Le thème de cette thèse est l’utilisation de méthodes probabilistes (plus spécifiquement de technique venant de la théorie de la percolation) pour mener une analyse non-perturbative de plusieurs modèles de physique statistique. La thèse est centrée sur les systèmes de spins et les modèles de percolation. Cette famille de modèle comprend le modèle d’Ising, le modèle de Potts, la percolation de Bernoulli, la percolation de Fortuin-Kasteleyn et les modèles de percolation continue. L’objectif principal de la thèse est de démontrer la décroissance exponentielle des corrélations au-dessus de la température critique et d’étudier les états de Gibbs des modèles en dessus. / The underlying theme of this thesis is using probabilistic methods and especially techniques of percolation theory to carry on a non-perturbative analysis of several models of statistical physics. The focus of this thesis is set on spin systems and percolation models including the Ising model, the Potts model, the Bernoulli percolation, the random-cluster model, and the continuum percolation models. The main objective of the thesis is to demonstrate exponential decay of correlations above the critical temperature and study the Gibbs states of the mentioned models.

This thesis generalizes and interprets Kesten\''s Incipient Infinite Cluster (IIC) measure in two ways. Firstly we generalize Járai\''s result which states that for planar lattices the local configurations around a typical point taken from crossing collection is described by IIC measure. We prove in Chapter 2 that for backbone, lowest crossing and set of pivotals, the same hold true with multiple armed IIC measures. We develop certain tools, namely Russo Seymour Welsh theorem and a strong variant of quasi-multiplicativity for critical percolation on 2-dimensional slabs in Chapters 3 and 4 respectively. This enables us to first show existence of IIC in Kesten\''s sense on slabs in Chapter 4 and prove that this measure can be interpreted as the local picture around a point of crossing collection in Chapter 5.

The effect of furrow spacing on infiltration should be determined in order to properly design an irrigation system. The blocked furrow infiltrometer (BFI) and the flowing furrow infiltrometer (FFI) methods were investigated for this purpose in two areas upon a precision field furrow. Three irrigations were performed in each method. The initial and final soil moisture contents (before and after irrigation), the furrow cross-section (before and after irrigation), the inflow volume and the furrow water surface elevations (during irrigation) were measured in each test furrow. Cumulative infiltration and infiltration rates were determined for each irrigation. The results indicate that the FFI test furrows infiltrated more water than did the BFI test furrows for the same infiltration time. The infiltration rates were higher in the FFI test furrows than in the BFI test furrows until they approach the basic intake rate. The infiltration rates were also higher during the 0.90 m spacing tests than during the 1.80 m spacing tests. The 0.90 m spacing test furrows infiltrated more water than did the 1.80 m spacing test furrows.

published_or_final_version / Civil Engineering / Master / Master of Science in Engineering

