The macroscopic shape of Gelfand-Tsetlin patterns and free probability

A Gelfand-Tsetlin function is a real-valued function $\phi:C \to \mathbb{R}$ defined on a finite subset $C$ of the lattice $\mathbb{Z}^2$ with the property that $\phi(x) \leq \phi(y)$ for every edge $\langle x,y \rangle$ directed north or east between two elements of $C$. We study the statistical physics properties of random Gelfand-Tsetlin functions from the perspective of random surfaces, showing in particular that the surface tension of Gelfand-Tsetlin functions at gradient $u = (u_1,u_2) \in \mathbb{R}_{>0}^2$ is given by \begin{align*} \sigma(u_1,u_2) = - \log (u_1 + u_2 ) - \log \sin (\pi u_1/(u_1+u_2)) -1 + \log \pi. \end{align*} A Gelfand-Tsetlin pattern is a Gelfand-Tsetlin function defined on the triangle $T_n = \{(x_1,x_2) \in \mathbb{Z}^2 : 1 \leq x_2 \leq x_1 \leq n \}$. We show that after rescaling, a sequence of random Gelfand-Tsetlin patterns with fixed diagonal heights approximating a probability measure $\mu$ satisfies a large deviation principle with speed $n^2$ and rate functional of the form \begin{align*} \mathcal{E}[\psi] := \int_{\blacktriangle} \sigma(\nabla \psi)\, \mathrm{d}s \,\mathrm{d}t - \chi[\mu] \end{align*} where $\chi[\mu]$ is Voiculescu's free entropy. We show that the Euler-Lagrange equations satisfied by the minimiser of the rate functional agree with those governing the free compression operation in free probability, thereby resolving a recent conjecture of Shlyakhtenko and Tao.