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Perturbation of Wigner matrices and a conjecture

Authors: Mark Fannes and Dénes Petz
Journal: Proc. Amer. Math. Soc. 131 (2003), 1981-1988
MSC (2000): Primary 15A15, 15A62, 46L54
Published electronically: February 20, 2003
MathSciNet review: 1963740
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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 $t\mapsto \hbox {Tr}\, \exp (H(n)-\text{i}\, tH_{0})$ 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.

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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

Keywords: Bessis-Moussa-Vilani conjecture, Gaussian random matrix, Wigner theorem, positive definite function, free random variables, semicircular element
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
Article copyright: © Copyright 2003 American Mathematical Society