The movement of salts in soils following application of feedlot wastes was studied experimentally and theoretically. The objectives of the study were (1) to evaluate the movement of salts in the soil following heavy application of animal wastes as related to the aggregate sizes of manure and water management practices and (2) to develop a mathematical model to predict the movement of salts within the soil and manure mixture. In the experimental study, air dried manure was formed into three distinct sizes, small (to pass 40 mesh sieve), medium (0.9 am in diameter, 1.2 cm in length), and large (1.8 cm in diameter, 1.8 cm in length). Soil columns of 15 cm inside diameter were packed with 30 cm of a soil-manure mixture over a 10 cm depth of soil. The manure application rate was equivalent to 100 metric tons/ha calculated on the basis of the cross-sectional area of the column. A total of five pore volumes of water was passed through the soil under continuous and intermittent water applications. The leachates were collected in 1/2 pore volume increments and the volume, EC, and pH determined immediately. Within 48 hours of the sampling, the leachates were analyzed for Na, K, Ca, Mg, Cl, and five trace elements (Fe, Mn, Zn, Cu, and Ni). The EC of the leachate for the first 1/2 pore volume was significantly (1% level) highest for the small-sized aggregates and the lowest for the large aggregate treatments under both moisture regimes. During the second increment, the order was reversed. In the later water applications, the EC for small aggregates was higher than the other sizes. There were no significant differences between the EC of the leachate from medium and large aggregate treatments during the later periods under either water treatments. Under both moisture regimes, the amount of Na removed from small aggregates decreased more rapidly than the other sizes. More than 90% of the total Na added to the system by manure was removed from the small aggregate treatment. In contrast, the highest amount of K removed (895 mg from small-sized aggregates under continuous leaching' represents about 35% of the total amount present In the manure applied. More Ca was removed from the small-sized aggregate treatments under both moisture regimes than was added by manure application. As for Mg the pattern of the removal was similar to that of Ca. No Cl was detected in the leachate after the fifth 1/2 pore volume was displaced. A mathematical model was developed to predict the movement of readily soluble ions such as Na, K, and Cl from different aggregate sizes of manure. The theoretical curves were compared with earlier experimental data and the parameters appearing in the mathematical equation were estimated. The results for Cl, Na, and K are presented graphically, and the estimated parameters as well as the values of the square roots of the sum of the squares between the theoretical and experimental values as percentage of the sum of the experimental values (SSR) are reported. From the low value of SSR's, it is evident that the model can predict the movement of the readily soluble ions from different aggregate sizes of manure quite accurately. A discussion on the suitability of the model for different sizes of manure aggregates and also a comparison between two different procedures to fit the model to the experimental data are given. In addition, a three phase theoretical model was developed to describe the movement of readily soluble ions from a soil-manure-water system. Experimental data were used in testing the model. The results for Na, K, and Cl are presented graphically. Estimated parameters for the experimental system and the values of SSR are presented. This model also can predict the movement of readily soluble ions from a soil-manure-water system.

