text: We consider the totally asymmetric simple exclusion process in a critical scaling parametrized by @math , which creates a shock in the particle density of order @math @math 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 @math one recovers the product limit law and the degeneration of the correlation length observed at shocks of order @math . This result is shown to apply to a general last-passage percolation model. We also obtain bounds on the two-point functions of several @math processes.
  @cite_1: 
  @cite_4: The Airy processes describe spatial fluctuations in wide range of growth models, where each particular Airy process arising in each case depends on the geometry of the initial profile. We show how the coupling method, developed in the last-passage percolation context, can be used to prove that several types of Airy processes have a continuous version, and behave locally like a Brownian motion. We further extend these results to an Airy sheet, by proving existence of a continuous version and local convergence to additive Brownian motion.
  @cite_11: We address the question of how a localized microscopic defect, especially if it is small with respect to certain dynamic parameters, affects the macroscopic behavior of a system. In particular we consider two classical exactly solvable models: Ulam's problem of the maximal increasing sequence and the totally asymmetric simple exclusion process. For the first model, using its representation as a Poissonian version of directed last passage percolation on @math , we introduce the defect by placing a positive density of extra points along the diagonal line. For the latter, the defect is produced by decreasing the jump rate of each particle when it crosses the origin. The powerful algebraic tools for studying these processes break down in the perturbed versions of the models. Taking a more geometric approach we show that in both cases the presence of an arbitrarily small defect affects the macroscopic behavior of the system: in Ulam's problem the time constant increases, and for the exclusion process the flux of particles decreases. This, in particular, settles the longstanding Slow Bond Problem.
summary: A result we shall use repeatedly in this paper concerns the transversal fluctuations of the maximizer @math when @math are i.i.d. and @math there are bounds on the probability that the maximizer @math from @math to a point @math deviates more than @math from the straight line @math , @math . This is a result from @cite_11 and cited here as Theorem 3.2. Furthermore, for line-to-point problems, we have detailed control over the distribution of the (random) starting point of @math , see @cite_4 , Lemma 1.1, Lemma 1.2 and also (4.18) in @cite_1 .