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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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  • 1. D. Bessis, P. Moussa, and M. Villani, Monotonic converging variational approximations to the functional integrals in quantum statistical mechanics, J. Mathematical Phys. 16 (1975), no. 11, 2318–2325. MR 0416396
  • 2. Nicole Berline, Ezra Getzler, and Michèle Vergne, Heat kernels and Dirac operators, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 298, Springer-Verlag, Berlin, 1992. MR 1215720
  • 3. B. D. Bojanov, H. A. Hakopian, and A. A. Sahakian, Spline functions and multivariate interpolations, Mathematics and its Applications, vol. 248, Kluwer Academic Publishers Group, Dordrecht, 1993. MR 1244800
  • 4. E. Brézin, Dyson’s universality in generalized ensembles of random matrices, The mathematical beauty of physics (Saclay, 1996) Adv. Ser. Math. Phys., vol. 24, World Sci. Publ., River Edge, NJ, 1997, pp. 1–11. MR 1490846
  • 5. M. Fannes, D. Petz, On the function ${\hbox {Tr\,}} e^{H+\text {i}\,tK}$, Int. J. Math. and Math. Sci. 29 (2002), 389-393.
  • 6. M. Gaudin, Sur la transformée de Laplace de 𝑡𝑟𝑒^{-𝐴} considérée comme fonction de la diagonale de 𝐴, Ann. Inst. H. Poincaré Sect. A (N.S.) 28 (1978), no. 4, 431–442 (French, with English summary). MR 511072
  • 7. Fumio Hiai and Dénes Petz, The semicircle law, free random variables and entropy, Mathematical Surveys and Monographs, vol. 77, American Mathematical Society, Providence, RI, 2000. MR 1746976
  • 8. Madan Lal Mehta, Random matrices, 2nd ed., Academic Press, Inc., Boston, MA, 1991. MR 1083764
  • 9. M. L. Mehta and Kailash Kumar, On an integral representation of the function 𝑇𝑟[𝑒𝑥𝑝(𝐴-𝜆𝐵)], J. Phys. A 9 (1976), no. 2, 197–206. MR 0429946
  • 10. Pierre Moussa, On the representation of 𝑇𝑟(𝑒^{(𝐴-𝜆𝐵)}) as a Laplace transform, Rev. Math. Phys. 12 (2000), no. 4, 621–655. MR 1763844, 10.1142/S0129055X00000204
  • 11. Dan Voiculescu, Limit laws for random matrices and free products, Invent. Math. 104 (1991), no. 1, 201–220. MR 1094052, 10.1007/BF01245072
  • 12. Dan Voiculescu, A strengthened asymptotic freeness result for random matrices with applications to free entropy, Internat. Math. Res. Notices 1 (1998), 41–63. MR 1601878, 10.1155/S107379289800004X

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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