Asymptotic fluctuations of geometric [formula omitted]-TASEP, geometric [formula omitted]-PushTASEP and [formula omitted]-PushASEP

Abstract

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–Péché distribution and that of the top eigenvalue of finite GUE matrices.