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