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We introduce Macdonald characters and use algebraic properties of Macdonaldpolynomials to study them. As a result, we produce several formulas forMacdonald characters, which are generalizations of those obtained by Gorin andPanova in [Ann. Probab. 43 (2015), 3052-3132, arXiv:1301.0634], and areexpected to provide tools for the study of statistical mechanical models,representation theory and random matrices. As first application of ourformulas, we characterize the boundary of the $(q,t)$-deformation of theGelfand-Tsetlin graph when $t=q^{\theta}$ and $\theta$ is a positive integer.

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