A teoria da percolação tem se revelado muito útil no tratamento de inúmeros fenômenos da natureza. Devido a sua grande versatilidade, esta teoria é objeto de intensa pesquisa. Aqui, propomos novas formas de percolação e as estudamos através de simulações numéricas. Na primeira parte de nosso trabalho, investigamos a estrutura dos aglomerados gerados pelo modelo de percolação por invasão múltipla. Estimamos os valores das dimensões fractais do esqueleto, do esqueleto elástico, dos pontos de estrangulamento e dos menores caminhos, como função dos parâmetros do modelo. Por ter uma estrutura geométrica bastante estabilizada, o modelo otimizado pode vir a ser muito útil no tratamento de problemas com diluição da mecânica estatística. O modelo de percolação atenuada foi concebido para permitir que, durante o processo de invasão, os poros grandes possam também ser ocupados. Esta ocupação ocorre com uma probabilidade que diminui quando o tamanho do poro aumenta.Estimamos cuidadosamente os limiares de percolação e construímos os diagramas de fase correspondentes. Verificamos que os limiares de percolação de nosso modelo não satisfazem a conjectura de Galam e Mauger. Estudamos o efeito da inércia em fluidos escoando através de meios porosos incorporando uma caminhada de N passos ao modelo de percolação por invasão. A magnitude da inércia é proporcional ao parâmetro N, que representa o número de poros seqüencialmente invadidos após a ruptura do perímetro, em cada etapa do processo. Investigamos este modelo em duas e três dimensões. Verificamos que no caso bidimensional, as caminhadas de N passos são facilmente bloqueadas o que leva ao surgimento de um limite superior para o número de passos efetivamente realizados. Nossas estimativas das dimensões fractais dos aglomerados (como função do parâmetro N), indicam que este modelo pertence a uma classe de universalidade diferente daquela da percolação por invasão ordinária. Propomos um modelo de percolação para tratar um processo de solidificação de dois fluidos imiscíveis na presença de impurezas móveis. O movimento das impurezas ocorre devido a uma interação repulsiva de curto alcance observada experimentalmente por Ulhmann, Chalmers e Jackson (UCJ). Dependendo das concentrações de fluidos e impurezas, pode haver a formação de uma fase sólida que percola todo o sistema. Construímos o diagrama de fases deste modelo no espaço das concentrações e calculamos seus expoentes críticos. Nossos resultados indicam que o modelo pertence à mesma classe de universalidade que a percolação ordinária. Finalmente, estudamos um processo de percolação por invasão na presença de impurezas que se movem segundo o mecanismo UCJ. Encontramos um valor crítico para a concentração de impurezas, acima do qual não mais existe percolação. O perfil de aceitação aproxima-se de uma função de Heavyside, com o ponto de descontinuidade dependendo da concentração de impurezas. / Percolation theory provides a quantitative and conceptual model for the understanding of many natural phenomena. Here, we present new kinds of percolation and study them using Monte Carlo simulation techniques. We start studying the cluster, the backbone and the elastic backbone structures of the multiple invasion percolation for both the perimeter and the optimized versions. The behavior of the mass, the number of cutting sites and loops are investigated and their corresponding scaling exponents are estimated. By construction, the mass of the optimized model scales exactly with the gyration radius of the cluster - we verify that this also happens with the backbone. Our simulation shows that the red sites almost disappear, indicating that the cluster has achieved a high degree of connectivity. We propose a new kind of invasion percolation, which permits that, besides small pores, large pores being also occupied. In our model, the occupation probability of a pore diminishes with the pore\'s size. We estimate their corresponding percolation thresholds and show that they do not satisfy the Galam and Mauger conjecture. In order to take into account the inertia of the invader fluid, a new kind of invasion percolation is introduced. In this model, which we named N-steps invasion percolation, the inertia forces are controlled by the number N of pores (or steps) invaded after the perimeter rupture. The new model belongs to a different class of universality and has its fractal dimension depending on N. A blocking phenomenon takes place in two dimensions. It imposes an upper bound value on N. For pores sizes larger than the critical threshold, the acceptance profile exhibits a permanent tail. We also introduce a model for the solidification process of two immiscible fluids interacting repulsively with mobile impurities on a two dimensional square lattice. In the space of the fluids and impurities concentrations the phase diagram exhibits a critical curve separating a percolating from a non-percolating phase. Estimated values for the fractal dimension and the exponent ? of the order parameter, reveal that the critical exponents do not vary along this curve, i.e., they are independent of the impurities concentration. The universality class we find is that of the ordinary percolation. Finally, based on the main ideas of the dynamic epidemic and invasion percolation models, we propose a model to describe the cleaning process of a dirty porous medium by fluid injection. An analysis of the acceptance profiles strongly indicates that this model is a kind of self-organized system.

This thesis studies networks of silver nanowires as a transparent conducting electrode material and presents an investigation into the relationship between electrical and optical properties in the networks. The work focusses on two main aspects: the production of networks via different deposition methods; and the development of a predictive model based on theory that relates the sheet resistance to the optical transmittance. The deposition methods of drop-casting, bar-coating and spray-coating are used to create networks and the randomness of these networks is compared using image analysis in ImageJ, a public domain image processing program, and Wolfram Mathematica, a computer algebra program. It is determined that spray-coating results in the most random networks, therefore all subsequent experiments are carried out using this as the deposition method. Annealing condition tests are carried out on the nanowire networks to determine the optimal annealing conditions required to burn off poly(vinyl pyrrolidone) (PVP) remaining from the nanowire synthesis process and sinter the nanowire junctions to improve network conductivity. The sheet resistances and optical transmittances of the networks are measured and compared to networks created by other research groups. It is found that the networks created in this study exhibited similar optical and electrical properties to those in the literature, obtaining Rs = 100 Ω/sq for T = 81%.The developed model is based on theory and relates the sheet resistance to the optical transmittance using only the length and width of the nanowires used in the network and the mean network coverage as variables. The model can be used to predict the properties of a network if these factors are known. The model is compared with experimental data both from this study and from the literature, along with simulated data from the literature that was obtained by Monte Carlo methods. It is shown that there is an excellent fit between the model and all data that it is compared with. It is demonstrated that < 1% of the network coverage is greater than 2 for typical nanowire networks, proving that the networks are two-dimensional and therefore do not require a bulk regime to describe the relationship, as has been suggested in prior work.

