Current fluctuations for TASEP: A proof of the Prähofer--Spohn conjecture

Abstract

We consider the family of two-sided Bernoulli initial conditions for TASEP which, as the left and right densities (${\rho }_{-},{\rho }_{+}$) 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 ${\rho }_{-},{\rho }_{+}$ 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: ${\rho }_{-}$ and $1-{\rho }_{+}$. 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 ${\rho }_{-}$ and $1-{\rho }_{+}$, and the scaling ratio ${\gamma }^2=M/N$. 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.

Published in at http://dx.doi.org/10.1214/10-AOP550 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org)

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• Large deviations and dynamical phase transitions in stochastic chemical networks
• Universality of slow decorrelation in KPZ growth
• Generic Dynamical Phase Transition in One-Dimensional Bulk-Driven Lattice Gases with Exclusion
• Pfaffian Schur processes and last passage percolation in a half-quadrant
• Limit processes for TASEP with shocks and rarefaction fans
• Scaling limit for Brownian motions with one-sided collisions
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• Finite GUE distribution with cut-off at a shock
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• Coupling derivation of optimal-order central moment bounds in exponential last-passage percolation