Determinantal Measures Related to Big $q$-Jacobi Polynomials

We define a novel combinatorial object—the extended Gelfand–Tsetlin graph with cotransition probabilities depending on a parameter $q$. The boundary of this graph admits an explicit description. We introduce a family of probability measures on the boundary and describe their correlation functions. These measures are a $q$-analogue of the spectral measures studied earlier in the context of the problem of harmonic analysis on the infinite-dimensional unitary group.