Nós apresentamos o processo K numa árvore, que é um processo de Markov com estados instantâneos e generaliza o processo K no grafo completo, como o limite do modelo de armadilha numa árvore, e aplicamos esse resultado para derivar um limite de escala para o modelo de armadilha do GREM. / We present the K process on a tree, which is a Markov process with instantaneous states and generalises the K process on the complete graph, as a limit of the trap model on a tree, and apply this result to derive a scaling limit to the GREM-like trap model.

Derivamos resultados para o Modelo de Armadilhas de Bouchaud no hipercubo a baixa temperatura. Este é um passeio aleatório simples simétrico em tempo contínuo que espera um tempo exponencial com taxa aleatória com distribuição no domínio de atração de uma lei estável de expoente menor do que 1. Os resultados recaem sobre o processo limite chamado K-processo, basicamente, um processo markoviano em um espaço de estados enumerável que entra em qualquer conjunto finito com distribuição uniforme. / We derive results for the Bouchaud trap model in the hypercube at low temperature. This is a continuous-time simple symmetric random walk on hypercube that waits a exponetial time with a random rate with distribution in the domain of attraction of a stable law of exponent lower than 1. The results arise to a scaling limit called k-process, roughly, a Markov process in a denumerable state space which enters finite sets with uniform distribution.

Introduzimos uma familia de trajetorias aleatorias coalescentes com certo tipo de comportamento radial a qual chamaremos de Teia Poissoniana radial discreta. Mostramos que o limite fraco na escala difusiva desta familia e uma familia de trajetorias aleatorias coalescentes que chamaremos de Teia Browniana radial. Por m, caraterizamos o objeto limite como um mapeamento continuo da Teia Browniana restrita num subconjunto de R2. / We introduce a family of coalescing random paths with certain kind of radial behavior. We call them the discrete radial Poisson Web. We show that under diusive scaling this family converges in distribution to a family of coalescing random paths which we call radial Brownian Web. Finally, we characterize the limiting object as a continuous mapping of the Brownian Web restricted to a subset of R2.

Introduzimos uma familia de trajetorias aleatorias coalescentes com certo tipo de comportamento radial a qual chamaremos de Teia Poissoniana radial discreta. Mostramos que o limite fraco na escala difusiva desta familia e uma familia de trajetorias aleatorias coalescentes que chamaremos de Teia Browniana radial. Por m, caraterizamos o objeto limite como um mapeamento continuo da Teia Browniana restrita num subconjunto de R2. / We introduce a family of coalescing random paths with certain kind of radial behavior. We call them the discrete radial Poisson Web. We show that under diusive scaling this family converges in distribution to a family of coalescing random paths which we call radial Brownian Web. Finally, we characterize the limiting object as a continuous mapping of the Brownian Web restricted to a subset of R2.

Nós apresentamos o processo K numa árvore, que é um processo de Markov com estados instantâneos e generaliza o processo K no grafo completo, como o limite do modelo de armadilha numa árvore, e aplicamos esse resultado para derivar um limite de escala para o modelo de armadilha do GREM. / We present the K process on a tree, which is a Markov process with instantaneous states and generalises the K process on the complete graph, as a limit of the trap model on a tree, and apply this result to derive a scaling limit to the GREM-like trap model.

Derivamos resultados para o Modelo de Armadilhas de Bouchaud no hipercubo a baixa temperatura. Este é um passeio aleatório simples simétrico em tempo contínuo que espera um tempo exponencial com taxa aleatória com distribuição no domínio de atração de uma lei estável de expoente menor do que 1. Os resultados recaem sobre o processo limite chamado K-processo, basicamente, um processo markoviano em um espaço de estados enumerável que entra em qualquer conjunto finito com distribuição uniforme. / We derive results for the Bouchaud trap model in the hypercube at low temperature. This is a continuous-time simple symmetric random walk on hypercube that waits a exponetial time with a random rate with distribution in the domain of attraction of a stable law of exponent lower than 1. The results arise to a scaling limit called k-process, roughly, a Markov process in a denumerable state space which enters finite sets with uniform distribution.

