We consider a new interacting particle system on the one-dimensional lattice that interpolates between TASEP and Toom's model: A particle cannot jump to the right if the neighboring site is occupied, and when jumping to the left it simply pushes all the neighbors that block its way. We prove that for flat and step initial conditions, the large time fluctuations of theheight function of the associated growth model along any space-like path are described by the Airy<sub>1</sub> and Airy<sub>2</sub> processes. This includes fluctuations of the height profile for a fixed time and fluctuations of a tagged particle's trajectory as special cases.
Alexei Borodin. Patrik Ferrari. "Large time asymptotics of growth models on space-like paths I: PushASEP." Electron. J. Probab. 13 1380 - 1418, 2008. https://doi.org/10.1214/EJP.v13-541