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Thesis Reference Thesis Dynamics and combinatorics of 2 and 3 dimensional triangulations YOUNAN, Maher Afif Abstract We study in detail two aspects of 2 and 3 dimensional triangulations of the sphere. In the "Dynamics" part, we construct two models in statistical mechanics, on the set of all triangulations of $S^2$ and $S^3$ respectively, and we show that these models behave as glasses, in the sense that these models show a tremendous slowing down of their dynamics once we lower the temperature. We call such models "topological glasses". In the "combinatorics" part, we study the question of counting the number of triangulations of the 3d sphere $S^3$. This is an old and difficult question and even though we do not solve it, we make some very interesting contributions and we find results we strongly believe are a good step towards finding a solution. Reference YOUNAN, Maher Afif. Dynamics and combinatorics of 2 and 3 dimensional triangulations. Thèse de doctorat : Univ. Genève, 2014, no. Sc. 4738 URN : urn:nbn:ch:unige-432696 DOI : 10.13097/archive-ouverte/unige:43269 Available at: http://archive-ouverte.unige.ch/unige:43269 Disclaimer: layout of this document may differ from the published version. 1 / 1 UNIVERSITÉ DE GENÈVE FACULTÉ DES SCIENCES Section de Physique Professeur Jean-Pierre ECKMANN Département de Physique théorique Professeur Peter WITTWER Dynamics And Combinatorics Of 2 and 3 Dimensional Triangulations THÈSE présentée à la Faculté des sciences de l’Université de Genève pour obtenir le grade de Docteur ès sciences, mention physique par Maher YOUNAN de Maghdouché (Liban) Thèse No 4738 GENÈVE Atelier d’impression de l’Université de Genève 2014 Remerciements Ces cinq dernières années de ma vie ont été très enrichissantes, et m’ont permis de grandir en tant que scientifique mais aussi en tant qu’être humain. Je tiens donc, avant tout autre propos, à consacrer ce chapitre à toutes celles et tous ceux qui m’ont aidé et accompagné durant ce parcours. Tout d’abord, un très grand merci au Professeur Jean-Pierre Eckmann, mon directeur de thèse qui a partagé avec moi bien plus que ses connaissances scientifiques. Je n’oublierai pas de sitôt les pauses café où nous parlions de science mais aussi de politique, d’actualité et de religion. Le Professeur Jean-Pierre Eckmann a toujours été disponible, et je le remercie pour sa direction, ses conseils et toute l’aide qu’il m’a apportée le long de ces années. Je tiens à remercier le Professeur Peter Wittwer pour avoir accepté d’être mon codirecteur de thèse ainsi que pour toutes les discussions enrichissantes que nous avons eues et tous les conseils et l’aide qu’il m’a apportés. Je remercie le Professeur Charles Radin de l’Université de Texas, le Professeur Pierre Collet de l’École Polytechnique de Paris et le Professeur Yvan Velenik de l’Université de Genève pour avoir accepté de faire partie de mon jury de thèse et pour avoir partagé avec moi leurs connaissances lors de diverses discussions que nous avons eues durant ces dernières années. Un merci particulier au Professeur Charles-Edouard Pfister de l’École Polytechnique Fédérale de Lausanne. C’est pendant ses cours que j’ai appris à aimer la physique mathéma- tique. Ce travail n’aurait pas vu le jour sans le soutien financier du Fonds National Suisse de la Recherche Scientifique (FNS) et du Conseil Européen de la Recherche (ERC). Sur un plan plus personnel, je tiens à remercier le Département de Physique Théorique de l’Université de Genève, et en particulier les secrétaires Francine Gennai-Nicole et Cécile Jaggi-Chevalley, ainsi que l’ingénieur informatique Andreas Malaspinas. Je souhaite remercier mes collègues et amis Cyrille Zbinden, Christoph Boeckle, Julien Guillod, Noé Cuneo, Andrea Agazzi et Jimmy Dubuisson pour toutes les discussions et les petites anecdotes qu’on a eues. C’est en grande partie grâce à eux que mon expérience à Genève fut aussi riche. Je remercie toute ma famille qui m’a toujours soutenu, ma mère Lina, mon père Afif, ma sœur Nathalie et mon frère Nader. Mes pensées vont aussi à tous mes amis. Je tiens à citer Anca Stérie, Marine Vidaud, Ludovic Pirl et la famille Kambly. Sans leur soutien, mes études n’auraient probablement pas été aussi fructueuses. Finalement, je tiens à remercier toutes les personnes qui liront ce document. i ii Résumé Cette thèse traite des triangulations topologiques, i.e., des décompositions en simplexes des sphères S 2 et S 3 de dimension 2 et 3. En particulier, nous nous intéressons à deux aspects de ces triangulations: leur nombre, qui est le sujet du deuxième et troisième chapitre, ainsi que la dynamique de verre qu’on peut avoir sur ces triangulations, traitée dans le premier et le dernier chapitre. En mécanique statistique hors équilibre, un verre est défini comme étant un solide amor- phe qui ne possède pas la périodicité