Universal Plücker positivity and the Octopus inequality
Abstract
We prove a positivity theorem for the universal Plücker coordinates of Karp and Purbhoo when exactly one parameter is negative, with a sharp uniform threshold governed by the largest Plancherel up-transition probability. At the critical specialisation, positivity of the normalised (2,2)-coordinate is equivalent to the Octopus inequality, the main technical tool in the proof of Aldous's spectral gap conjecture. By connecting the inequality with Schubert calculus, this answers questions raised by Caputo and Aldous. For a Wronskian with distinct real zeros, we determine the maximal common half-line on which all branching-balanced coordinates are positive semidefinite. We also give Plücker-theoretic proofs of two hypergraph inequalities of Alon, Kozma, and Puder.
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BibTeX
@misc{zhu2026universal,
title = {{Universal Plücker positivity and the Octopus inequality}},
author = {Haoran Zhu},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.03746},
eprint = {2609.03746},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.03746},
}
RIS
TY - UNPB
TI - Universal Plücker positivity and the Octopus inequality
AU - Haoran Zhu
PY - 2026
DA - 2026/09/03
PB - arXiv
DO - 10.48550/arxiv.2609.03746
UR - https://arxiv.org/abs/2609.03746
AB - We prove a positivity theorem for the universal Plücker coordinates of Karp and Purbhoo when exactly one parameter is negative, with a sharp uniform threshold governed by the largest Plancherel up-transition probability. At the critical specialisation, positivity of the normalised (2,2)-coordinate is equivalent to the Octopus inequality, the main technical tool in the proof of Aldous's spectral gap conjecture. By connecting the inequality with Schubert calculus, this answers questions raised by Caputo and Aldous. For a Wronskian with distinct real zeros, we determine the maximal common half-line on which all branching-balanced coordinates are positive semidefinite. We also give Plücker-theoretic proofs of two hypergraph inequalities of Alon, Kozma, and Puder.
ER -