  We consider the PushTASEP (pushing totally asymmetric simple exclusion
process, also sometimes called long-range TASEP) with the step initial
configuration evolving in an inhomogeneous space. That is, the rate of each
particle's jump depends on the location of this particle. We match the
distribution of the height function of this PushTASEP with Schur processes.
Using this matching and determinantal structure of Schur processes, we obtain
limit shape and fluctuation results which are typical for stochastic particle
systems in the Kardar-Parisi-Zhang universality class. PushTASEP is a close
relative of the usual TASEP. In inhomogeneous space the former is integrable,
while the integrability of the latter is not known.
