Abstract
We show that for sufficiently large \(n\geq 1\) and \(d=C n^{3/4}\) for some universal constant \(C>0\), a random spectrahedron with matrices drawn from Gaussian orthogonal ensemble has Gaussian surface area \(\Theta (n^{1/8})\) with high probability.
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Notes
- 1.
A d-facet polytope is the special case of a spectrahedron when the matrices, \(A^{(1)},\ldots ,A^{(n)},B\) are diagonal.
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Acknowledgements
We thank Daniel Kane, Assaf Naor, Fedor Nazarov, and Yiming Zhao for useful correspondence. O.R. is supported by the Simons Collaboration on Algorithms and Geometry, a Simons Investigator Award, and by the National Science Foundation (NSF) under Grant No. CCF-1814524. P.Y. is supported by the National Key R&D Program of China 2018YFB1003202, National Natural Science Foundation of China (Grant No. 61972191), the Program for Innovative Talents and Entrepreneur in Jiangsu and Anhui Initiative in Quantum Information Technologies Grant No. AHY150100.
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Arunachalam, S., Regev, O., Yao, P. (2023). On the Gaussian Surface Area of Spectrahedra. In: Eldan, R., Klartag, B., Litvak, A., Milman, E. (eds) Geometric Aspects of Functional Analysis. Lecture Notes in Mathematics, vol 2327. Springer, Cham. https://doi.org/10.1007/978-3-031-26300-2_2
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