In 1999 Schramm introduced the one-parameter family of random planar chords known as Schramm-Loewner evolution (SLE(kappa)). More recently,Wang defined a functional on (deterministic) planar chords and loops called Loewner energy. The Loewner energy is the rate function of a large deviation principle on SLE(kappa) as kappa tends to 0. Curves of finite energy are more regular than SLE(kappa) and have several interesting properties. For example, there is a link to Teichmüller theory; the family of finite energy loops coincides with the class of Weil-Petersson quasicircles. In this thesis we study natural generalizations of the chordal Loewner energy. We define a two-sided radial Loewner energy, corresponding to the process of a chordal SLE conditioned to hit a marked interior point. We characterize curves of finite two-sided radial energy and show that there is a unique curve of minimal energy. We then move on to discuss a generalization of the multichordal Loewner energy, introduced by Peltola and Wang, to chords with fused endpoints. First, we construct a multichordal Loewner energy on curves which have not yet reached their respective endpoints, agreeing with the energy defined by Peltola and Wang in the limit. We then generalize this energy to two curves which aim at the same point and define the fused multichordal Loewner energy by taking the limit. / År 1999 introducerade Schramm enparameterfamiljen av stokastiska planara kordor som kallas Schramm-Loewner evolution (SLE(kappa)). På senare tid har Wang definierat en funktional på (deterministiska) planara kordor och slingor kallad Loewnerenergi. Loewnerenergin är hastighetsfunktionen av enstora avvikelserprincip på SLE(kappa) då kappa går mot 0. Kurvor med ändlig energi är mer reguljära än SLE(kappa) och har flera intressanta egenskaper. Till exempel finns det en koppling till Teichmullerteori; familjen av slingor med ändlig energi sammanfaller med klassen av Weil-Petersson-kvasicirklar. I den har masteruppsatsen studerar vi naturliga generaliseringar av den kordala Loewnerenergin. Vi definierar en tvåsidig radiell Loewnerenergi som motsvarar processen där en kordal SLE är betingad att träffa en markerad inre punkt. Vi karaktäriserar kurvor med ändlig energi och visar att det finns en unik kurva med minimal energi. Vi går sedan vidare till att diskutera en generalisering av den multikordala Loewnerenergin, introducerad av Peltola och Wang, till kordor som möts i ändpunkterna. Först konstruerar vi en en energi på kurvor som inte ännu har träffat sina ändpunkter, som överensstämmer med energin definierad av Peltola och Wang då vi går i gräns. Vi generaliserar sedan den energin till två kurvor som siktar mot samma punkt och definierar den fusionerade multikordala energin genomatt gå i gräns.

Schramm-Loewner evolution (SLE(kappa)) is an important contemporary tool for identifying critical scaling limits of two-dimensional statistical systems. The SLE(kappa) one-parameter family of processes can be viewed as a special case of a more general, two-parameter family of processes we denote SLE(kappa, mu). The SLE(kappa, mu) process is defined for kappa>0 and real numbers mu; it represents the solution of the chordal Loewner equations under special conditions on the driving function parameter which require that it is a Brownian motion with drift mu and variance kappa. We derive properties of this process by use of methods applied to SLE(kappa) and application of Girsanov's Theorem. In contrast to SLE(kappa), we identify stationary asymptotic behavior of SLE(kappa, mu). For kappa in (0,4] and mu > 0, we present a pathwise construction of a process with stationary temporal increments and stationary imaginary component and relate it to the limiting behavior of the SLE(kappa, mu) generating curve. Our main result is a spatial invariance property of this process achieved by defining a top-crossing probability for points in the upper half plane with respect to the generating curve.

Doctor en Ciencias de la Ingeniería, Mención Modelación Matemática / En esta tesis doctoral se abordan cuatro tópicos diferentes pero mutuamente relacionados: Problemas variacionales sobre álgebras de Jordan Euclideanos, problemas de complementariedad sobre espacios de matrices simétricas, análisis angular entre dos conos convexos y cerrados, y el camino central en programación cónica simétrica. La primera parte de este trabajo corresponde al estudio del concepto de operator commutation en álgebras de Jordan Euclideanos por medio del establecimiento de un principio de conmutación para problemas variacionales los cuales poseen datos espectrales. El principal enfoque de la segunda parte es el análisis y resolución numérica de una amplia clase de problemas de complementariedad formuladas en espacios de matrices simétricas. Las condiciones de complementariedad son expresadas en términos de la ordenación de Loewner o, mas general, con respecto a un par dual de conos Loewnerianos. En la tercera parte presentamos una construcción de la teoría general de ángulos críticos para pares de conos convexos y cerrados. El análisis angular de pares de conos con estructuras especiales es también abordada. Por ejemplo, en nuestro estudio incluimos: subespacios lineales, conos poliedrales, conos de revolución, conos topheavy y conos de matrices. La última parte de este trabajo está dedicada al estudio de la convergencia del camino central y del comportamiento de su punto límite en programación cónica simétrica. Esto es hecho por medio del uso de herramientas de álgebras de Jordan.

