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

## TL;DR

This paper develops new formulas for Macdonald characters using algebraic properties of Macdonald polynomials, enabling analysis of the boundary of the $(q,t)$-Gelfand-Tsetlin graph and impacting areas like statistical mechanics, representation theory, and random matrices.

## Contribution

It introduces Macdonald characters and derives formulas that generalize previous results, providing new tools for studying complex algebraic and probabilistic models.

## Key findings

- Characterization of the boundary of the $(q,t)$-Gelfand-Tsetlin graph for $t=q^{ heta}$
- New formulas for Macdonald characters generalizing prior work
- Potential applications to statistical mechanics, representation theory, and random matrices

## Abstract

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, arXiv:1301.0634], 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.