This work presents a numerical investigation of self-avoiding walks (SAWs) on percolation clusters, a canonical model for polymers in disordered media. A new algorithm has been developed allowing exact enumeration of over ten thousand steps. This is an increase of several orders of magnitude compared to previously existing enumeration methods, which allow for barely more than forty steps. Such an increase is achieved by exploiting the fractal structure of critical percolation clusters: they are hierarchically organized into a tree of loosely connected nested regions in which the walks segments are enumerated separately. After the enumeration process, a region is \"decimated\" and behaves in the following effectively as a single point. Since this method only works efficiently near the percolation threshold, a chain-growth Monte Carlo algorithm has also been used. Main focus of the investigations was the asymptotic scaling behavior of the average end-to-end distance as function of the number of steps on critical clusters in different dimensions. Thanks the highly efficient new method, existing estimates of the scaling exponents could be improved substantially. Also investigated were the number of possible chain conformation and the average entropy, which were found to follow an unusual scaling behavior. For concentrations above the percolation threshold the exponent describing the growth of the end-to-end distance turned out to differ from that on regular lattices, defying the prediction of the accepted theory. Finally, SAWs with short range attractions on percolation clusters are discussed. Here, it emerged that there seems to be no temperature-driven collapse transition as the asymptotic scaling behavior of the end-to-end distance even at zero temperature is the same as for athermal SAWs. / Die vorliegenden Arbeit präsentiert eine numerische Studie von selbstvermeidenden Zufallswegen (SAWs) auf Perkolationsclustern, ein kanonisches Modell für Polymere in stark ungeordneten Medien. Hierfür wurde ein neuer Algorithmus entwickelt, welcher es ermöglicht SAWs von mehr als zehntausend Schritten exakt auszuzählen. Dies bedeutet eine Steigerung von mehreren Größenordnungen gegenüber der zuvor existierenden Methode, welche kaum mehr als vierzig Schritte zulässt. Solch eine Steigerung wird erreicht, indem die fraktale Struktur der Perkolationscluster geziehlt ausgenutzt wird: Die Cluster werden hierarchisch in lose verbundene Gebiete unterteilt, innerhalb welcher Wegstücke separat ausgezählt werden können. Nach dem Auszählen wird ein Gebiet \"dezimiert\" und verhält sich während der Behandlung größerer Gebiete effektiv wie ein Gitterpunkt. Da diese neue Methode nur nahe der Perkolationsschwelle funktioniert, wurde zum Erzielen der Ergebnisse zudem ein Kettenwachstums-Monte-Carlo-Algorithmus (PERM) eingesetzt. Untersucht wurde zunächst das asymptotische Skalenverhalten des Abstands der beiden Kettenenden als Funktion der Schrittzahl auf kritischen Clustern in verschiedenen Dimensionen. Dank der neuen hochperformanten Methode konnten die bisherigen Schätzer für den dies beschreibenden Exponenten signifikant verbessert werden. Neben dem Abstand wurde zudem die Anzahl der möglichen Konformationen und die mittlere Entropie angeschaut, für welche ein ungewöhnliches Skalenverhalten gefunden wurde. Für Konzentrationen oberhalb der Perkolationsschwelle wurde festgestellt, dass der Exponent, welcher das Wachstum des Endabstands beschreibt, nicht dem für freie SAWs entspricht, was nach gängiger Lehrmeinung der Fall sein sollte. Schlussendlich wurden SAWs mit Anziehung zwischen benachbarten Monomeren untersucht. Hier zeigte sich, dass es auf kritischen Perkolationsclustern keinen Phasenübergang zu geben scheint, an welchem die Ketten kollabieren, sondern dass das Skalenverhalten des Endabstands selbst am absoluten Nullpunkt der Temperatur unverändert ist.

