Curve separation in supercritical half-space last passage percolation

2025/10/08 by Evgeni Dimitrov, Dimitrov, Evgeni, Zhengye Zhou +1
Physics and Astronomy · #05E05 #60B20 #60G55 #60K35 #82C23 #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #NMR spectroscopy and applications #Probability (math.PR) #Quantum chaos and dynamical systems #Theoretical and Computational Physics

paper · pdf · doi:10.48550/arxiv.2510.07508

openalex publication_date 2025/10/08 · openalex created_date 2025/10/18 · openalex updated_date 2026/07/28

Abstract

We study line ensembles arising naturally in symmetrized/half-space geometric last passage percolation (LPP) on the N × N square. The weights of the model are geometrically distributed with parameter q² off the diagonal and cq on the diagonal, where q ∈ (0,1) and c ∈ [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.

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