The frequency-parametrization conjecture for finite-rank Gelfand–Tsetlin type graphs

Let $\mathcal{C}$ be the cone of admissible frequency vectors, let $x(n)=(x_1(n),\ldots,x_d(n))$ be a path, and let $\mu$ be the central measure it determines. A frequency is a limit

$$\lambda_i=\lim_{n\to\infty}\frac{x_i(n)}{n},\qquad i=1,\ldots,d.$$

Frequency-parametrization conjecture. In the finite-rank case, these limits are finite for every $i$, and conversely, for every $\lambda=(\lambda_1,\ldots,\lambda_d)\in\mathcal{C}$, every path with frequencies $\lambda_i$ determines a central measure $\mu=\mu(\lambda)$ depending only on $\lambda$.

This is the proposed general phenomenon underlying the explicit examples in the paper, where central measures are parametrized by limiting frequencies. Its status beyond the cases established in the paper is open.