This thesis consists of two papers devoted to the asymptotics of random matrix ensembles and measure valued stochastic processes which can be considered as generalizations of the Gaussian unitary ensemble (GUE) of Hermitian matrices H=A+A†, where the entries of A are independent identically distributed (iid) centered complex Gaussian random variables. In the first paper, a system of interacting diffusing particles on the real line is studied; special cases include the eigenvalue dynamics of matrix-valued Ornstein-Uhlenbeck processes (Dyson's Brownian motion). It is known that the empirical measure process converges weakly to a deterministic measure-valued function and that the appropriately rescaled fluctuations around this limit converge weakly to a Gaussian distribution-valued process. For a large class of analytic test functions, explicit formulae are derived for the mean and covariance functionals of this fluctuation process. The second paper concerns a family of random matrix ensembles interpolating between the GUE and the Ginibre ensemble of n x n matrices with iid centered complex Gaussian entries. The asymptotic spectral distribution in these models is uniform in an ellipse in the complex plane, which collapses to an interval of the real line as the degree of non-Hermiticity diminishes. Scaling limit theorems are proven for the eigenvalue point process at the rightmost edge of the spectrum, and it is shown that a non-trivial transition occurs between Poisson and Airy point process statistics when the ratio of the axes of the supporting ellipse is of order n -1/3. / Denna avhandling består av två vetenskapliga artiklar som handlar om gränsvärdessatser för slumpmatriser och måttvärda stokastiska processer. De modeller som studeras kan betraktas som generaliseringar av den gaussiska unitära ensembeln (GUE) av hermiteska n x n-matriser H=A+A†, där A är en matris vars element är oberoende, likafördelade, centrerade, komplexa normalfördelade stokastiska variabler. I artikel I betraktas ett system av växelverkande diffunderande partiklar på reella linjen, vissa specialfall av denna modell kan tolkas som egenvärdesdynamiken för matrisvärda Ornstein-Uhlenbeck-processer (Dysons brownska rörelse). Sedan tidigare är det känt att den empiriska måttprocessen konvergerar svagt mot en deterministisk måttvärd funktion och att fluktuationerna runt denna gräns, i lämplig skalning, konvergerer svagt mot en distributionsvärd gaussisk process. För en stor klass av analytiska testfunktioner härleds explicita formler för medelvärdes- och kovariansfunktionalerna för denna fluktuationsprocess. Artikel II behandlar en familj av slumpmatrisensembler som interpolerar mellan GUE och Ginibre-ensembeln, bestående av matriser A som ovan. För denna modell är egenvärdena komplexa och asymptotiskt likformigt fördelade i en ellips i komplexa planet. Skalningsgränsvärdessatser för egenvärdet med maximal realdel och för egenvärdespunktprocessen kring detta visas för ett allmänt val av interpolationsparametern i modellen. Då förhållandet mellan axlarna i den asymptotiska ellipsen är av storleksordning n-1/3 uppträder en övergångsfas mellan Airypunktprocess- och Poissonprocessbeteendena, typiska för GUE respektive Ginibre-ensembeln. / QC 20100705

Submitted by Maike Costa (maiksebas@gmail.com) on 2016-03-28T13:00:07Z No. of bitstreams: 1 arquivo total.pdf: 809257 bytes, checksum: 7406ef37d18bbaf1d9cdd5649f5cff19 (MD5) / Made available in DSpace on 2016-03-28T13:00:07Z (GMT). No. of bitstreams: 1 arquivo total.pdf: 809257 bytes, checksum: 7406ef37d18bbaf1d9cdd5649f5cff19 (MD5) Previous issue date: 2015-12-11 / Let X = fX 0;X0 = 0g be a mean zero -stable random walk on Z with inhomogeneous jump rates f 􀀀1 i ; i 2 Zg, with 2 (1; 2] and f i : i 2 Zg is a family of independent random walk variables with common marginal distribution in the basis of attraction of an -stable law with 2 (0; 2]. In this paper we derive results about the long time behavior of this process, we obtain the scaling limit. To this end, rst we will approach probability on metric spaces, speci cally treat the D space of the functions that are right-continuous and have left-hand limits. We will also expose some results dealing with stable laws that are directly related to the above problem. / Seja X = fX 0;X0 = 0g um passeio aleat orio de m edia zero -est avel sobre Z com taxas de saltos n~ao homog^eneas f 􀀀1 i ; i 2 Zg, com 2 (1; 2] e f i : i 2 Zg uma fam lia de vari aveis aleat orias independentes com distribui c~ao marginal comum na bacia de atra c~ao de uma lei -est avel com 2 (0; 2]. Neste trabalho, obtemos resultados sobre o comportamento a longo prazo deste processo obtendo seu limite de escala. Para isso, faremos previamente um estudo sobre probabilidade em espa cos m etricos, mais especi camente sobre o espa co D das fun coes cont nuas a direita com limite a esquerda. Tamb em iremos expor alguns resultados que tratam de leis est aveis que est~ao relacionadas diretamente ao problema supracitado.

