# Limit Processes for TASEP with Shocks and Rarefaction Fans

@article{Corwin2010LimitPF, title={Limit Processes for TASEP with Shocks and Rarefaction Fans}, author={Ivan Corwin and Patrik L. Ferrari and Sandrine P{\'e}ch{\'e}}, journal={Journal of Statistical Physics}, year={2010}, volume={140}, pages={232-267} }

We consider the totally asymmetric simple exclusion process (TASEP) with two-sided Bernoulli initial condition, i.e., with left density ρ− and right density ρ+. We study the associated height function, whose discrete gradient is given by the particle occurrences. Macroscopically one has a deterministic limit shape with a shock or a rarefaction fan depending on the values of ρ±. We characterize the large time scaling limit of the multipoint fluctuations as a function of the densities ρ± and of…

## 45 Citations

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## References

SHOWING 1-10 OF 73 REFERENCES

Current fluctuations for TASEP: A proof of the Pr\

- Mathematics
- 2009

We consider the family of two-sided Bernoulli initial conditions for TASEP which, as the left and right densities (p-, ρ+) are varied, give rise to shock waves and rarefaction fans—the two phenomena…

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…

Shock fluctuations in the asymmetric simple exclusion process

- Mathematics
- 1994

SummaryWe consider the one dimensional nearest neighbors asymmetric simple exclusion process with ratesq andp for left and right jumps respectively;q<p. Ferrari et al. (1991) have shown that if the…

Scaling Limit for the Space-Time Covariance of the Stationary Totally Asymmetric Simple Exclusion Process

- Mathematics
- 2005

The totally asymmetric simple exclusion process (TASEP) on the one-dimensional lattice with the Bernoulli ρ measure as initial conditions, 0<ρ<1, is stationary in space and time. Let Nt(j) be the…

Two Speed TASEP

- Mathematics
- 2009

We consider the TASEP on ℤ with two blocks of particles having different jump rates. We study the large time behavior of particles’ positions. It depends both on the jump rates and the region we…

Limit process of stationary TASEP near the characteristic line

- Mathematics
- 2010

The totally asymmetric simple exclusion process (TASEP) on\input amssym ${\Bbb Z}$ with the Bernoulli‐ρ measure as an initial condition, 0 < ρ < 1, is stationary. It is known that along the…

Fluctuation Properties of the TASEP with Periodic Initial Configuration

- Mathematics
- 2007

Abstract
We consider the joint distributions of particle positions for the continuous time totally asymmetric simple exclusion process (TASEP). They are expressed as Fredholm determinants with a…

Slow decorrelations in KPZ growth

- Mathematics
- 2008

For stochastic growth models in the Kardar-Parisi-Zhang (KPZ) class in 1 + 1 dimensions, fluctuations grow as t1/3 during time t and the correlation length at a fixed time scales as t2/3. In this…

Dynamics of a Tagged Particle in the Asymmetric Exclusion Process with the Step Initial Condition

- Mathematics
- 2007

The one-dimensional totally asymmetric simple exclusion process (TASEP) is considered. We study the time evolution property of a tagged particle in the TASEP with the step initial condition.…

Transition between Airy1 and Airy2 processes and TASEP fluctuations

- Computer Science
- 2007

In this paper the totally asymmetric simple exclusion process is considered, a model in the KPZ universality class, and its one‐point distribution is a new interpolation between GOE and GUE edge distributions.