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

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.

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