Fluctuations in the discrete TASEP with periodic initial configurations and the Airy1 process

We consider the totally asymmetric simple exclusion process (TASEP) in discrete time with sequential update. The joint distribution of the positions of selected particles is expressed as a Fredholm determinant with a kernel defining a signed determinantal point process. We focus on periodic initial conditions where particles occupy dZ, d >= 2. In the proper large time scaling limit, the fluctuations of particle positions are described by the Airy(1) process. Interpreted as a growth model, this confirms universality of fluctuations with flat initial conditions for a discrete set of slopes.