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Uniform convergence of Pfaffian point process to the Airy line ensemble

Zhengye Zhou

TL;DR

This work proves that line ensembles arising from half-space Pfaffian Schur processes, which encode half-space geometric LPP, converge uniformly on compact sets to the Airy line ensemble in the bulk and to the Airy wanderer in critical/spiked regimes. The authors obtain sharp kernel convergence via steepest-descent analysis of double-contour Pfaffian kernels, then upgrade finite-dimensional convergence to uniform convergence by leveraging tightness for interlacing geometric line ensembles. A central contribution is the demonstration that Pfaffian line ensembles exhibit the same universal Airy-type limits as determinantal models within KPZ, while also accommodating spiked deformations that yield Airy wanderer limits. The results advance the understanding of universality in half-space KPZ models and provide a robust framework for uniform convergence in the Pfaffian setting, with potential applications to spiked boundary phenomena and related line ensembles.

Abstract

We consider a family of Pfaffian Schur processes whose first coordinate marginal relates to the half--space geometric last passage percolation. We show that the line ensembles corresponding to the Pfaffian Schur processes with geometric weights converge uniformly over compact sets to the Airy line ensemble. By detailed asymptotic analysis of the kernels, we can verify the conditions for the finite dimensional weak convergence introduced in arXiv:2408.08445. By utilizing the tightness criteria of the line ensembles established in arXiv:2410.23899, we can further improve the finite dimensional convergence to the uniform convergence over compact sets. Moreover, using the same methodology we also show that sequences of spiked Pfaffian Schur processes converge uniformly over compact sets to the Airy wanderer line ensembles constructed in arXiv:2408.08445.

Uniform convergence of Pfaffian point process to the Airy line ensemble

TL;DR

This work proves that line ensembles arising from half-space Pfaffian Schur processes, which encode half-space geometric LPP, converge uniformly on compact sets to the Airy line ensemble in the bulk and to the Airy wanderer in critical/spiked regimes. The authors obtain sharp kernel convergence via steepest-descent analysis of double-contour Pfaffian kernels, then upgrade finite-dimensional convergence to uniform convergence by leveraging tightness for interlacing geometric line ensembles. A central contribution is the demonstration that Pfaffian line ensembles exhibit the same universal Airy-type limits as determinantal models within KPZ, while also accommodating spiked deformations that yield Airy wanderer limits. The results advance the understanding of universality in half-space KPZ models and provide a robust framework for uniform convergence in the Pfaffian setting, with potential applications to spiked boundary phenomena and related line ensembles.

Abstract

We consider a family of Pfaffian Schur processes whose first coordinate marginal relates to the half--space geometric last passage percolation. We show that the line ensembles corresponding to the Pfaffian Schur processes with geometric weights converge uniformly over compact sets to the Airy line ensemble. By detailed asymptotic analysis of the kernels, we can verify the conditions for the finite dimensional weak convergence introduced in arXiv:2408.08445. By utilizing the tightness criteria of the line ensembles established in arXiv:2410.23899, we can further improve the finite dimensional convergence to the uniform convergence over compact sets. Moreover, using the same methodology we also show that sequences of spiked Pfaffian Schur processes converge uniformly over compact sets to the Airy wanderer line ensembles constructed in arXiv:2408.08445.

Paper Structure

This paper contains 16 sections, 17 theorems, 142 equations, 3 figures.

Table of Contents

  1. Introduction and main results
  2. Half-space geometric last passage percolation
  3. Main results
  4. Main ideas and paper outline
  5. Pfaffian Schur processes
  6. Definitions
  7. Pfaffian point process structure
  8. Parameter scaling for the curves
  9. Kernel convergence
  10. Preliminary estimates
  11. Function bounds
  12. Proof of Proposition \ref{['Prop.KernelConvsub']}
  13. Proof of Proposition \ref{['Prop.KernelConvcritical']}
  14. Weak convergence of the line ensembles
  15. Finite-dimensional convergence
  16. ...and 1 more sections

Key Result

Theorem 1.7

Assume the same notation as in Definition Def.CriticalScaledLPP, further assume $c \in (0, z_c)$ and $a_i=q$ for all $i\ge 1$. Then, $\mathcal{U}^{N} \Rightarrow \mathcal{U}^{\infty}$ in $C\left(\mathbb{N} \times \mathbb{R} \right)$. Here, $\mathcal{U}^{\infty} = \{\mathcal{U}^{\infty}_i\}_{i \geq where $\mathcal{A} = \{\mathcal{A}_i\}_{i \geq 1}$ is the Airy line ensemble from Definition Def.Ai

Figures (3)

  • Figure 1: The array $W = (w_{i,j}: i,j \geq 1)$ with symmetric $w_{i,j}$ as defined in \ref{['eq: weights']} and an up-right path $\pi$ (in gray) that connects $(1,1)$ to $(7,5)$.
  • Figure 2: The top several curves of $\{\lambda_i(m,n): i \geq 1, 0 \leq m \leq n\}$ for $q = 0.5$, $c=0.8$ and $n = 500$ concentrate around the function $n h (m/n)$ on $[0,n]$, where $h (\kappa) = \frac{ q (q + 2 \sqrt{\kappa} + q \kappa)}{1-q^2}$.
  • Figure 3: The left side depicts the contours $C(x,\theta, R,r)$ with $r<0$ from Definition \ref{['Def.ContoursBulk']}. The right side depicts the contours $\mathcal{C}_a^{\varphi}[r]$ with $r<0$, defined above (\ref{['Eq.I11Vanish']}).

Theorems & Definitions (42)

  • Definition 1.1
  • Definition 1.2
  • Remark 1.3
  • Remark 1.4
  • Remark 1.5
  • Remark 1.6
  • Theorem 1.7
  • Theorem 1.8
  • Remark 1.9
  • Definition 2.1
  • ...and 32 more