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

TL;DR

This paper analyzes the universal scaling limits of line ensembles in half-space geometric last passage percolation near the diagonal boundary, revealing phase-dependent convergence to the pinned half-space Airy line ensemble or Brownian motion.

Contribution

It establishes the convergence of the line ensemble to the pinned half-space Airy line ensemble in subcritical and supercritical regimes, completing the asymptotic analysis of the model.

Findings

In subcritical phase, the line ensemble converges to the pinned half-space Airy line ensemble.

In supercritical phase, the lower lines converge to the same Airy limit, while the top line becomes Brownian motion.

Provides a rigorous example of the pinned half-space Airy line ensemble as a universal limit.

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.