Transition to Shocks and Decoupling of Last Passage Times

We consider the totally asymmetric simple exclusion process (TASEP) in a critical scaling parametrized by ,which creates a shock in the particle density of order , 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 one recovers the product limit law and the degeneration of the correlation length observed earlier at shocks of order 1. This result can be phrased in terms of a general last passage percolation (LPP) model, which allows us to study the decoupling of Airy processes and LPP times in the time-like direction.