which arises in a scaling limit of the system as $q\to1$. We parallel this limit here, and discover a corresponding beta polymer-like system. The main new feature is that in our model, the distribution of the weights depends on the ratio of the partition functions immediately to the left and below (see ). We had initially expected that the inverse beta polymer of [@thieryLD2015integrable] would arise from our pushing system, but presently this does not seem to be the case (though perhaps the inverse beta polymer could be included into a 4-parameter family of pushing systems whose existence we speculate on below). However, in the $q\to 1$ limit we do arrive (see and ) at a solution to the following recursion relation, which bears great similarity to that satisfied by the inverse beta polymer: When $\tilde{Z}(i,t-1)>\tilde{Z}(i-1,t)$, $$\tilde{Z}(i,t) = \tilde{Y} \tilde{Z}(i,t-1) + (1-\tilde{Y}) \tilde{Z}(i-1,t)$$ where $\tilde{Y}$ is $\mathscr{Beta}^{-1}(\bar{\mu},\tfrac{1}{2}-\bar{\mu})$-distributed (see ); and when $\tilde{Z}(i,t-1)<\tilde{Z}(i-1,t)$, $$\tilde{Z}(i,t) = \tilde{Y} \tilde{Z}(i-1,t) + (1-\tilde{Y}) \tilde{Z}(i,t-1)$$ where $\tilde{Y}$ is $\mathscr{Beta}^{-1}(\bar{\mu},\tfrac{1}{2})$-distributed. For this random recursion (with suitable boundary data) we compute moment formulas () and conjecture a Laplace transform formula ().

Besides the $q\to 1$ limits, there are other new limit processes which arise from our work. In 