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The problem of the description of finite factor representations of theinfinite-dimensional unitary group, investigated by Voiculescu in 1976, isequivalent to the description of all totally positive Toeplitz matrices.Vershik-Kerov showed that this problem is also equivalent to the description ofthe simplex of central (i.e. possessing a certain Gibbs property) measures onpaths in the Gelfand-Tsetlin graph. We study a quantum version of the latterproblem. We introduce a notion of a q-centrality and describe the simplex ofall q-central measures on paths in the Gelfand-Tsetlin graph. Conjecturally,q-central measurets are related to representations of the quantized universalenveloping algebra U_\epsilon(gl_\infty). We also define a class of q-Toeplitzmatrices and show that every extreme q-central measure corresponds to aq-Toeplitz matrix with non-negative minors. Finally, our results can be viewedas a classification theorem for certain Gibbs measures on rhombus tilings ofthe halfplane.  We use a class of q-interpolation polynomials related to Schur functions. Oneof the key ingredients of our proofs is the binomial formula for thesepolynomials proved by Okounkov.

The q-Gelfand-Tsetlin graph, Gibbs measures and q-Toeplitz matrices