Limit of the octahedron recurrence with GUE boundary data
Abstract
We study the tropical octahedron recurrence on the lattice tetrahedron {(x1,x2,x3,x4)∈ℤ≥04: x1+x2+x3+x4=n} with an intrinsic double-hive boundary law. Let M1(n),M2(n),M3(n) be independent standard n× n GUE matrices and, for fixed ℓ1,ℓ2,ℓ3>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 X1(n),X2(n),X3(n), and X1(n)+X2(n)+X3(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 {(x1,x2,x3,x4)∈ℝ≥04: x1+x2+x3+x4=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 X1(n),X2(n),X3(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.
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BibTeX
@misc{narayanan2026limit,
title = {{Limit of the octahedron recurrence with GUE boundary data}},
author = {Hariharan Narayanan},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.09121},
eprint = {2610.09121},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.09121},
}
RIS
TY - UNPB
TI - Limit of the octahedron recurrence with GUE boundary data
AU - Hariharan Narayanan
PY - 2026
DA - 2026/10/06
PB - arXiv
DO - 10.48550/arxiv.2610.09121
UR - https://arxiv.org/abs/2610.09121
AB - We study the tropical octahedron recurrence on the lattice tetrahedron {(x_1,x_2,x_3,x_4)∈ℤ_≥0^4: 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× n GUE matrices and, for fixed ℓ_1,ℓ_2,ℓ_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)∈ℝ_≥0^4: 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.
ER -