particular, for $q=0$ we find a geometric-Bernoulli generalization of the PushTASEP (see ).

We do not perform any asymptotics in this paper. In fact, the key result towards such an aim would be a Fredholm determinant formula for a suitable $q$-Laplace transform. However, due to the pushing mechanism, the moments generally used to derive such a formula become infinite past a certain power. Still, we present which contains what we believe is a correct formula based on previous analogous works.

Our present investigation suggests that there should be strong parallels between pushing and non-pushing integrable particle systems. In the non-pushing (e.g. standard TASEP) context, [@CorwinPetrov2015] introduced a 4-parameter family of stochastic vertex models, which recovers the 3-parameter $q$-Hahn TASEP in a special analytic continuation and degeneration. It would be interesting to develop a parallel 4-parameter family of pushing systems, and obtain the corresponding duality relations and contour integral formulas. We expect that this can be done in the following manner. Consider the stochastic higher spin vertex model with horizontal spin $J\in \mathbb{Z}_{\ge1}$. Subtract $J$ from the number of arrows along each horizontal edge, and interpret the negative arrow numbers as counting arrows pointing in the opposite direction. This minor modification 