L'objet de cette thèse est l'étude d'une classe importante de marches aléatoires en milieu aléatoire, appelée marches aléatoires avec conductances aléatoires. Nous présentons trois principaux résultats montrant des comportements opposés, irrégulier et standard du noyau de la chaleur des marches aléatoires avec conductances aléatoires à queue polynômiale. Les deux premiers (cf. Chapitre 2) portent sur les marches aléatoires simples dans $\Z^d, d>1$, gouvernées par une famille de conductances aléatoires i.i.d. à valeurs dans l'intervalle $[0,1]$, avec une queue polynomiale d'exposant $\gamma$ au voisinage de $0$. Nous montrons en premier lieu pour toute dimension supérieure à $4$ que la probabilité de retour après $2n$ sauts décroit de façon irrégulière en ce sens qu'elle admet une borne inférieure que l'on peut rendre, à un terme sous-polynomial près, aussi proche que l'on veut de $1/n^{2}$ en laissant le paramètre $\gamma$ tendre vers $0$. En considérant le même modèle et à l'opposé du premier résultat, nous montrons en second lieu pour toute dimension $d$ supérieure à $2$ que le noyau de la chaleur de la marche aléatoire admet une borne supérieure que l'on peut rendre, à un terme sous-polynomial près, aussi proche que l'on veut de la borne standard $1/n^{d/2}$ en laissant le paramètre $\gamma$ tendre vers l'infini. Nous considérons dans le troisième résultat (cf. Chapitre 3) les mêmes chaînes de Markov mais en temps continu et étudions la décroissance de la probabilité de retour asymptotique. Nous prouvons pour tout $\gamma> d/2$ que la dimension spectrale est standard, i.e. égale à $d$. Une conséquence prévisible de ce résultat est que ceci reste tout aussi vrai en temps discret.

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La diffusion des ions et des radionucléides au sein des matériaux cimentaires est l'un des facteurs les plus importants qui déterminent la durabilité et les propriétés de confinement de ces matériaux. Cette étude s'inscrit, en particulier, dans le domaine de confinement des déchets radioactifs de faible et moyenne activité. Elle consiste à mettre en évidence l'influence de la microstructure des mortiers, notamment la présence des granulats, sur la diffusion de l'eau tritiée au sein de ces matériaux. La démarche consiste, dans un premier temps, à sélectionner des formulations de mortiers à base de CEM I afin d'étudier l'influence de la teneur en granulats, de la granulométrie et du rapport eau/ciment sur les paramètres de diffusion. Des différentes techniques expérimentales complémentaires ont été utilisées afin de caractériser la structure poreuse : porosimétrie à l'eau, porosimétrie mercure, perte au feu et imagerie MEB associée à l'analyse d'image. Dans ce contexte, un protocole d'analyse d'images a été mis en place afin de quantifier la porosité à l'interface granulat/pâte. Le lien entre les propriétés de la microstructure et les paramètres de transport a été ensuite examiné. Pour cela, des essais de diffusion à l'eau tritiée (HTO) ont été conduits et des corrélations entre les paramètres de la microstructure et le transport ont été réalisées. Enfin, afin de mettre en avant le rôle des phases mésoscopiques (Matrice/granulats/ITZ) dans le mécanisme de diffusion un modèle 3D a été développé et des calculs de diffusivités équivalentes ont été effectués. La présente étude confirme la présence d'une interface granulat/pâte au voisinage des grains de sable siliceux. Cette auréole de transition (ITZ) se caractérise par une épaisseur qui varie entre 10 et 20 µm et une porosité environ trois fois plus grande que celle de la matrice cimentaire. En dessous de 55% de sable normalisé, l'effet de cette interface sur les propriétés macroscopiques de transport est faible. En effet, l'effet de dilution et de tortuosité liés aux granulats reste dominant. Par conséquent, les données acquises à l'échelle de pâte de ciment restent valables et sont extrapolable à l'échelle des mortiers. Ces résultats ont été confirmés par les calculs analytiques et numériques de la diffusivité homogénéisée. Au-delà de 55% de sable normalisé, d'autres effets liés au grands nombre de grains de sable rentrent en jeu comme les bulles d'air et les taches poreuses dus principalement à la difficulté d'obtenir des matériaux bien compactés. Ceci rend ces formulations extrêmes et ne permettent pas d'approfondir notre compréhension du lien entre la microstructure et les propriétés de transport au-delà de cette teneur en sable

Renormalization group analysis is a useful tool for studying critical behaviour of stochastic systems. In this thesis, field-theoretic renormalization group will be applied to the scalar model representing directed percolation, known as Gribov model, in presence of the random velocity field. Turbulent mixing will be modelled by the compressible form of stochastic Navier-Stokes equation where the compressibility is described by an additional field related to the density. The task will be to find corresponding scaling properties.

