Year: 2026
Type: preprint
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.
Source: arXiv (Cornell University)
Author: Zhu, Haoran
Subfield: Discrete Mathematics and Combinatorics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7208760822
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