### Abstract

We consider the family of two-sided Bernoulli initial conditions for TASEP which, as the left and right densities (ρ-,ρ+) are varied, give rise to shock waves and rarefaction fans-the two phenomena which are typical to TASEP. We provide a proof of Conjecture 7.1 of [Progr. Probab. 51 (2002) 185-204] which characterizes the order of and scaling functions for the fluctuations of the height function of two-sided TASEP in terms of the two densities ρ-,ρ+ and the speed y around which the height is observed. In proving this theorem for TASEP, we also prove a fluctuation theorem for a class of corner growth processes with external sources, or equivalently for the last passage time in a directed last passage percolation model with two-sided boundary conditions: ρ- and 1-ρ+. We provide a complete characterization of the order of and the scaling functions for the fluctuations of this model's last passage time L(N,M) as a function of three parameters: the two boundary/source rates ρ- and 1 - ρ+, and the scaling ratio γ^{2} = M/N. The proof of this theorem draws on the results of [Comm. Math. Phys. 265 (2006) 1-44] and extensively on the work of [Ann. Probab. 33 (2005) 1643-1697] on finite rank perturbations of Wishart ensembles in random matrix theory.

Original language | English (US) |
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Pages (from-to) | 104-138 |

Number of pages | 35 |

Journal | Annals of Probability |

Volume | 39 |

Issue number | 1 |

DOIs | |

State | Published - Jan 1 2011 |

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### Keywords

- Asymmetric simple exclusion process
- Interacting particle systems
- Last passage percolation

### ASJC Scopus subject areas

- Statistics and Probability
- Statistics, Probability and Uncertainty

### Cite this

*Annals of Probability*,

*39*(1), 104-138. https://doi.org/10.1214/10-AOP550