Perturbation of Wigner matrices and a conjecture
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- by Mark Fannes and Dénes Petz PDF
- Proc. Amer. Math. Soc. 131 (2003), 1981-1988 Request permission
Abstract:
Let $H_{0}$ be an arbitrary self-adjoint $n\times n$ matrix and $H(n)$ be an $n\times n$ (random) Wigner matrix. We show that $t\mapsto \operatorname {Tr} \exp (H(n)-\mathrm {i} tH_{0})$ is positive definite in the average. This partially answers a long-standing conjecture. On the basis of asymptotic freeness our result implies that $t\mapsto \tau (\exp (a- \text {i} tb))$ is positive definite whenever the noncommutative random variables $a$ and $b$ are in free relation, with $a$ semicircular.References
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Additional Information
- Mark Fannes
- Affiliation: Instituut voor Theoretische Fysica, K.U. Leuven, B-3001 Leuven, Belgium
- Dénes Petz
- Affiliation: Department for Mathematical Analysis, Budapest University of Technology and Economics, H–1521 Budapest XI., Hungary
- Received by editor(s): July 6, 2001
- Published electronically: February 20, 2003
- Additional Notes: The second author was partially supported by OTKA T 032662
- Communicated by: David R. Larson
- © Copyright 2003 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 131 (2003), 1981-1988
- MSC (2000): Primary 15A15, 15A62, 46L54
- DOI: https://doi.org/10.1090/S0002-9939-03-06813-8
- MathSciNet review: 1963740