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.
"Current fluctuations for TASEP: A proof of the Prähofer–Spohn conjecture." Ann. Probab. 39 (1) 104 - 138, January 2011. https://doi.org/10.1214/10-AOP550