A phase transition for q-TASEP with a few slower particles

We consider a q-TASEP model started from step initial condition where all but finitely many particles have speed 1 and a few particles are slower. It is shown in Ferrari and Veto (2013) that the rescaled particles position of q-TASEP with identical hopping rates obeys a limit theorem à la Tracy–Widom. We adapt this work to the case of different hopping rates and show that one observes the so-called BBP transition. Our proof is a refinement of Ferrari–Vető’s and does not require any condition on the parameter q nor the macroscopic position of particles.