Lozenge tilings of a hexagon and q -Racah ensembles

We study the limiting behavior of random lozenge tilings of the hexagon with a $q$-Racah weight as the size of the hexagon grows large. Based on the asymptotic behavior of the recurrence coefficients of the $q$-Racah polynomials, we give a new proof for the fact that the height function for a random tiling concentrates near a deterministic limit shape and that the global fluctuations are described by the Gaussian Free Field. These results were recently proved using (dynamic) loop equation techniques. In this paper, we extend the recurrence coefficient approach that was developed for (dynamic) orthogonal polynomial ensembles to the setting of $q$-orthogonal polynomials. An interesting feature is that the complex structure is easily found from the limiting behavior of the (explicitly known) recurrence coefficients. A particular motivation for studying this model is that the variational characterization of the limiting height function has an inhomogeneous term. The study of the regularity properties of the minimizer for general variational problems with such inhomogeneous terms is a challenging open problem. In a general setup, we show that the variational problem gives rise to a natural complex structure associated with the same Beltrami equation as in the homogeneous situation. We also derive a relation between the complex structure and the complex slope. In the case of the $q$-Racah weighting of lozenge tilings of the hexagon, our representation of the limit shape and their fluctuations in terms of the recurrence coefficients allows us to verify this relation explicitly.