  We consider the q-TASEP that is a q-deformation of the totally asymmetric
simple exclusion process (TASEP) on Z for q in [0,1) where the jump rates
depend on the gap to the next particle. For step initial condition, we prove
that the current fluctuation of q-TASEP at time t are of order t^{1/3} and
asymptotically distributed as the GUE Tracy-Widom distribution, which confirms
the KPZ scaling theory conjecture.
