Hall-Littlewood-PushTASEP and its KPZ limit

We study a new model of interactive particle systems which we call the randomly activated cascading exclusion process (RACEP). Particles wake up according to exponential clocks and then take a geometric number of steps. If another particle is encountered during these steps, the first particle goes to sleep at that location and the second is activated and proceeds accordingly. We consider a totally asymmetric version of this model which we refer as Hall-Littlewood-PushTASEP (HL-PushTASEP) on $\mathbb{Z}_{\geq 0}$ lattice where particles only move right and where initially particles are distributed according to Bernoulli product measure on $\mathbb{Z}_{\geq 0}$. We prove KPZ-class limit theorems for the height function fluctuations. Under a particular weak scaling, we also prove convergence to the solution of the KPZ equation.