Hives from deformed GUE minor processes

We construct random hives from deformed GUE minor processes. Starting from two independent diagonally deformed GUE matrices $$X=\sqrt{n}(wG+uD),Y=\sqrt{n}(w^{\mathrm{\prime }}{G}^{\mathrm{\prime }}+u^{\mathrm{\prime }}{D}^{\mathrm{\prime }}),$$ where $D,D^{\mathrm{\prime }}$ are diagonal and have GUE spectra, we use their minor processes to form a double hive and then apply the octahedron recurrence. Under the matching condition $$\frac{u}{w^2}=\frac{u^{\mathrm{\prime }}}{(w^{\mathrm{\prime }}{)}^2},$$ we prove that the resulting hive law is close, in relative entropy, to a GUE hive law. More precisely, if $$a^2=w^2+u^2,b^2=(w^{\mathrm{\prime }}{)}^2+(u^{\mathrm{\prime }}{)}^2,$$ then the produced hive density $q_n$ satisfies $$D_{\mathrm{K}\mathrm{L}}\text{\hspace{0.17em}⁣}(q_n\parallel \mathrm{Density}(H_n(a\sqrt{n},b\sqrt{n},c_{\ast \ast }\sqrt{n})\left)\right)=O(n\log n)\mathrm{.}$$ The third scale $c_{\ast \ast }$ is determined by a limiting tetrahedral optimization problem; equivalently, writing $\delta =u+u^{\mathrm{\prime }}$, $${\delta }^2=\frac{2c_{\ast \ast }^4(c_{\ast \ast }^2-a^2-b^2)}{(c_{\ast \ast }^2-a^2+b^2)(c_{\ast \ast }^2+a^2-b^2)}\mathrm{.}$$ Thus the construction realizes GUE hive laws, up to subleading relative entropy, throughout the right-angled and obtuse regime. The appendix records two explicit surface-tension approximations and numerical comparisons which motivated the construction.