# Competition interfaces and second class particles

@article{Ferrari2004CompetitionIA, title={Competition interfaces and second class particles}, author={Pablo A. Ferrari and Leandro P. R. Pimentel}, journal={Annals of Probability}, year={2004}, volume={33}, pages={1235-1254} }

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 second class particle in the exclusion process can be linearly mapped into the competition interface between two growing clusters in the last-passage percolation model. Using technology built up for geodesics in percolation, we show that the competition interface converges almost surely to an…

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## References

SHOWING 1-10 OF 36 REFERENCES

### 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…

### Shock fluctuations in asymmetric simple exclusion

- Mathematics
- 1992

SummaryThe one dimensional nearest neighbors asymmetric simple exclusion process in used as a microscopic approximation to the Burgers equation. We study the process with rates of jumpsp>q to the…

### Density and uniqueness in percolation

- Mathematics
- 1989

Two results on site percolation on thed-dimensional lattice,d≧1 arbitrary, are presented. In the first theorem, we show that for stationary underlying probability measures, each infinite cluster has…

### First passage percolation and a model for competing spatial growth

- MathematicsJournal of Applied Probability
- 1998

An interacting particle system modelling competing growth on the ℤ2 lattice is defined as follows. Each x ∈ ℤ2 is in one of the states {0,1,2}. 1's and 2's remain in their states for ever, while a 0…

### Non-equilibrium behaviour of a many particle process: Density profile and local equilibria

- Mathematics
- 1981

SummaryOne considers a simple exclusion particle jump process on ℤ, where the underlying one particle motion is a degenerate random walk that moves only to the right. One starts with the…

### Shocks in one-Dimensional Processes with Drift

- Mathematics
- 1994

The local structure of sh ock s in one-dimensional, nearest neighbor attractive systems with drift and conserved density is reviewed. The systems include the asymmetric simple exclusion, the zero…

### A constructive approach to Euler hydrodynamics for attractive processes. Application to k-step exclusion

- Mathematics
- 2002

### A Surface View of First-Passage Percolation

- Mathematics
- 1995

Let \(\tilde B\)(t) be the set of sites reached from the origin by time t in standard first-passage percolation on Z d , and let B0 (roughly lim \(\tilde B\) (t)/t) be its deterministic asymptotic…

### Geodesics and spanning trees for Euclidean first-passage percolation

- Mathematics
- 2000

The metric D α (q, q') on the set Q of particle locations of a homogeneous Poisson process on R d , defined as the infimum of (Σ i |q i -q i+1 | α ) 1/α over sequences in Q starting with q and ending…

### FIRST-PASSAGE PERCOLATION ON VORONOI TILINGS

- Mathematics
- 2004

In this paper we consider first-passage percolation models on Voronoi tilings of the plane and present a sufficient condition for the compactness of the limit set. This result is based on a static…