Asymptotic Formulas for Macdonald Polynomials and the Boundary of the $(q, t)$-Gelfand–Tsetlin Graph

We introduce Macdonald characters and use algebraic properties of Macdonald polynomials to study them. As a result, we produce several formulas for Macdonald characters, which are generalizations of those obtained by Gorin and Panova in [Ann. Probab. 43 (2015), 3052–3132], and are expected to provide tools for the study of statistical mechanical models, representation theory and random matrices. As first application of our formulas, we characterize the boundary of the $(q,t)$-deformation of the Gelfand–Tsetlin graph when $t = q^{\theta}$ and $\theta$ is a positive integer.