Limit of the octahedron recurrence with GUE boundary data

Source abstract

We study the tropical octahedron recurrence on the lattice tetrahedron $\{(x_1,x_2,x_3,x_4)\in\mathbb Z_{\geq0}^4: x_1+x_2+x_3+x_4=n\}$ with an intrinsic double-hive boundary law. Let $M_1^{(n)},M_2^{(n)},M_3^{(n)}$ be independent standard $n\times n$ GUE matrices and, for fixed $\ell_1,\ell_2,\ell_3>0$, set $X_r^{(n)}=\ell_r\sqrt n\,M_r^{(n)},\qquad r=1,2,3.$ The boundary law on the two upper panels is the Gibbs density on the cone of double hives whose four exterior sides correspond, in the prescribed orientations, to the spectra of $X_1^{(n)},X_2^{(n)},X_3^{(n)}$, and $X_1^{(n)}+X_2^{(n)}+X_3^{(n)}$. We then perform an octahedron-recurrence sweep to obtain the value at every lattice point of the tetrahedron. After scaling lattice positions by $n^{-1}$ and recurrence values by $n^{-2}$, we prove that the resulting random field converges uniformly in probability to a deterministic Lipschitz function on $\{(x_1,x_2,x_3,x_4)\in\mathbb R_{\geq0}^4: x_1+x_2+x_3+x_4=1\}$. For every strictly interior target lattice point we construct a canonical planar bipartite graph supported on a union of three triangular panels and prove an exact perfect-matching formula for the recurrence value. Its variable weights are explicit interlacing gaps from the minor processes of $X_1^{(n)},X_2^{(n)},X_3^{(n)}$, together with explicit seam and matching-independent terms. The asymptotic analysis of this formula yields a surface-tension variational characterization of the deterministic limit. This matching rule supplies, in the present four-simplex setting, the kind of higher-dimensional combinatorial formula identified as a major cluster-algebra problem by Henriques and Speyer.