des cristaux. Certains verres ont même une structure similaire à celle des liquides telle qu’on peut leur associer un coefficient de viscosité. Dans cette thèse, nous nous intéressons à un type de modèles de verre très particulier que nous appellons les verres topologiques. Ils sont définis de la façon suivante: L’espace de phase est l’ensemble de toutes les triangulations de la sphère S d de dimension d = 2, 3. Le cas d = 2 est résolu dans le premier chapitre tandis que le cas d = 3 est le sujet du dernier chapitre. Les mouvements élémentaires de la dynamique sont donnés par les mouvements de Pachner qui conservent le nombre de noeuds, à savoir le mouvement T1 en 2 dimensions et les mouvement 2-3 et 3-2 en 3 dimensions. Etant donné une triangulation A, son énergie est locale, donc une somme des contributions de chaque noeud v: E = f (I(v)) , v∈VX(A) où v ∈ V(A) est un noeud de A, I(v) est le voisinage de v, i.e., la sous-triangulation de A contenant uniquement les voisins de v et f (·) est une fonction positive qui donne la contribution en énergie de v. Finalement, la dynamique est donnée par un algorithme de Metropolis dont le seul paramètre libre est la tempérture T. Ces systèmes se comportent comme des verres; en particulier, leur dynamique ralentit énormément quand la tempérture T s’approche de zéro. Par exemple, nous observons que l’énergie s’approche de sa valeur stationnaire à T ≪ 1 de façon polynomiale et non pas expo- nentielle. De plus, la fonction de corrélation du temps décroît avec un temps de relaxation τr, qu’on peut interpréter comme un coefficient de viscosité, qui croît exponentiellement en fonction de l’inverse de la température. Dans le deuxième et troisième chapitre, nous étudions en détail une propriété de l’espace de phase 3d, à savoir sa taille: quelle est le nombre de triangulations de S 3 avec t tétrahèdres? La question équivalente en 2d a été résolue par W. Tutte en 1962; sa téchnique est basée sur une variation de la méthode permettant de compter les arbres binaires. Une généralisation simple de cette méthode au cas 3d n’aboutit pas, à cause de la richesse de la topology de S 3. En particulier, il est possible de trianguler la sphère S 3 de telle façon à ce qu’elle contienne une arête nouée. iii iv Etant donné une triangulation arbitraire A de S 3, notre méthode consiste à la décomposer en un arbre de triangulations élémentaires que nous appelons noyaux. Un noyau est une triangulation 3d d’un polytope (donc avec bord), sans noeud interne, tel que chaque face interne possède au plus une arête externe (sur le bord). Un tétrahèdre (le simplexe 3d) est un noyau, mais contrairement au cas 2d, il existe des noyaux plus complexes. Nous montrons que si le nombre de noyaux avec t tétrahèdres admet une borne exponentielle en t, alors le nombre de triangulations de S 3 admet une borne exponentielle similaire. Dans le troisième chapitre, nous réduisons ces noyaux à des triangulations encore plus petites, en identifiant certains noeuds adjacents sur le bord. Nous obtenons ce que nous appelons des atomes. Nous montrons que si leur nombre admet une borne exponentielle, alors le nombre de noyaux et par suite, celui de toutes les triangulations de S 3 admet une borne similaire. L’intérêt de cette décomposition en atomes est le suivant: le nombre de triangulations contenenant une arête nouée d’une façon particulière, comme par exemple en un noeud de trèfle, est très grand. Par contre, un atome contient l’essentiel de l’information topologique. Nous postulons que le nombre d’atomes contenant une arête avec un noeud particulier est très petit, peut-être même égal à un. Démontrer un tel résultat permtettra de résoudre le problème consistant à compter les triangulations de S 3. Contents Remerciements i Résumé iii 1 Introduction 1 2 2D Topological Glass 11 2.1 Introduction,themodel. ...... 11 2.2 Equilibrium and the approximation of the dynamics . ........... 12 2.3 Descriptionofthestationarystate . ......... 12 2.3.1 Energyofthestationarystate . ..... 13 2.3.2 Distributionofthecolors . .... 13 2.3.3 Energycostofflips .............................. 14 2.3.4 The number of local defect configurations . ....... 15 2.4 Dynamics of the system (at equilibrium) . ......... 17 2.4.1 Themostprobableflips. .. 18 2.4.2 Lifetimeofpairs............................... .. 21 2.5 The Geometry of pair-defect collisions . .......... 21 2.5.1 Definition..................................... 22 2.5.2 Collisiontypes................................ .. 22 2.6 Relevantandirrelevantpairs . ....... 26 2.7 Timecorrelationsatequilibrium . ........ 27 2.8 Theagingprocess ................................. ... 29 2.8.1 Threetimescales............................... .. 30 2.8.2 The quasistationarity assumption and the density of pairs ........ 30 2.8.3 The diffusion constant of single defects .
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