This thesis contains four papers and two introductory chapters. It is mainly devoted to problems concerning random growth models related to the Loewner differential equation. In Paper I we derive a rate of convergence of the Loewner driving function for loop-erased random walk to Brownian motion with speed 2 on the unit circle, the Loewner driving function for radial SLE(2). Thereby we provide the first instance of a formal derivation of a rate of convergence for any of the discrete models known to converge to SLE. In Paper II we use the known convergence of (radial) loop-erased random walk to radial SLE(2) to prove that the scaling limit of loop-erased random walk excursion in the upper half plane is chordal SLE(2). Our proof relies on a version of Wilson’s algorithm for weighted graphs together with a Beurling-type hitting estimate for random walk excursion. We also establish and use the convergence of the radial SLE path to the chordal SLE path as the bulk point tends to a boundary point. In the final section we sketch how to extend our results to more general domains. In Paper III we prove an upper bound on the optimal Hölder exponent for the chordal SLE path parameterized by capacity and thereby establish the optimal exponent as conjectured by J. Lind. We also give a new proof of the lower bound. Our proofs are based on sharp estimates of moments of the derivative of the inverse SLE map. In particular, we improve an estimate of G. F. Lawler. In Paper IV we consider radial Loewner evolutions driven by unimodular Lévy processes. We rescale the hulls of the evolution by capacity, and prove that the weak limit of the rescaled hulls exists. We then study a random growth model obtained by driving the Loewner equation with a compound Poisson process with two real parameters: the intensity of the underlying Poisson process and a localization parameter of the Poisson kernel which determines the jumps. A particular choice of parameters yields a growth process similar to the Hastings-Levitov HL(0) model. We describe the asymptotic behavior of the hulls with respect to the parameters, showing that growth tends to become localized as the jump parameter increases. We obtain deterministic evolutions in one limiting case, and Loewner evolution driven by a unimodular Cauchy process in another. We also show that the Hausdorff dimension of the limiting rescaled hulls is equal to 1.

京都大学 / 新制・課程博士 / 博士(理学) / 甲第22977号 / 理博第4654号 / 新制||理||1669(附属図書館) / 京都大学大学院理学研究科数学・数理解析専攻 / (主査)教授 日野 正訓, 教授 泉 正己, 准教授 楠岡 誠一郎 / 学位規則第4条第1項該当 / Doctor of Science / Kyoto University / DFAM

Made available in DSpace on 2014-06-12T18:06:31Z (GMT). No. of bitstreams: 2 arquivo904_1.pdf: 2034290 bytes, checksum: 6c7a1aa3574036c31382491866de428a (MD5) license.txt: 1748 bytes, checksum: 8a4605be74aa9ea9d79846c1fba20a33 (MD5) Previous issue date: 2010 / Universidade Federal de Pernambuco / Padrões complexos são frequentemente observados em diferentes fenômenos físicos, tais como, o movimento de uma interface entre dois fluidos não miscíveis, eletrodeposição, etc, onde a dinâmica da interface é controlada pelo gradiente de uma função potencial, a qual satisfaz a equação de Laplace. Recentemente, uma ferramenta importante da análise complexa, a equação de Loewner, tem sido utilizada para estudar problemas de crescimento laplaciano em duas dimensões. Em poucas palavras, a equação de Loewner é uma equação diferencial de primeira ordem para a evolução temporal da transformação conforme que leva o domínio físico , onde se dá o crescimento, em um domínio matemático que se asemelha ao domínio físico inicial (ou seja, aquele existente antes de começar o processo de crescimento). Nesta tese, primeiramente apresentamos uma dedução alternativa da equação de Loewner para dois casos considerados recentemente na literatura em que curvas simples crescem no semiplano superior ou na geometria do canal. Nosso método de obtenção da equação de Loewner é baseado na transformação de Schwarz-Christoffel entre os planos matemáticos em dois instantes de tempo infinitesimalmente próximos. Em seguida, estendemos o formalismo da equação de Loewner para estudar uma clase mais geral de problemas de crescimento, em que agora tem-se o avanço de uma interface envolvendo uma região de área crescente. Em nosso modelo de crescimento, a interface possui certos pontos especiais, chamados de cristas e vales, onde o fator de crescimento é um máximo e um mínimo local, respectivamente. A regra de crescimento do modelo é definida em termos de certas curvas poligonais que crescem no plano matemático. Para as duas geometrias de interesse, o semiplano superior e o canal, deduzimos a correspondente equação de Loewner que governa a dinâmica da interface. Vários exemplos de evolução temporal de interfaces são discutidos, tanto no caso em que se tem uma única interface, seja com uma ou várias cristas, quanto no caso de múltiplas interfaces crescendo simultaneamente. Em particular, o conhecido efeito de blindagem, onde uma das crista avança bem mais que as outras, é normalmente observado para o caso de interfaces não simétricas. Uma breve comparação qualitativa é feita entre nossos resultados e alguns padrões observados em experimento

