# Intégrales matricielles et Probabilités Non-Commutatives

Abstract : This dissertation splits into three part. In the first one, we give an explicit formula for the explicit computation of all moments of the Haar measure on the unitary group, in terms of path enumeration on the Cayley graph of the symmetric group. This result yields a general theorem of asymptotic freeness for random matrices, as well as results of convergences of unitary matrix integrals. In particular, we give a combinatorial interpretation of the limit of the Itzykson-Zuber integral as well as a link with Voiculescu's R-transform. In a second part, completely different, we define a non commutative probabilistic framework in which we prove that Martin boundary theory extends and allows an integral representation of any positive harmonic function. As an application of these purely quantum results, we compute Martin boundaries of some classical random walks in a Weyl chamber. The example of a random walk on $SU_q(2)$ is treated as well. In the third part, we propose an analytic approach to asymptotics of Haar measure on a compact group. We compute the image of the Haar measure of the unitary groupo imder contraction by a projector. This allows us to recover and interpret in a combinatorial way some asymptotics obtained in the first part. Besides, we establish that the square of the radial part of a contraction of a unitary random matrix is a Jacobi ensemble. An orthogonal polynomial machinery then allows to strenghten results of asymptotic convergences knows in free probabilty, and establish universality results for eigenvalues.
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Theses
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https://tel.archives-ouvertes.fr/tel-00004306
Contributor : Benoit Collins <>
Submitted on : Tuesday, January 27, 2004 - 1:37:09 PM
Last modification on : Thursday, December 10, 2020 - 10:52:45 AM
Long-term archiving on: : Friday, April 2, 2010 - 8:04:24 PM

### Identifiers

• HAL Id : tel-00004306, version 1

### Citation

Benoit Collins. Intégrales matricielles et Probabilités Non-Commutatives. Mathématiques [math]. Université Pierre et Marie Curie - Paris VI, 2003. Français. ⟨tel-00004306⟩

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