Inhomogeneous Geometric Last Passage Percolation and the $q$-deformed Jacobi Unitary Ensemble

We study directed last passage percolation with geometric weights in a spatially inhomogeneous environment. We derive an explicit limit shape and establish fluctuation limits for the passage time, showing that, depending on the strength of the inhomogeneity, the limiting fluctuations are governed by Fredholm determinants associated with either the Airy or Bessel kernel. Our approach is based on an exact relation between the passage time and the smallest particle of the $q$-deformed Jacobi unitary ensemble ($q$-JUE). As part of our analysis, we derive the macroscopic limiting density of the $q$-JUE, which provides a natural $q$-deformation of the classical Wachter distribution, as well as of the limiting distribution of the Meixner ensemble. We further establish Airy and Bessel scaling limits for the smallest particle, which are of independent interest.