\nu \mu^{-1} q^{p}, \nu q^{g}, \nu^{2} q^{g+p}}(L-p)
    \,\big\vert_{\nu=0}=\mu^{L-p}\frac{(\mu;q)_{\infty}}{(q;q)_{L-p}}.$$ This is the $q$-geometric distribution. The first particle jumps according to this $q$-geometric distribution (note that $p=0$ for the first particle). For the update probabilities of all other particles we have $$\label{deg_P_nu0}
    \mathbf{P}_{\ell,g}(L)\,\big\vert_{\nu=0}
    =
    (\mu;q)_{\infty}
    \sum_{p=0}^{\min\left\{ \ell,L \right\}}
    q^{gp}\mu^{L-p}
    \frac{(q^g;q^{-1})_{\ell-p}(q^{-1};q^{-1})_{\ell}}
    {(q;q)_{L-p}(q^{-1};q^{-1})_{p}(q^{-1};q^{-1})_{\ell-p}}.$$ We see that the $\nu=0$ process coincides with the geometric $q$-PushTASEP introduced in [@MatveevPetrov2014 Section 6.3], and our parameter $\mu$ corresponds to $\alpha a_i$ (specific to each particle).

Further setting $q=0$ in the geometric $q$-PushTASEP reduces it to a geometric PushTASEP (that is, a discrete time PushTASEP with geometrically distributed jumps). In the sum only one summand will be nonzero, and[^3] $$\label{geom_push_q0_nu0}
    \mathbf{P}_{\ell,g}(L)\,\big\vert_{\nu=q=0}
    =
    \sum_{p=0}^{\min\left\{ \ell,L \right\}}
    (1-\mu)\,\mu^{L-p}\,\mathbf{1}_{p=\max\left\{ 0,\ell-g \right\}}
    =
    (1-\mu)\,\mu^{L-\max\{0,\ell-g\}}.$$ In words, the first particle jumps according to a geometric distribution with parameter $\mu$, and for each $i=2,3,\ldots$, the particle $x_{i-1}$ pushes $x_i$ by the minimal possible distance which preserves the order of the particles (cf. ), and after this push the particle $x_i$ makes an independent jump according to the geometric distribution. The geometric PushTASEP is well-known, see, e.g., [@BorFerr2008DF], [@warrenwindridge2009some] for its connections to Schur processes and dynamics on them.

Finally, in both the geometric $q$-PushTASEP and 