We consider the $q^\text{Volume}$ lozenge tiling model on a large, finite
hexagon. It is well-known that random lozenge tilings of the hexagon correspond
to a two-dimensional determinantal point process via a bijection with ensembles
of non-intersecting paths. The starting point of our analysis is a formula for
the correlation kernel due to Duits and Kuijlaars which involves the
Christoffel-Darboux kernel of a particular family of non-Hermitian orthogonal
polynomials. Our main results are split into two parts: the first part concerns
the family of orthogonal polynomials, and the second concerns the behavior of
the boundary of the so-called arctic curve. In the first half, we identify the
orthogonal polynomials as a non-standard instance of little $q$-Jacobi
polynomials and compute their large degree asymptotics in the $q \to 1$ regime.
A consequence of this analysis is a proof that the zeros of the orthogonal
polynomials accumulate on an arc of a circle and an asymptotic formula for the
Christoffel-Darboux kernel. In the second half, we use these asymptotics to
show that the boundary of the liquid region converges to the Airy process, in
the sense of finite dimensional distributions, away from the boundary of the
hexagon. At inflection points of the arctic curve, we show that we do not need
to subtract/add a parabola to the Airy line ensemble, and this effect persists
at distances which are $o(N^{-2/9})$ in the tangent direction.

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