We consider the finite two-dimensional Ising model on a lattice with periodic boundaryconditions. Kaufman determined the spectrum of the transfer matrix on the finite,periodic lattice, and her derivation was a simplification of Onsager's famous result onsolving the two-dimensional Ising model. We derive and rework Kaufman's resultsby applying representation theory, which give us a more direct approach to computethe spectrum of the transfer matrix. We determine formulas for the spin correlationfunction that depend on the matrix elements of the induced rotation associated withthe spin operator. The representation of the spin matrix elements is obtained byconsidering the spin operator as an intertwining map. We wrap the lattice aroundthe cylinder taking the semi-infinite volume limit. We control the scaling limit of themulti-spin Ising correlations on the cylinder as the temperature approaches the criticaltemperature from below in terms of a Bugrij-Lisovyy conjecture for the spin matrixelements on the finite, periodic lattice. Finally, we compute the matrix representationof the spin operator for temperatures below the critical temperature in the infinite-volume limit in the pure state defined by plus boundary conditions.

Le modèle d'accrochage de polymère décrit le comportement d'une chaîne de Markov en interaction avec un état donné. Cette interaction peut attirer ou repousser la chaîne de Markov et elle est modulée par deux paramètres, h et β. Quand β = 0 on parle de modèle homogène, qui est complètement solvable. Le modèle désordonné, i.e. quand β > 0, est mathématiquement le plus intéressant. Dans ce cas, l'interaction dépend d'une source d'aléa extérieur indépendant de la chaîne de Markov, appelée désordre. L'interaction est réalisée en modifiant la loi originelle de la chaîne de Markov par une mesure de Gibbs et la probabilité obtenue définit le modèle d'accrochage de polymère. Le but principal est d'étudier et de comprendre la structure des trajectoires typiques de la chaîne de Markov sous cette nouvelle probabilité. Le premier sujet de recherche concerne le modèle d'accrochage de polymère où le désordre est à queues lourdes et où le temps de retour de la chaîne de Markov suit une distribution sous-exponentielle. Dans notre deuxième résultat nous étudions le modèle d'accrochage de polymère avec un désordre à queues légères et le temps de retour de la chaîne de Markov avec une distribution à queues polynomiales d'exposant α > 0. On peut démontrer qu'il existe un point critique, h(β). Notre but est comprendre le comportement du point critique quand β -> 0. La réponse dépend de la valeur de α. Dans la littérature on a des résultats précis pour α < ½ et α > 1. Nous montrons que α ∈ (1/2, 1) le comportement du modèle dans la limite du désordre faible est universel et le point critique, opportunément changé d'échelle, converge vers la même quantité donnée par un modèle continu / The pinning model describes the behavior of a Markov chain in interaction with a distinguished state. This interaction can attract or repel the Markov chain path with a force tuned by two parameters, h and β. If β = 0 we obtain the homogeneous pinning model, which is completely solvable. The disordered pinning model, i.e. when β > 0, is most challenging and mathematically interesting. In this case the interaction depends on an external source of randomness, independent of the Markov chain, called disorder. The interaction is realized by perturbing the original Markov chain law via a Gibbs measure, which defines the Pinning Model. Our main aim is to understand the structure of a typical Markov chain path under this new probability measure. The first research topic of this thesis is the pinning model in which the disorder is heavy-tailed and the return times of the Markov chain have a sub-exponential distribution. In our second result we consider a pinning model with a light-tailed disorder and the return times of the Markov chain with a polynomial tail distribution, with exponent α > 0. It is possible to show that there exists a critical point, h(β). Our goal is to understand the behavior of the critical point when β -> 0. The answer depends on the value of α and in the literature there are precise results only for the case α < ½ et α > 1. We show that for α ∈ (1/2, 1) the behavior of the pinning model in the weak disorder limit is universal and the critical point, suitably rescaled, converges to the related quantity of a continuum model