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Paths in Weyl chambers and random matrices
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  • Published: December 2002

Paths in Weyl chambers and random matrices

  • Philippe Bougerol1 &
  • Thierry Jeulin2 

Probability Theory and Related Fields volume 124, pages 517–543 (2002)Cite this article

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  • 32 Citations

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Abstract.

 Baryshnikov [4] and Gravner, Tracy & Widom [15] have shown that the largest eigenvalue of a random matrix of the G.U.E. of order d has the same distribution as

where is a d-dimensional Brownian motion. We provide a generalization of this formula to all the eigenvalues and give a geometric interpretation. For any Weyl chamber a+ of an Euclidean finite-dimensional space a, we define a natural continuous path transformation T which associates to a path w in a a path Tw in ¯a+. This transformation occurs in the description of the asymptotic behaviour of some deterministic dynamical systems on the symmetric space G/K where G is the complex group with chamber a+. When and if W is the Euclidean Brownian motion on a then T W is the process of the eigenvalues of the Dyson Brownian motion on the set of Hermitian matrices and (T W)(1) is distributed as the eigenvalues of the G.U.E.

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Authors and Affiliations

  1. Université Paris 6, Probabilités et modèles aléatoires (UMR 7599), 4 Place Jussieu, 75232 Paris Cedex 05, France. e-mail: bougerol@ccr.jussieu.fr, , , , , , FR

    Philippe Bougerol

  2. Université Paris 7, Probabilités et modèles aléatoires (UMR 7599), 2 Place Jussieu, 75251 Paris Cedex 05, France. e-mail: jeulin@math.jussieu.fr, , , , , , FR

    Thierry Jeulin

Authors
  1. Philippe Bougerol
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  2. Thierry Jeulin
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Additional information

Received: 9 January 2002 / Revised version: 1 April 2002 / Published online: 30 September 2002

Subject Mathematics Classification (2000): Primary 15A52, 17B10, 60B99, 60J65; Secondary 22E30, 22E46, 43A85

Key words or phrases: Random matrix – Gaussian Unitary Ensemble – Symmetric space – Weyl chamber – Brownian motion – Complex semisimple group – Representation theory – Pitman's theorem

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Bougerol, P., Jeulin, T. Paths in Weyl chambers and random matrices. Probab Theory Relat Fields 124, 517–543 (2002). https://doi.org/10.1007/s004400200221

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  • Issue Date: December 2002

  • DOI: https://doi.org/10.1007/s004400200221

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Keywords

  • Dynamical System
  • Brownian Motion
  • Asymptotic Behaviour
  • Symmetric Space
  • Random Matrice
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