location of the poles in (\ref{['S1EAKS']}) and (\ref{['DefPhiS11']}) when shifted by $t_1$ (for the poles in $z$) and by $t_2$ (for the poles in $w$).
• Figure 2: The figure depicts the top three curves in $\mathcal{L}^{a,b,c}$ from Theorem \ref{['T3']}. The slopes $\beta_i$ are the sorted in ascending order numbers $\{\sqrt{2} (b_i^+)^{-1}\}_{ i\geq 1}$ and the non-zero elements in $\{\sqrt{2} b_i^-\}_{i \geq 1}$ (counted with multiplicities and with $+\infty$ allowed). The slopes $\alpha_i$ are the sorted in descending order numbers $\{-\sqrt{2} (a_i^+)^{-1}\}_{ i\geq 1}$ and the non-zero elements in $\{- \sqrt{2} a_i^-\}_{i \geq 1}$ (counted with multiplicities and with $-\infty$ allowed).
• Figure 3: 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 4: The NE chain $(3,2), (3,3), (5,3), (5,4), (6,4), (8,6)$
• Figure 5: The left part depicts the contours $\Gamma_{\alpha_1 + t_1}^+, \Gamma_{\beta_1 + t_2}^-, \Gamma_{\beta_3 + t_2}^-$ and their intersection points $v_{\pm}$, $u_{\pm}$. The right part depicts the contours $\Gamma_{\alpha_1}^+, \Gamma_{\alpha_1}^{+,1}$, where the latter consists of the two segments connecting $v_{\pm} - t_1$ with $u_{\pm} - t_1$.

Theorems & Definitions (85)

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