Curve separation in supercritical half-space last passage percolation

We study line ensembles arising naturally in symmetrized/half-space geometric last passage percolation (LPP) on the $N\times N$ square. The weights of the model are geometrically distributed with parameter $q^2$ off the diagonal and $cq$ on the diagonal, where $q\in (0,1)$ and $c\in [0,q^{-1})$. In the supercritical regime $c>1$, we show that the ensembles undergo a phase transition: the top curve separates from the rest and converges to a Brownian motion under $N^{1\mathrm{/}2}$ fluctuations and $N$ spatial scaling, while the remaining curves converge to the Airy line ensemble under $N^{1\mathrm{/}3}$ fluctuations and $N^{2\mathrm{/}3}$ spatial scaling. Our analysis relies on a distributional identity between half-space LPP and the Pfaffian Schur process. The latter exhibits two key structures: (1) a Pfaffian point process, which we use to establish finite-dimensional convergence of the ensembles, and (2) a Gibbsian line ensemble, which we use to extend convergence uniformly over compact sets.