Since their discovery in the 1950's by Erdos and Renyi, network theory (the study of objects and their associations) has blossomed into a full-fledged branch of mathematics. Due to the network's flexibility, diverse scientific problems can be reformulated as networks and studied using a common set of tools. I define a network G = (V,E) composed of two parts: (i) the set of objects V, called nodes, and (ii) set of relationships (associations) E, called links, that connect objects in V. We can extend the classic network of nodes and links by describing the intensity of these associations with weights. More formally, weighted networks augment the classic network with a function f(e) from links to the real line, uncovering powerful ways to model real-world applications. This thesis studies new ways to construct robust micro powergrids, mine people's perceptions of causality on a social network, and proposes a new way to analyze crowdsourcing all in the context of the weighted network model. The current state of Earth's ecosystem and intensifying climate calls on scientists to find new ways to harvest clean affordable energy. A microgrid, or neighborhood-scale powergrid built using renewable energy sources attached to personal homes, suggest one way to ameliorate this energy crisis. We can study the stability (robustness) of such a small-scale system with weighted networks. A novel use of weighted networks and percolation theory guides the safe and efficient construction of power lines (links, E) connecting a small set of houses (nodes, V) to one another and weights each power line by the distance between houses. This new look at the robustness of microgrid structures calls into question the efficacy of the traditional utility. The next study uses the twitter social network to compare and contrast causal language from everyday conversation. Collecting a set of 1 million tweets, we find a set of words (unigrams), parts of speech, named entities, and sentiment signal the use of informal causal language. Breaking a problem difficult for a computer to solve into many parts and distributing these tasks to a group of humans to solve is called Crowdsourcing. My final project asks volunteers to 'reply' to questions asked of them and 'supply' novel questions for others to answer. I model this 'reply and supply' framework as a dynamic weighted network, proposing new theories about this network's behavior and how to steer it toward worthy goals. This thesis demonstrates novel uses of, enhances the current scientific literature on, and presents novel methodology for, weighted networks.

Mémoire numérisé par la Direction des bibliothèques de l'Université de Montréal.

Associamos independentemente a cada vértice v de un grafo infinito G um raio de infecção aleatório R_v e definimos um modelo de percolação sujeito às seguintes regras: (1) no tempo zero só a raiz é declarada infectada, (2) um vértice é declarado infectado em um instante t, t>0, se está a uma distância no maximo R_v de algum vértice v previamente infectado, e (3) vértices infectados permanecem infectados para sempre. Dizemos que há sobrevivência em uma realização particular do modelo se o número final de vértices infectados é infinito. Neste trabalho damos condições suficientes sobre o grafo G para a transição de fase deste modelo, estabelecendo limitantes não triviais para o parâmetro crítico quando os raios R_v têm distribuição geometrica de parâmetro 1-p. Além disto, restringindo nosso estudo para o caso das árvores esfericamente simétricas, obtemos um melhor limitante superior para este parâmetro. Finalmente, concluímos que o parâmetro crítico para o modelo nas árvores homogêneas de grau d+1 se comporta assintoticamente como 1/(2d). / We assign independently to each vertex v of an infinite graph G, a random radius of infection R_v and define a percolation model subject to the following rules: (1) at time zero, only the root is declared infected, (2) a vertex is declared infected at time t, t>0, if it is at distance at most R_v of some vertex v previously infected, and (3) infected vertices stay infected forever. We say that there is survival in a particular realization of the model if the final number of infected vertices is infinite. In this work, we give sufficient conditions on the graph G for the phase transition of this model, by stating non-trivial bounds for the critical parameter when the radii have geometrical distribution with parameter 1-p. In addition, restricting our study to the case of the spherically symmetric trees, we obtain an improved upper bound for this critical parameter. Finally, we conclude that the critical parameter for the model on homogeneous trees of degree (d+1) behaves asymptotically as 1/(2d).

