Numerical investigation of a MST conjecture for random Haar matrices
Year: 2026
Type: preprint
Abstract: The state-to-state control time of finite-dimensional conservative bilinear quantum systems is bounded by the weight of minimum spanning trees (MST) derived from the drift matrices. For a spectrum scaling as k^3/2 , determining whether the control time vanishes as the dimension d tends to infinity remains an open problem due to the global correlations imposed by the Haar measure. We formulate two conjectures regarding the asymptotic behavior of the MST weight\,: a general decay to zero, and a scaling in 1/sqrt(ln d) derived from an independent Gaussian surrogate model. Numerical investigations up to dimension d = 18101 validate the general decay, confirm that the exact Haar model and the independent surrogate are statistically indistinguishable in mean at all tested dimensions, and extract the thermodynamic limit $α$ = 10/3, which both models approach from below once the scaling variable 1/ ln d is adopted.
Source: arXiv (Cornell University)
Authors: Chambrion, Thomas, Mendico, Cristian
Institution: Institut de Recherche Mathématique Avancée
Language: English
Subfield: Atomic and Molecular Physics, and Optics
Field: Physics and Astronomy
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7225832238
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