• Corpus ID: 245123952

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

@inproceedings{Rahman2021InfiniteGC,
  title={Infinite geodesics, competition interfaces and the second class particle in the scaling limit},
  author={Mustazee Rahman and B{\'a}lint Vir{\'a}g},
  year={2021}
}
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 show 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 in tasep, and… 

Figures from this paper

The stationary horizon and semi-infinite geodesics in the directed landscape
. The stationary horizon is a stochastic process consisting of coupled Brownian motions, indexed by their real-valued drifts. It was recently constructed by the first author as the scaling limit of
Diffusive scaling limit of the Busemann process in Last Passage Percolation
In exponential last passage percolation, we consider the rescaled Busemann process x 7→ N−1/2B 0,[xN ]e1 (x ∈ R), as a process parametrized by the scaled density ρ = 1/2+μ4N −1/2, and taking values

References

SHOWING 1-10 OF 39 REFERENCES
Competition interfaces and second class particles
The one-dimensional nearest-neighbor totally asymmetric simple exclusion process can be constructed in the same space as a last-passage percolation model in Z 2 . We show that the trajectory of a
Geodesics and the competition interface for the corner growth model
We study the directed last-passage percolation model on the planar integer lattice with nearest-neighbor steps and general i.i.d. weights on the vertices, outside the class of exactly solvable
The scaling limit of the longest increasing subsequence
We provide a framework for proving convergence to the directed landscape, the central object in the Kardar-Parisi-Zhang universality class. For last passage models, we show that compact convergence
A phase transition for competition interfaces
We study the competition interface between two growing clusters in a growth model associated to last-passage percolation. When the initial unoccupied set is approximately a cone, we show that this
The directed landscape
The conjectured limit of last passage percolation is a scale-invariant, independent, stationary increment process with respect to metric composition. We prove this for Brownian last passage
Three-halves variation of geodesics in the directed landscape
We show that geodesics in the directed landscape have $3/2$-variation and that weight functions along the geodesics have cubic variation. We show that the geodesic and its landscape environment
Hausdorff dimensions for shared endpoints of disjoint geodesics in the directed landscape
Within the Kardar-Parisi-Zhang universality class, the space-time Airy sheet is conjectured to be the canonical scaling limit for last passage percolation models. In recent work arXiv:1812.00309 of
Second class particles as microscopic characteristics in totally asymmetric nearest-neighbor K-exclusion processes
Summary. We study aspects of the hydrodynamics of one-dimensional totally asym- metric K-exclusion, building on the hydrodynamic limit of Seppalainen (1999). We prove that the weak solution chosen by
Roughening and inclination of competition interfaces.
TLDR
It is demonstrated that a phase transition occurs for the asymptotic inclination of this interface when the final macroscopic shape goes from curved to noncurved, and that the flat case (stationary growth) is a critical point for the fluctuations.
Renormalization Fixed Point of the KPZ Universality Class
The one dimensional Kardar–Parisi–Zhang universality class is believed to describe many types of evolving interfaces which have the same characteristic scaling exponents. These exponents lead to a
...
...