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We have investigated gradient models, one of them was a model with double-well potential and the other one a so called extended model. In dimension two we have calculated exact free energies of the disseminated edge configurations for the extended model and for arbitrary dimension we have derived bounds on these free energies. Combining these bounds with an argument on exstince of bad contours together with the estimate of the number of these contours and using the method of reflection positivity we have been able to show that at low temperatures there is a phase transition in the extended model. We have further shown that the phase transition exists also in the double-well model as long as a conjecture on estimates of mean energy holds. Besides these results the thesis also contains basic tools of statistical physics and facts from related fields, as well as basic results on gradient models, so that our work can serve as an introduction into these areas.

O objetivo desta tese é demonstrar que para um subshift de Markov topologicamente transitivo com alfabeto enumerável e um potencial &#402 com pressão de Gurevic finita e variação limitada (&#402) < &#8734, existe um único estado de equilíbrio &#181t&#402 para cada t > 1, e a família (&#181t&#402)t>1 tem um ponto de acumulação quando t > &#8734. Além disso se também supomos que o &#402 é um potencial de Markov, demonstramos que a família de estados de equilíbrio (&#181t&#402)t>1 converge quando t > &#8734. Finalmente demonstramos a continuidade em &#8734 da entropia com respeito ao parâmetro t. Estes resultados não dependem da hipótese de existência de medidas de Gibbs. / The aim of this thesis is to prove that for a topologically transitive Markov subshift with countable alphabet and a summable potential &#402 with finite topological pressure Gurevic and bounded variation (&#402) < &#8734, there exists an equilibrium state &#181t&#402 tf for each t > 1 and the family of equilibrium states (&#181t&#402)t>1 associated to each potential tf has an accumulation point at t > &#8734. Moreover if we also assume that &#402 is a Markov potential we prove that the equilibrium states family (&#181t&#402)t>1 converges when t > &#8734. Finally we prove the continuity at &#8734 of the entropy with respect to the parameter t. These results do not depend on assuming the existence of Gibbs measures.

O objetivo desta tese é demonstrar que para um subshift de Markov topologicamente transitivo com alfabeto enumerável e um potencial &#402 com pressão de Gurevic finita e variação limitada (&#402) < &#8734, existe um único estado de equilíbrio &#181t&#402 para cada t > 1, e a família (&#181t&#402)t>1 tem um ponto de acumulação quando t > &#8734. Além disso se também supomos que o &#402 é um potencial de Markov, demonstramos que a família de estados de equilíbrio (&#181t&#402)t>1 converge quando t > &#8734. Finalmente demonstramos a continuidade em &#8734 da entropia com respeito ao parâmetro t. Estes resultados não dependem da hipótese de existência de medidas de Gibbs. / The aim of this thesis is to prove that for a topologically transitive Markov subshift with countable alphabet and a summable potential &#402 with finite topological pressure Gurevic and bounded variation (&#402) < &#8734, there exists an equilibrium state &#181t&#402 tf for each t > 1 and the family of equilibrium states (&#181t&#402)t>1 associated to each potential tf has an accumulation point at t > &#8734. Moreover if we also assume that &#402 is a Markov potential we prove that the equilibrium states family (&#181t&#402)t>1 converges when t > &#8734. Finally we prove the continuity at &#8734 of the entropy with respect to the parameter t. These results do not depend on assuming the existence of Gibbs measures.

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.

[pt] Neste trabalho apresentamos a teoria de Tomita-Takesaki para uma álgebra de Von Neumann M com vetor cíclico separante u. Usamos o caso finito dimensional para motivar a teoria, depois prosseguimos para os argumentos analíticos geralmente empregados para provar o caso infinito dimensional. Também calculamos os operadores modulares da teoria para três exemplos padrão. Na mecânica estatística quântica, os estados de equilíbrio termodinâmico de um sistema físico com um número de partículas e volume finito são modelados pelos estados de Gibbs, enquanto no caso infinito eles são modelados pelos chamados estados KMS através da abordagem de álgebra de operadores. Mostramos como a teoria de Tomita-Takesaki fornece estados KMS naturais e a unicidade da evolução temporal do sistema físico para esses estados. / [en] In this work we present the Tomita-Takesaki theory for a Von Neumann algebra M with cyclic separating vector u. We use the finite-dimensional case to motivate the theory, and then proceed to the analytical arguments usually employed to prove the infinite dimensional case. Also, we calculate the modular operators from the theory for three standard examples. In quantum statistical mechanics, the thermodynamic equilibrium states of a physical system with finitely many particles and finite volume are modeled by Gibbs states, while in the infinite case they are modeled by the so called KMS states through the operator-algebraic approach.We show how Tomita-Takesaki theory provides natural KMS states and the uniqueness of the time evolution of the physical system for those states.