Probability Theory and Related Fields

, Volume 108, Issue 4, pp 517-542

First online:

Large deviations for Wigner's law and Voiculescu's non-commutative entropy

  • G. Ben ArousAffiliated withURA 762, CNRS, DMI, Ecole Normale Superieure, F-75230 Paris, France
  • , A. GuionnetAffiliated withURA 743, CNRS, Bat. 425, Université de Paris Sud, F-91405 Orsay, France

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Summary.

We study the spectral measure of Gaussian Wigner's matrices and prove that it satisfies a large deviation principle. We show that the good rate function which governs this principle achieves its minimum value at Wigner's semicircular law, which entails the convergence of the spectral measure to the semicircular law. As a conclusion, we give some further examples of random matrices with spectral measure satisfying a large deviation principle and argue about Voiculescu's non commutative entropy.

Mathematics Subject of Classification: 60F10 15A18 15A52