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Université Paris-Sud – Paris XI

1. Le Masson, Etienne. Ergodicité et fonctions propres du laplacien sur les grands graphes réguliers : Ergodicity and eigenfunctions of the Laplacian on large regular graphs.

Degree: Docteur es, Mathématiques, 2013, Université Paris-Sud – Paris XI

Dans cette thèse, nous étudions les propriétés de concentration des fonctions propres du laplacien discret sur des graphes réguliers de degré fixé dont le nombre de sommets tend vers l'infini. Cette étude s'inspire de la théorie de l'ergodicité quantique sur les variétés. Par analogie avec cette dernière, nous développons un calcul pseudo-différentiel sur les arbres réguliers : nous définissons des classes de symboles et des opérateurs associés, et nous prouvons un certain nombre de propriétés de ces classes de symboles et opérateurs. Nous montrons notamment que les opérateurs sont bornés dans L², et nous donnons des formules de l'adjoint et du produit. Nous nous servons ensuite de cette théorie pour montrer un théorème d'ergodicité quantique pour des suites de graphes réguliers dont le nombre de sommets tend vers l'infini. Il s'agit d'un résultat de délocalisation de la plupart des fonctions propres dans la limite des grands graphes réguliers. Les graphes vérifient une hypothèse d'expansion et ne comportent pas trop de cycles courts, deux hypothèses vérifiées presque sûrement par des suites de graphes réguliers aléatoires.

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.

Advisors/Committee Members: Anantharaman, Nalini (thesis director).

Subjects/Keywords: Fonctions propres du laplacien; Ergodicité quantique; Analyse semi-classique; Opérateurs pseudo-différentiels; Graphes réguliers; Grands graphes aléatoires; Laplacian eigenfunctions; Quantum ergodicity; Semi-classical analysis; Pseudo-differential operators; Regular graphs; Large random graphs

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APA · Chicago · MLA · Vancouver · CSE | Export to Zotero / EndNote / Reference Manager

APA (6th Edition):

Le Masson, E. (2013). Ergodicité et fonctions propres du laplacien sur les grands graphes réguliers : Ergodicity and eigenfunctions of the Laplacian on large regular graphs. (Doctoral Dissertation). Université Paris-Sud – Paris XI. Retrieved from http://www.theses.fr/2013PA112179

Chicago Manual of Style (16th Edition):

Le Masson, Etienne. “Ergodicité et fonctions propres du laplacien sur les grands graphes réguliers : Ergodicity and eigenfunctions of the Laplacian on large regular graphs.” 2013. Doctoral Dissertation, Université Paris-Sud – Paris XI. Accessed July 11, 2020. http://www.theses.fr/2013PA112179.

MLA Handbook (7th Edition):

Le Masson, Etienne. “Ergodicité et fonctions propres du laplacien sur les grands graphes réguliers : Ergodicity and eigenfunctions of the Laplacian on large regular graphs.” 2013. Web. 11 Jul 2020.

Vancouver:

Le Masson E. Ergodicité et fonctions propres du laplacien sur les grands graphes réguliers : Ergodicity and eigenfunctions of the Laplacian on large regular graphs. [Internet] [Doctoral dissertation]. Université Paris-Sud – Paris XI; 2013. [cited 2020 Jul 11]. Available from: http://www.theses.fr/2013PA112179.

Council of Science Editors:

Le Masson E. Ergodicité et fonctions propres du laplacien sur les grands graphes réguliers : Ergodicity and eigenfunctions of the Laplacian on large regular graphs. [Doctoral Dissertation]. Université Paris-Sud – Paris XI; 2013. Available from: http://www.theses.fr/2013PA112179


University of Queensland

2. Hodgkinson, Liam. Approximations for finite spin systems and occupancy processes.

Degree: School of Mathematics & Physics, 2019, University of Queensland

Subjects/Keywords: central limit theorem; dynamic random graphs; interacting particle systems; microscale models; network models; occupancy processes; quantitative law of large numbers; simulation; Stein’s method; tau-leaping; 0102 Applied Mathematics; 0104 Statistics

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APA · Chicago · MLA · Vancouver · CSE | Export to Zotero / EndNote / Reference Manager

APA (6th Edition):

Hodgkinson, L. (2019). Approximations for finite spin systems and occupancy processes. (Thesis). University of Queensland. Retrieved from http://espace.library.uq.edu.au/view/UQ:4925754

Note: this citation may be lacking information needed for this citation format:
Not specified: Masters Thesis or Doctoral Dissertation

Chicago Manual of Style (16th Edition):

Hodgkinson, Liam. “Approximations for finite spin systems and occupancy processes.” 2019. Thesis, University of Queensland. Accessed July 11, 2020. http://espace.library.uq.edu.au/view/UQ:4925754.

Note: this citation may be lacking information needed for this citation format:
Not specified: Masters Thesis or Doctoral Dissertation

MLA Handbook (7th Edition):

Hodgkinson, Liam. “Approximations for finite spin systems and occupancy processes.” 2019. Web. 11 Jul 2020.

Vancouver:

Hodgkinson L. Approximations for finite spin systems and occupancy processes. [Internet] [Thesis]. University of Queensland; 2019. [cited 2020 Jul 11]. Available from: http://espace.library.uq.edu.au/view/UQ:4925754.

Note: this citation may be lacking information needed for this citation format:
Not specified: Masters Thesis or Doctoral Dissertation

Council of Science Editors:

Hodgkinson L. Approximations for finite spin systems and occupancy processes. [Thesis]. University of Queensland; 2019. Available from: http://espace.library.uq.edu.au/view/UQ:4925754

Note: this citation may be lacking information needed for this citation format:
Not specified: Masters Thesis or Doctoral Dissertation

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