Mistura compactada de materiais inconsolidados das formações Serra Geral e Botucatu foram percoladas por soluções com diversas concentrações de K+, Cl- e Cu++, em testes de coluna, para a avaliar a potencialidade da sua utilização como liner. Na modelagem dos resultados adotaram-se resoluções analíticas e semi-analíticas (programa POLLUTE v6). Estudo complementar sobre o comportamento eletroquímico destes solos foi realizado pelos ensaios de titulação potenciométrica, capacidade de tamponamento, pH em água e KCI, CTC e análise mineralógica por difração de Raios-X e térmica diferencial. Os resultados mostraram uma forte influência das características da carga elétrica superficial do solo no comportamento competitivo dos íons. / A compacted mixture of soils from Serra Geral and Botucatu formations were percolated by chemical solutions with several concentrations of K+, Cu++ e Cl-, in column tests, in order to assess the potential of this mixture as a liner-building material. In the modeling procedures for the results, analytical and semi-analytical solutions (POLLUTE v.6 software) were used. Additional studies about the electrochemical behavior of these soils were performed, using potenciometric titration, soil buffer capacity, pH in water and KCI and mineralogical assessment by X-Ray Diffraction and differential thermal analysis. The results showed a very strong influence of the soil superficial charge in the competitive ion behavior.

O estudo de Teoria de Jogos tem se expandido para diversas áreas, tendo sua aplicação inicial na economia, hoje é utilizado na psicologia, na filosofia e tem um papel importantíssimo na biologia evolutiva. O seu sucesso está ligado ao fato de que os jogos têm o poder de prever interações usando conceitos simples como a cooperação e a competição. Dentre os jogos há o famoso de Dilema do Prisioneiro (PD), em que indivíduos completamente racionais devem optar entre cooperar ou trair (desertar) seu companheiro de jogo. A estratégia dominante e o equilíbrio de Nash, para o PD, é a deserção mútua visto que os indivíduos são sempre tentados a não cooperar. O dilema é que eles obteriam um ganho melhor se cooperassem mutuamente. Na vida real os indivíduos se encontram em várias situações nas quais eles devem optar entre ser egoístas ou altruístas e, frequentemente, acabam optando pelo altruísmo. Mesmo com a previsão da deserção na teoria clássica dos jogos, em 1992 Nowak e May (NOWAK; MAY, 1992) mostraram que cooperação é mantida em jogos com interação espacial e evolutivos A partir dessa descoberta, estudos de jogos em diversos tipos de rede foram propostos, entre eles as redes diluídas (que possuem sítios vacantes). Nesse tipo de rede foi observado que certas densidades favorecem a cooperação, particularmente próximo ao limiar de percolação para regras de atualização estocásticas (com ruído). Porém a probabilidade de troca do Replicador, mesmo sendo estocástica, não se encaixa nesse padrão observado. Descobrimos que esse comportamento anômalo está relacionado com estruturas formadas entre buracos e desertores que impedem alguns indivíduos de ter acesso ao ruído, assim a informação não flui livremente na rede. Consequentemente o sistema fica preso em um estado congelado, que pode ser quebrado com algum tipo de perturbação. Também abordamos a relação entre o limiar de percolação por sítio e a cooperação de uma forma mais quantitativa do que já foi apresentada até então, acompanhamos o desenvolvimento da cooperação dentro dos clusters e mostramos como o limiar de percolação afeta as estruturas básicas da rede. / The study of Game Theory, having its initial application in economics, has expanded to several areas and is now used in psychology, philosophy and plays a major role in evolutionary biology. Its success is related to the fact that games have the power to predict and study interactions using simple concepts such as cooperation and competition. Among the games there is the famous Prisoner Dilemma (PD), where completely rational individuals have to choose between cooperating or betraying their game partner. The dominant strategy and the Nash equilibrium for PD is mutual desertion as individuals are always tempted to not cooperate. The dilemma is that they would get a higher payoff if they mutually cooperated. In real life, individuals find themselves in various situations where they must choose to be selfish or altruistic, and often they choose altruism. Even with the prediction of defection in classical game theory, in 1992, Nowak and May (NOWAK; MAY, 1992) showed that cooperation is maintained in evolutionary spatial games. With this discovery, the study of games on several types of networks was proposed, among them the diluted networks (which have vacant sites) In this type of lattice, it was observed that at certain densities cooperation is promoted, particularly close to the percolation threshold for stochastic updating rules. However, the exchange probability of the Replicator dynamics, despite being stochastic, does not obey this observed pattern. We found that this anomalous behavior is related to structures formed between holes and defectors that prevent some individuals from having access to noise, so information does not flow freely in the network. Consequently the system becomes trapped in a frozen state, but this state can be broken by perturbing the system. We also address the relationship between the percolation threshold and cooperation in a more quantitative way than has been presented lately, by following the development of cooperation within clusters and showing how the percolation threshold affects the basic structures of the lattice.