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Processus de Dunkl, matrices aléatoires, et marches aléatoires sur des espaces non-commutatifs

Abstract : There are four independent parts composing this thesis. The aim of the first part is the construction of the affine Dunkl process, which is a càdlàg Markov process whose infinitesimal generator is given by the Dunkl Laplacian in the case of an affine root system. The construction of the process is achieved by a skew-product representation by means of its radial part and a jump process on the associated Weyl group. The second part is the study of right eigenvalues of Gaussian quaternionic random matrices, where we prove the almost surely convergence of the empirical spectral distribution. The third part deals with random walks in noncommutative spaces, which are discrete time approximations of some eigenvalues processes of minors of Hermitian Brownian motion. The natural context for this study is invariant theory which allows to characterize the Markovian property of some of these processes. Finally, in the last part we prove a Courant theorem on the interlacing property of zeros of eigenfunctions of a Schrödinger operator on a finite tree.
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Contributor : Francois Chapon <>
Submitted on : Monday, December 13, 2010 - 3:54:37 PM
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  • HAL Id : tel-00546082, version 1


Francois Chapon. Processus de Dunkl, matrices aléatoires, et marches aléatoires sur des espaces non-commutatifs. Mathématiques [math]. Université Pierre et Marie Curie - Paris VI, 2010. Français. ⟨tel-00546082⟩



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