Abstract
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 Ξ 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 Ξ direction. In Ξ directions the Busemann difference profile is distributed like Brownian local time. We describe the point process of directions and spatial locations where the Busemann functions separate.
Funding Statement
The work of O. Busani was funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under Germany’s Excellence Strategy–GZ 2047/1, projekt-id 390685813 and partly performed at University of Bristol.
T. Seppäläinen was partially supported by National Science Foundation Grants DMS-1854619 and DMS-2152362 and by the Wisconsin Alumni Research Foundation.
E. Sorensen was partially supported by T. Seppäläinen under National Science Foundation Grants DMS-1854619 and DMS-2152362.
Acknowledgments
Duncan Dauvergne explained the mixing of the directed landscape, recorded as Lemma B.3. E.S. thanks also Erik Bates, Shirshendu Ganguly, Jeremy Quastel, Firas Rassoul-Agha and Daniel Remenik for helpful discussions. We also thank the two anonymous referees for incredibly helpful comments that have greatly improved the organization and exposition of this paper.
Citation
Ofer Busani. Timo Seppäläinen. Evan Sorensen. "The stationary horizon and semi-infinite geodesics in the directed landscape." Ann. Probab. 52 (1) 1 - 66, January 2024. https://doi.org/10.1214/23-AOP1655
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