Hives from deformed GUE minor processes
arXiv:2607.04138
The paper builds random hive structures from diagonally deformed GUE minor processes, applies the octahedron recurrence, and proves that the resulting hive distribution is close (up to O(n log n) relative entropy) to the standard GUE hive law.
Abstract
We construct random hives from deformed GUE minor processes. Starting from two independent diagonally deformed GUE matrices \[ X=\sqrt{n}(wG+uD),\qquad Y=\sqrt{n}(w'G'+u'D'), \] where \(D,D'\) 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 \[ \frac{u}{w^2}=\frac{u'}{(w')^2}, \] we prove that the resulting hive law is close, in relative entropy, to a GUE hive law. More precisely, if \[ a^2=w^2+u^2,\qquad b^2=(w')^2+(u')^2, \] then the produced hive density satisfies \[ D_{\mathrm{KL}}\!\left( q_n\, \middle\|\, \operatorname{Density}\bigl(H_n(a\sqrt n,b\sqrt n,c_{**}\sqrt n)\bigr) \right) = O(n\log n). \] The third scale is determined by a limiting tetrahedral optimization problem; equivalently, writing \(δ=u+u'\), \[ δ^2 = \frac{ 2c_{**}^4(c_{**}^2-a^2-b^2) }{ (c_{**}^2-a^2+b^2)(c_{**}^2+a^2-b^2) }. \] 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.