  We investigate the asymptotic fluctuation of three interacting particle
systems: the geometric q-TASEP, the geometric q-PushTASEP and the q-PushASEP.
We prove that the rescaled particle position converges to the GUE Tracy-Widom
distribution in the homogeneous case. If the jump rates of the first finitely
many particles are perturbed in the first two models, we obtain that the
limiting fluctuations are governed by the Baik-Ben Arous-Peche distribution and
that of the top eigenvalue of finite GUE matrices.