Generalizamos o modelo de percolação por invasão de maneira que vários sítios possam ser simultaneamente invadidos. Propomos dois tipos de generalização: na primeira, o fluxo de fluido invasor e controlado através do perímetro do aglomerado, enquanto que na segunda modificação, o crescimento e governado pela relação de escala entre a massa e o raio de giração dos aglomerados. Estudamos cuidadosamente tanto o perfil de aceitação quanto as dimensões fractais (\'D IND.F\') dos aglomerados assim crescidos. No modelo baseado nas relações de escala, \'D IND.F\' pode ser tratado como um mero parâmetro real que pode assumir qualquer valor no intervalo (0, ?). Nos intervalos (0, \'91 SOB.48\') e (2, ?), o sistema e frustrado. Para \'D IND.F\' > 2, o modelo exibe um fenômeno interessante: em algumas etapas ocorrem explosões no crescimento da massa dos aglomerados (bursts). Na região [\'91 SOB.48\',2], os aglomerados obedecem exatamente e em qualquer escala a relação M ~ RgDF entre a massa m e o raio de giração \'RG\'. Acreditamos que estes fractais cuja estrutura completamente e estabilizada possam ser muito úteis no tratamento de problemas de diluição da mecânica estatística. / We generalize the standard site invasion percolation model to permit simultaneous invasion of several sites. We propose two kinds of generalizations: one in which the invasion flux is controlled by the perimeter size and another where the scaling properties command the growth process. The acceptance profile as well as the fractal dimension \'D IND.F\' are carefully studied. In the model based on scaling relation, \'D IND.F\' can be treated as a mere real parameter in the range (0, ?). In the intervals (0, \'91 SOB.48\') and (2, ?) the system is frustrated. For \'D IND.F\' > 2 the model exhibits also an interesting burst phenomenon that is explained in the text. In the region [\'91 SOB.48\',2], the clusters obey exactly and in any scale the relation M ~ RgDF between the mass M and gyration radius Rg. These stable random fractals may be very useful in the study of dilute systems.

In the first part of this thesis I consider site and bond percolation on a Random Connection Model and prove that for a wide range of connection functions the critical site probability is strictly greater than the critical bond probability and use this fact to improve previously known non-strict inequalities to strict inequalities. In the second part I consider percolation on the even phase of a Random Sequential Adsorption model and prove that the critical intensity is finite and strictly bigger than 1. Both of these main results make use of an enhancement technique.

Utilizando uma técnica de matriz de transferência, expandimos em série a probabilidade de percolação P(q) para o problema da percolação por sítio na rede quadrada direcionada. Nosso método revela uma inesperada conexão entre este problema e o da enumeração dos modos de se dissecar uma bola. Mostramos que o método pode também ser usado para se expandir em série o tamanho médio do cluster S (p) . Uma análise baseada nos aproximantes de Padé fornece estimativas do valor crítico pc, e também do expoente crítico &#946. / Using a transfer matrix technique we obtain an extended series expansion of the percolation probability P(q) for the directed site percolation problem on the square lattice. Our method reveals an up to now unsuspected connection between this problem and the enumeration of the ways of dissecting a ball. We show that the method can also be used to determine a series expansion for the mean cluster size S(p). An analysis based on Padé approximants gives estimates of the critical threshold pc, and also of the critical exponent &#946.