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

TL;DR

This paper introduces free Askey--Wilson functionals to extend explicit formulas for geometric last passage percolation on a strip, enabling a comprehensive phase diagram analysis and Poisson approximation results.

Contribution

It develops a new functional framework that broadens the parameter range for explicit formulas and analyzes the phase diagram of the stationary measure.

Findings

Explicit formulas valid over broader boundary parameters

Full phase diagram characterization of asymptotics

Poisson approximation for varying parameters

Abstract

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.