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…
2 Citations
Diffusive scaling limit of the Busemann process in Last Passage Percolation
- Mathematics
- 2021
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…
The stationary horizon and semi-infinite geodesics in the directed landscape
- Mathematics
- 2022
. 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…
References
SHOWING 1-10 OF 39 REFERENCES
Competition interfaces and second class particles
- Mathematics
- 2005
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
- Mathematics
- 2015
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…
A phase transition for competition interfaces
- Mathematics
- 2009
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
- Mathematics
- 2018
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
- Mathematics
- 2020
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…
Second class particles as microscopic characteristics in totally asymmetric nearest-neighbor K-exclusion processes
- Mathematics
- 2000
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.
- PhysicsPhysical review. E, Statistical, nonlinear, and soft matter physics
- 2006
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
- Mathematics
- 2011
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…
The Kardar-Parisi-Zhang Equation and Universality Class
- Mathematics
- 2011
Brownian motion is a continuum scaling limit for a wide class of random processes, and there has been great success in developing a theory for its properties (such as distribution functions or…
Strong hydrodynamic limit for attractive particle systems on Z
- Mathematics
- 2010
We prove almost sure Euler hydrodynamics for a large class
of attractive particle systems on $Z$ starting from an arbitrary
initial profile. We generalize earlier works by Sepp"al"ainen
(1999) and…