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Pinning in non-critical half-space geometric last passage percolation

Sayan Das, Evgeni Dimitrov, Zongrui Yang

Abstract

We study a symmetrized (half-space) version of geometric last passage percolation with a boundary parameter $c$ that interpolates between subcritical, critical, and supercritical behavior. This model gives rise to a family of interlacing random curves, or a line ensemble, which encode both the usual last passage time and its higher-rank analogues. Although these ensembles are understood in most space-time regions, their behavior near the diagonal -- where the boundary effects are strongest -- has remained unclear outside the critical regime. We determine the universal scaling limits of the line ensemble in this near-diagonal region for both subcritical ($c < 1$) and supercritical ($c > 1$) phases. In the subcritical case, after appropriate centering and scaling, the entire line ensemble converges to the pinned half-space Airy line ensemble, a universal Brownian Gibbsian object recently constructed as a canonical limit for half-space models in the KPZ universality class in arXiv:2601.04546. In the supercritical case, we prove an analogous convergence together with a curve-separation phenomenon: the lower curves converge to the same pinned half-space Airy limit, while the top curve decouples and converges to Brownian motion. These results essentially complete the asymptotic description of half-space geometric last passage percolation and provide a new rigorous instance of the pinned half-space Airy line ensemble as a universal scaling limit.

Pinning in non-critical half-space geometric last passage percolation

Abstract

We study a symmetrized (half-space) version of geometric last passage percolation with a boundary parameter that interpolates between subcritical, critical, and supercritical behavior. This model gives rise to a family of interlacing random curves, or a line ensemble, which encode both the usual last passage time and its higher-rank analogues. Although these ensembles are understood in most space-time regions, their behavior near the diagonal -- where the boundary effects are strongest -- has remained unclear outside the critical regime. We determine the universal scaling limits of the line ensemble in this near-diagonal region for both subcritical () and supercritical () phases. In the subcritical case, after appropriate centering and scaling, the entire line ensemble converges to the pinned half-space Airy line ensemble, a universal Brownian Gibbsian object recently constructed as a canonical limit for half-space models in the KPZ universality class in arXiv:2601.04546. In the supercritical case, we prove an analogous convergence together with a curve-separation phenomenon: the lower curves converge to the same pinned half-space Airy limit, while the top curve decouples and converges to Brownian motion. These results essentially complete the asymptotic description of half-space geometric last passage percolation and provide a new rigorous instance of the pinned half-space Airy line ensemble as a universal scaling limit.

Paper Structure

This paper contains 38 sections, 46 theorems, 513 equations, 6 figures.

Table of Contents

  1. Introduction and main results
  2. Half-space geometric last passage percolation
  3. Main results
  4. Key ideas and paper outline
  5. Related works and future directions
  6. Gibbsian line ensembles
  7. Brownian line ensembles
  8. Geometric line ensembles
  9. Properties of discrete Gibbsian ensembles
  10. Properties of geometric line ensembles
  11. Convergence of geometric line ensembles
  12. Modulus of continuity estimates
  13. A general tightness criterion
  14. Proof of Theorem \ref{['Thm.Tightness']}
  15. Pfaffian Schur processes
  16. ...and 23 more sections

Key Result

Proposition 1.3

There exists a unique line ensemble $\mathcal{A}^{\mathrm{hs}; \infty} = \{\mathcal{A}^{\mathrm{hs}; \infty}_{i}\}_{i \geq 1}$ on $[0, \infty)$, which satisfies the following properties. Firstly, the line ensemble is ordered, meaning that almost surely In addition, if $\mathsf{S} = \{s_1, \dots, s_m\} \subset (0,\infty)$, then the random measure is a Pfaffian point process on $\mathbb{R}^2$ with

Figures (6)

  • Figure 1: The left side depicts the array $W = (w_{i,j}: i,j \geq 1)$ and an up-right path $\pi$ (in gray) that connects $(1,1)$ to $(7,5)$. The right side depicts $k = \min(m,n)$ pairwise disjoint up-right paths, with $\pi_i$ connecting $(1,i)$ to $(m, n-k+i)$ that cover the whole $n \times m$ rectangle.
  • Figure 2: The figure depicts the top three curves in $\{\lambda_{k}(\cdot,n)\}_{k \geq 1}$.
  • Figure 3: Phase diagram for $\{\lambda_{k}(t,n)\}_{k \geq 1}$. The top and bottom panels correspond to the subcritical $c \in [0,1)$ and supercritical $c \in (1, q^{-1})$ regimes, respectively.
  • Figure 4: The figure depicts simulations of the curves of $\mathfrak{L}^{\mathrm{ LPP}; N}$ for $N = 500$, $q = 0.5$, $c = 1.4$ (top) and $c = 0.8$ (bottom).
  • Figure 5: The figure depicts the contours $\gamma^+_N(a) = \gamma^+_N(a, 0) \cup \gamma^+_N(a,1)$ and $\gamma^-_N(-a) = \gamma^-_N(-a, 0) \cup \gamma^-_N(-a,1)$ for $a > 0$ from Definition \ref{['Def.ContoursBulk']}.
  • ...and 1 more figures

Theorems & Definitions (136)

  • Definition 1.1
  • Definition 1.2
  • Proposition 1.3
  • proof
  • Definition 1.4
  • Theorem 1.5
  • Remark 1.6
  • Remark 1.7
  • Remark 1.8
  • Remark 1.9
  • ...and 126 more