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Ergodicité et fonctions propres du laplacien sur les grands graphes réguliers

Abstract : N this thesis, we study concentration properties of eigenfunctions of the discrete Laplacian on regular graphs of fixed degree, when the number of vertices tend to infinity. This study is made in analogy with the Quantum Ergodicity theory on manifolds. We construct a pseudo-differential calculus on regular trees by defining symbol classes and associated operators and proving some properties of these classes of symbols and operators. In particular we prove that the operators are bounded on L² and give adjoint and product formulas. We then use this theory to prove a Quantum Ergodicity theorem on large regular graphs. This is a property of delocalization of most eigenfunctions in the large scale limit. We consider expander graphs with few short cycles (for instance random large regular graphs). These hypothesis are almost surely satisfied by sequences of random regular graphs.
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Submitted on : Friday, September 27, 2013 - 11:22:12 AM
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Etienne Le Masson. Ergodicité et fonctions propres du laplacien sur les grands graphes réguliers. Mathématiques générales [math.GM]. Université Paris Sud - Paris XI, 2013. Français. ⟨NNT : 2013PA112179⟩. ⟨tel-00866843⟩



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