L'objet de cette thèse est d'étudier la géométrie des immeubles à angles droits. Ces espaces, définis par J. Tits sont des espaces singuliers qui peuvent être vus comme des généralisations des arbres en dimension supérieure. La thèse est divisée en deux parties. Dans la première partie, nous décrivons comment la notion de résidus parallèles permet de comprendre l'action d'un groupe sur un immeuble. En corollaire nous retrouvons que dans un groupe de Coxeter et dans un produit graphé les intersections de sous-groupes paraboliques sont paraboliques. Dans la seconde partie, nous abordons la structure quasi-conforme du bord des immeubles hyperboliques à angles droits. En particulier, nous trouvons des exemples d'immeubles de dimension 3 et 4 dont le bord vérifie la propriété combinatoire de Loewner. Cette propriété est une version faible de la propriété de Loewner. Cette partie est motivée par le fait que, depuis G.D. Mostow, la structure quasi-conforme au bord a mené à plusieurs résultats de rigidités dans les espaces hyperboliques. Dans le cas des immeubles de dimension 2, M. Bourdon et H. Pajot ont prouvé la rigidité des quasi-isométries en utilisant la propriété de Loewner au bord. / The object of this thesis is to study the geometry of right-angled buildings. These spaces, defined by J. Tits, are singular spaces that can be seen as trees of higher dimension. The thesis is divided in two parts. In the first part, we describe how the notion of parallel residues allows to understand the action of a group on the building. As a corollary we recover that in Coxeter groups and in graph products intersections of parabolic subgroups are parabolic. In the second part, we discuss the quasiconformal structure of boundaries of right-angled hyperbolic buildings thanks to combinatorial tools. In particular, we exhibit some examples of buildings of dimension 3 and 4 whose boundary satisfy the combinatorial Loewner property. This property is a weak version of the Loewner property. This part is motivated by the fact that the quasiconformal structure of the boundary led to many results of rigidity in hyperbolic spaces since G.D. Mostow. In the case of buildings of dimension 2, many works have been done by M. Bourdon and H. Pajot. In particular, the Loewner property on the boundary permitted them to prove the quasi-isometry rigidity for some buildings of dimension 2.

No description available.

The Möbius energy and the Loewner energy are two Möbius invariant quantaties defined for Jordan curves. We start by introducing some of the basic properties of these two energies. Both are finite if and only if the curves belong to a class called Weil-Petersson. The Weil-Petersson class does not contain curves with corners. In part motivated by recent work of Johansson and Viklund we introduce regularized versions of both the Mövius and Loewner energy which allow for certain curves with isolated corners. We also look at the derivative of the Loewner energy. / Möbiusenergin och Loewnerenergin är två Möbius-invarianta kvantiteter definerade  för Jordan-kurvor. Vi börjar med att presentera några av de grundläggande egenskaperna hos dessa två energier. Båda är ändliga om och endast om kurvorna tillhör en klass som heter Weil-Petersson. Weil-Petersson-klassen innehåller inte kurvor med hörn. Delvis motiverad av nytt arbete av Johansson och Viklund introducerar vi regulariserade versioner av både Möbius- och Loewnerenergin som tillåter vissa kurvor med isolerade hörn. Vi tittar också på derivatan av Loewnerenergin.

The first part of the thesis deals with aspects of Loewner theory in several complex variables. First we show that a Loewner chain with minimal regularity assumptions (Df(0,t) of local bounded variation) satisfies an associated Loewner equation. Next we give a way of renormalizing a general Loewner chain so that it corresponds to the same increasing family of domains. To do this we will prove a generalization of the converse of Carathéodory's kernel convergence theorem. Next we address the problem of finding a Loewner chain solution to a given Loewner chain equation. The main result is a complete solution in the case when the infinitesimal generator satisfies Dh(0,t)=A where inf {Re<Az,z>: ||z| =1}> 0. We will see that the existence of a bounded solution depends on the real resonances of A, but there always exists a polynomially bounded solution. Finally we discuss some properties of classes of biholomorphic mappings associated to A-normalized Loewner chains. In particular we give a characterization of the compactness of the class of spirallike mappings in terms of the resonance of A. The second part of the thesis deals with the problem of finding examples of extreme points for some classes of mappings. We see that straightforward generalizations of one dimensional extreme functions give examples of extreme Carathéodory mappings and extreme starlike mappings on the polydisc, but not on the ball. We also find examples of extreme Carathéodory mappings on the ball starting from a known example of extreme Carathéodory function in higher dimensions.