distributions on $\mathbb{Z}$ are summarized in .

Acknowledgments {#acknowledgments .unnumbered}
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The authors thank Guillaume Barraquand for pointing out the simplification which led to . The authors acknowledge that part of this work was done while in attendance of the ICERM conference on Limit Shapes, held from April 13-17, 2015. IC was partially supported by the NSF grants DMS-1208998, DMS-1811143 and DMS-1664650, as well as a Packard Fellowship in Science and Engineering and a Clay Research Fellowship. LP was partially supported by the NSF grant DMS-1664617.

$q$-Hahn TASEP {#sec:TASEP}
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Here we briefly recall the definition and duality properties of the $q$-Hahn TASEP introduced and studied in [@Povolotsky2013], [@Corwin2014qmunu].

Assume that the parameters $0<q<1$, $0\le {\boldsymbol\upnu}\le {\boldsymbol\upmu}<1$ are fixed.[^1] The $q$-Hahn TASEP is a discrete time (with $t\in \mathbb{Z}_{\ge0}$) Markov process on configurations $\vec x(t)=(x_1(t)>x_2(t)>\ldots )$, $x_i\in\mathbb{Z}$, with at most one particle per site and a rightmost particle $x_1$. The evolution of the $q$-Hahn TASEP is as follows. At each discrete time step $t\to t+1$, each particle $x_i(t)$ jumps in parallel and independently to $x_i(t+1)=x_{i}(t)+v_i$, where $v_i$ is sampled from the probability distribution $\varphi_{q,{\boldsymbol\upmu},{\boldsymbol\upnu}}(v_i\mid x_{i-1}(t)-x_i(t)-1)$. Note for the update of $x_1$, we assume a virtual particle $x_0\equiv +\infty$, by agreement. 