Infinite geodesics, competition interfaces and the second class particle in the scaling limit

2021/12/13 by Mustazee Rahman, Bálint Virág, Rahman, Mustazee +1 · 3 citations
Mathematics · Physics and Astronomy · #FOS: Mathematics #FOS: Physical sciences #Mathematical Dynamics and Fractals #Mathematical Physics (math-ph) #Probability (math.PR) #Stochastic processes and statistical mechanics #Theoretical and Computational Physics

paper · pdf · doi:10.48550/arxiv.2112.06849

openalex publication_date 2021/12/13 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We establish fundamental properties of infinite geodesics and competition interfaces in the directed landscape. We construct infinite geodesics in the directed landscape, establish their uniqueness and coalescence, and define Busemann functions. We then define competition interfaces in the directed landscape. We prove the second class particle in tasep converges under KPZ scaling to a competition interface. Under suitable conditions, we show the competition interface has an asymptotic direction, analogous to the speed of a second class particle, and determine its law. Moreover, we prove the competition interface has an absolutely continuous law on compact sets with respect to infinite geodesics.

Cited by

• Radon-Nikodym derivative of inhomogeneous Brownian last passage percolation
• Limiting one-point fluctuations of the geodesic in the directed landscape near the endpoints when the geodesic length goes to infinity
• Revisiting the temporal law in KPZ random growth

Related

• Geodesics and the competition interface for the corner growth model
• Non-existence of three non-coalescing infinite geodesics with the same direction in the directed landscape
• The scaling limit of the longest increasing subsequence
• Busemann functions, geodesics, and the competition interface for directed last-passage percolation
• Competition interfaces and second class particles