Free Askey--Wilson functionals and geometric last passage percolation on a strip

Barraquand, Corwin, and Yang arXiv:2306.05983 established that geometric last passage percolation (LPP) on a strip of $\mathbb{Z}^2$ has a unique stationary measure. Building on this, Barraquand arXiv:2409.08927 derived explicit contour integral formulas for the model's multipoint probability generating function. In this paper, we introduce free Askey--Wilson functionals and use them to extend these generating function formulas. Our framework yields explicit expressions valid over a broader range of boundary parameters than previously accessible. This generalization allows us to determine the full phase diagram that characterizes how the large-scale asymptotics of the stationary measure depend on the boundary conditions. In addition, we prove a Poisson approximation for the stationary measure when the parameters vary with the strip width.