We consider the totally asymmetric simple exclusion process with initial conditions and/or jump rates such that shocks are generated. If the initial condition is deterministic, then the shock at time t will have a width of order t 1/3 . We determine the law of particle positions in the large time limit around the shock in a few models. In particular, we cover the case where at both sides of the shock the process of the particle positions is asymptotically described by the Airy 1 process. The limiting distribution is a product of two distribution functions, which is a consequence of the fact that at the shock two characteristics merge and of the slow decorrelation along the characteristics. We show that the result generalizes to generic last passage percolation models.

We investigate the free boundary Schur process, a variant of the Schur process introduced by Okounkov and Reshetikhin, where we allow the first and the last partitions to be arbitrary (instead of empty in the original setting). The pfaffian Schur process, previously studied by several authors, is recovered when just one of the boundary partitions is left free. We compute the correlation functions of the process in all generality via the free fermion formalism, which we extend with the thorough treatment of "free boundary states". For the case of one free boundary, our approach yields a new proof that the process is pfaffian. For the case of two free boundaries, we find that the process is not pfaffian, but a closely related process is. We also study three different applications of the Schur process with one free boundary: fluctuations of symmetrized last passage percolation models, limit shapes and processes for symmetric plane partitions, and for plane overpartitions.

We consider the totally asymmetric simple exclusion process (TASEP) with non-random initial condition and density λ on Z − and ρ on Z + , and a second class particle initially at the origin. For λ < ρ, there is a shock and the second class particle moves with speed 1 − λ − ρ. For large time t, we show that the position of the second class particle fluctuates on the t 1/3 scale and determine its limiting law. We also obtain the limiting distribution of the number of steps made by the second class particle until time t.

We consider the asymmetric simple exclusion process (ASEP) on Z with a single second class particle initially at the origin. The first class particles form two rarefaction fans which come together at the origin, where the large time density jumps from 0 to 1. We are interested in X (t), the position of the second class particle at time t. We show that, under the KPZ 1/3 scaling, X (t) is asymptotically distributed as the difference of two independent, GUE-distributed random variables. The key part of the proof is to show that X (t) equals, up to a negligible term, the difference of a random number of holes and particles, with the randomness built up by ASEP itself. This provides a KPZ analogue to the 1994 result of Ferrari and Fontes (Probab Theory Relat Fields 99:305-319, 1994), where this randomness comes from the initial data and leads to Gaussian limit laws.

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 at a = 0 one has step initial data, we provide bounds on the limiting law of particle positions for a > 0, which in particular imply that in the double limit lim a→∞ lim T →∞ one recovers the product limit law and the degeneration of the correlation length observed at shocks of order 1. This result can be phrased in terms of a general last passage percolation model. We also obtain bounds on the decoupling of twopoint functions of several Airy processes.