A renormalization analysis is presented for a generalized Harper equation (1 + α cos(2π(ω(i + 1/2) + φ)))ψi+1 + (1 + α cos(2π(ω(i − 1/2) + φ)))ψi−1 +2λ cos(2π(iω + φ))ψi = Eψi. (0.1) For values of the parameter ω having periodic continued-fraction expansion, we construct the periodic orbits of the renormalization strange sets in function space that govern the wave function fluctuations of the solutions of the generalized Harper equation in the strong-coupling limit λ→∞. For values of ω with non-periodic continued fraction expansions, we make some conjectures based on work of Mestel and Osbaldestin on the likely structure of the renormalization strange set.

Thesis (Ph. D.)--University of Texas at Austin, 2001. / Vita. Includes bibliographical references (leaves 150-160). Available also in a digital version from Dissertation Abstracts.

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Thesis (Ph. D.)--University of Texas at Austin, 2006. / Vita. Includes bibliographical references.

Ph.D., Faculty of Science, University of Witwatersrand, 2011 / Operators in N = 4 super Yang-Mills theory with an R-charge of O(N2) are dual to backgrounds which are asymtotically AdS5 S5. In this thesis we develop e cient techniques that allow the computation of correlation functions in these backgrounds. We nd that (i) contractions between elds in the string words and elds in the operator creating the background are the eld theory accounting of the new geometry, (ii) correlation functions of probes in these backgrounds are given by the free eld theory contractions but with rescaled propagators and (iii) in these backgrounds there are no open string excitations with their special end point interactions; we have only closed string excitations. Furthermore, these correlation functions are not well approximated by the planar limit. The non-planar diagrams, which in the bulk spacetime correspond to string loop corrections, are enhanced by huge combinatorial factors. We show how these loop corrections can be resummed. As a typical example of our results, in the half-BPS background of M maximal giant gravitons we nd the usual 1=N expansion is replaced by a 1=(M +N) expansion. Further, we nd that there is a simple exact relationship between amplitudes computed in the trivial background and amplitudes computed in the background of M maximal giant gravitons. We also nd strong evidence for the BMN-type sectors suggested in arXiv:0801.4457. The problem of computing the anomalous dimensions of (nearly) half-BPS operators with a large R-charge is reduced to the problem of diagonalizing a Cuntz oscillator chain. Due to the large dimension of the operators we consider, non-planar corrections must be summed to correctly construct the Cuntz oscillator dynamics. These non-planar corrections do not represent quantum corrections in the dual gravitational theory, but rather, they account for the backreaction from the heavy operator whose dimension we study. Non-planar corrections accounting for quantum corrections seem to spoil integrability, in general. It is interesting to ask if non-planar corrections that account for the backreaction also spoil integrability. We nd a limit in which our Cuntz chain continues to admit extra conserved charges suggesting that integrability might survive.

Thesis (Ph. D.)--University of Oregon, 1999. / Typescript. Includes vita and abstract. Includes bibliographical references (leaves 165-166). Also available for download via the World Wide Web; free to University of Oregon users. Address: http://wwwlib.umi.com/cr/uoregon/fullcit?p9948028.

Le problème de la limite continue de la gravitation quantique à boucle est encore ouvert. En effet, la dynamique précise n’est pas connue et nous ne disposons pas des outils nécessaires à l’étude de cette limite le cas échéant. Dans cette thèse, nous étudions quelques méthodes de coarse-graining (étude à gros grains) qui devraient contribuer à cette entreprise. Nous nous concentrons sur deux aspects du flot: la détermination d’observables naturelles à grandes échelles d’un côté et la manière de s’abstraire du problème de la dynamique à graphe variable en la projetant sur des graphes fixes de l'autre.Pour déterminer les observables aux grandes distances, nous étudions le cas des tétraèdres hyperboliques et leur description naturelle dans un langage proche de celui de la gravitation quantique à boucle. Les holonomies de surface en particulier jouent un rôle important. Cela dégage la structure des double spin networks constitués d'un graphe et de son dual, structure qui semble aussi apparaître dans les travaux de Freidel et al. Pour résoudre le problème des graphes variables, nous considérons et définissons les loopy spin networks. Ils encodent par des boucles la courbure locale d'un vertex effectif et permettent ainsi de décrire différents graphes en les masquant via le processus de coarse-graining. De plus, leur définition donne un procédé naturel systématique de coarse-graining pour passer d'une échelle à une autre.Ensemble, ces deux principaux résultats posent le fondement d'un programme de coarse-graining pour les théories invariantes sous difféomorphismes. / The continuum limit of loop quantum gravity is still an open problem. Indeed, no proper dynamics in known to start with and we still lack the mathematical tools to study its would-be continuum limit. In the present PhD dissertation, we will investigate some coarse-graining methods that should become helpful in this enterprise. We concentrate on two aspects of the theory's coarse-graining: finding natural large scale observables on one hand and studying how the dynamics of varying graphs could be cast onto fixed graphs on the other hand.To determine large scale observables, we study the case of hyperbolic tetrahedra and their natural description in a language close to loop quantum gravity. The surface holonomies in particular play an important role. This highlights the structure of double spin networks, which consist in a graph and its dual, which seems to also appear in works from Freidel et al. To solve the problem of varying graphs, we consider and define loopy spin networks. They encode the local curvature with loops around an effective vertex and allow to describe different graphs by hidding them in a coarse-graining process. Moreover, their definition gives a natural procedure for coarse-graining allowing to relate different scales.Together, these two results constitute the foundation of a coarse-graining programme for diffeomorphism invariant theories.