THE STATIONARY HORIZON AND SEMI-INFINITE GEODESICS IN THE DIRECTED LANDSCAPE

• 成果类型: Article
• 署名作者: Busani, Ofer; Seppalainen, Timo; Sorensen, Evan
• 署名单位: University of Bonn; University of Wisconsin System; University of Wisconsin Madison
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• DOI: 10.1214/23-AOP1655
• 发表日期: 2024
• 页码: 1-66
• 关键词: corner growth-model; busemann functions; infinite geodesics; polymers

摘要

The stationary horizon (SH) is a stochastic process of coupled Brownian motions indexed by their real-valued drifts. It was first introduced by the first author as the diffusive scaling limit of the Busemann process of exponential last-passage percolation. It was independently discovered as the Busemann process of Brownian last-passage percolation by the second and third authors. We show that SH is the unique invariant distribution and an attractor of the KPZ fixed point under conditions on the asymptotic spatial slopes. It follows that SH describes the Busemann process of the directed landscape. This gives control of semi-infinite geodesics simultaneously across all initial points and directions. The countable dense set E of directions of discontinuity of the Busemann process is the set of directions in which not all geodesics coalesce and in which there exist at least two distinct geodesics from each initial point. This creates two distinct families of coalescing geodesics in each Xi direction. In E directions the Busemann difference profile is distributed like Brownian local time. We describe the point process of directions xi is an element of Xi and spatial locations where the xi +/- Busemann functions separate.