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 $\phi (x)\le \phi (y)$ for every edge $⟨x,y⟩$ 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*} σ(u_1,u_2) = - \log (u_1 + u_2 ) - \log \sin (πu_1/(u_1+u_2)) -1 + \log π. \end{align*} A Gelfand-Tsetlin pattern is a Gelfand-Tsetlin function defined on the triangle $T_n=\{(x_1,x_2)\in {\mathbb{Z}}^2:1\le x_2\le x_1\le n\}$. 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*} $\mathcal{E}$[ψ] := \int_{\blacktriangle} σ(\nabla ψ), $\mathrm{d}$s ,$\mathrm{d}$t - χ[μ] \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.