### 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) |
---|---|

Pages (from-to) | 104-138 |

Number of pages | 35 |

Journal | Annals of Probability |

Volume | 39 |

Issue number | 1 |

DOIs | |

State | Published - Jan 2011 |

### Fingerprint

### 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

**Current fluctuations for TASEP : A proof of the Prähofer-Spohn conjecture.** / Ben Arous, Gerard; Corwin, Ivan.

Research output: Contribution to journal › Article

*Annals of Probability*, vol. 39, no. 1, pp. 104-138. https://doi.org/10.1214/10-AOP550

}

TY - JOUR

T1 - Current fluctuations for TASEP

T2 - A proof of the Prähofer-Spohn conjecture

AU - Ben Arous, Gerard

AU - Corwin, Ivan

PY - 2011/1

Y1 - 2011/1

N2 - 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 γ2 = 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.

AB - 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 γ2 = 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.

KW - Asymmetric simple exclusion process

KW - Interacting particle systems

KW - Last passage percolation

UR - http://www.scopus.com/inward/record.url?scp=78650289779&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=78650289779&partnerID=8YFLogxK

U2 - 10.1214/10-AOP550

DO - 10.1214/10-AOP550

M3 - Article

VL - 39

SP - 104

EP - 138

JO - Annals of Probability

JF - Annals of Probability

SN - 0091-1798

IS - 1

ER -