Transition to Shocks in TASEPand Decoupling of Last Passage Times

We consider the totally asymmetric simple exclusion process in a critical scaling parametrized by a≥0, which creates a shock in the particle density of order aT-1/3, T the observation time. When starting from step initial data, we provide bounds on the limiting law which in particular imply that in the double limit lima → ∞limT → ∞ one recovers the product limit law and the degeneration of the correlation length observed at shocks of order $1$. This result is shown to apply to a general last-passage percolation model. We also obtain bounds on the two-point functions of several Airy processes.