Curve separation in supercritical half-space last passage percolation
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arXiv
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| Formato: | Preprint |
| Publicado: |
2025
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| author | Dimitrov, Evgeni Zhou, Zhengye |
| author_facet | Dimitrov, Evgeni Zhou, Zhengye |
| contents | 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 $N^{1/2}$ fluctuations and $N$ spatial scaling, while the remaining curves converge to the Airy line ensemble under $N^{1/3}$ fluctuations and $N^{2/3}$ 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_07508 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Curve separation in supercritical half-space last passage percolation Dimitrov, Evgeni Zhou, Zhengye Probability Mathematical Physics 60K35, 60G55, 82C23, 05E05, 60B20 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 $N^{1/2}$ fluctuations and $N$ spatial scaling, while the remaining curves converge to the Airy line ensemble under $N^{1/3}$ fluctuations and $N^{2/3}$ 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. |
| title | Curve separation in supercritical half-space last passage percolation |
| topic | Probability Mathematical Physics 60K35, 60G55, 82C23, 05E05, 60B20 |
| url | https://arxiv.org/abs/2510.07508 |