Large deviation conjecture for Gelfand–Tsetlin interfaces

Let $\mu$ be a probability measure, and let $\rho_\mu:[0,1] \to \mathbb{R}$ denote the right-continuous inverse function of $F_\mu(s):= \int_{-\infty}^s \mu(\mathrm{d}x)$. Let $\phi_n:\Delta \to \mathbb{R}$ be the linear interpolation of the stochastic interface associated with a uniformly chosen Gelfand–Tsetlin pattern whose bottom row is $\rho_\mu(1/n),\ldots,\rho_\mu(n/n)$. Large deviation conjecture. Under a suitable topology, the random function $\phi_n$ satisfies a large deviation principle with speed $n^2$ and rate function

$$\mathcal{R}_\mu[\phi] = \begin{cases} \mathcal{I}[\phi] + \mathcal{F}[\mu] & \phi \in \mathcal{C}^1(\mathring{\Delta})\text{ with }\phi(s,s) = \rho_\mu(s),\\ +\infty & \text{otherwise}. \end{cases}$$

Here $\mathring{\Delta}$ denotes the interior of $\Delta$, and $\mathcal{I}$ and $\mathcal{F}$ are the functionals defined in the source. This conjecture would give a large-deviation description of stochastic Gelfand–Tsetlin interfaces and connect their asymptotic fluctuations to the free-energy functional; the suitable topology and the claimed large deviation principle remain to be established.