Global structure of semi-infinite geodesics and competition interfaces in Brownian last-passage percolation

This paper studies the large scale limits of multi-type invariant distributions and Busemann functions of planar stochastic growth models in the Kardar–Parisi–Zhang (KPZ) class. We identify a set of sufficient hypotheses for convergence of multi-type invariant measures of last-passage percolation (LPP) models to the stationary horizon (SH), which is the unique multi-type stationary measure of the KPZ fixed point. Our limit theorem utilizes conditions that are expected to hold broadly in the KPZ class, including convergence of the scaled last-passage process to the directed landscape. We verify these conditions for the six exactly solvable models whose scaled bulk versions converge to the directed landscape, as shown by Dauvergne and Virág. We also present a second, more general, convergence theorem with future applications to polymer models and particle systems. Our paper is the first to show convergence to the SH without relying on information about the structure of the multi-type invariant measures of the prelimit models. These results are consistent with the conjecture that the SH is the universal scaling limit of multi-type invariant measures in the KPZ class.

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The stationary horizon and semi-infinite geodesics in the directed landscape

Busani,

Seppäläinen,

Sorensen

2024

Ann. Probab.

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Global structure of semi-infinite geodesics and competition interfaces in Brownian last-passage percolation

Cited by 4 publications

References 23 publications

Jointly stationary solutions of periodic Burgers flow

Jointly stationary solutions of periodic Burgers flow

Scaling Limit of Multi-Type Invariant Measures via the Directed Landscape

The stationary horizon and semi-infinite geodesics in the directed landscape

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