We study the properties of Gibbs measures for functions with d-summable variation defined on a subshift X. Based on Meyerovitch\'s work from 2013, we prove that if X is a subshift of finite type (SFT), then any equilibrium measure is also a Gibbs measure. Although the definition provided by Meyerovitch does not make any mention to conditional expectations, we show that in the case where X is a SFT it is possible to characterize these measures in terms of more familiar notions presented in the literature. / Nós estudamos as propriedades de medidas de Gibbs para funções com variação d-somável definidas em um subshift X. Baseado no trabalho de Meyerovitch de 2013, provamos que se X é um subshift de tipo finito (STF), então qualquer medida de equilíbrio é também uma medida de Gibbs. Embora a definição fornecida por Meyerovitch não faz qualquer menção à esperanças condicionais, mostramos que no caso em que X é um STF, é possível caracterizar estas medidas em termos de noções mais familiares apresentadas na literatura.

In this work we provide several improvements in the study of phase transitions of interacting particle systems: - We determine a quantitative relation between non-extremality of the limiting Gibbs measure of a tree-based spin system, and the temporal mixing of the Glauber Dynamics over its finite projections. We define the concept of 'sensitivity' of a reconstruction scheme to establish such a relation. In particular, we focus on the independent sets model, determining a phase transition for the mixing time of the Glauber dynamics at the same location of the extremality threshold of the simple invariant Gibbs version of the model. - We develop the technical analysis of the so-called spatial mixing conditions for interacting particle systems to account for the connectivity structure of the underlying graph. This analysis leads to improvements regarding the location of the uniqueness/non-uniqueness phase transition for the independent sets model over amenable graphs; among them, the elusive hard-square model in lattice statistics, which has received attention since Baxter's solution of the analogous hard-hexagon in 1980. - We build on the work of Montanari and Gerschenfeld to determine the existence of correlations for the coloring model in sparse random graphs. In particular, we prove that correlations exist above the 'clustering' threshold of such a model; thus providing further evidence for the conjectural algorithmic 'hardness' occurring at such a point.

In this thesis we study various properties of the spins models, in particular, Ising and Dyson models. We study the stability of the phase transition of the nearest-neighbor ferromagnetic Ising model when we add a perturbation to the critical external field that becomes weaker far from the root of the Cayley tree. We also study the relation between g-measures and Gibbs measures, showing that the Dyson model at sufficiently low temperature is not a g-measure. Counting contours on trees is also studied, showing the characterization of the trees that have infinite number of contours, and comparisons between various definitions of contours. We also study the measures of the spatial Gibbs random graphs, and their local convergence. / Nesta tese estudamos diversas propriedades dos modelos de spins, em particular, os modelos de Ising e Dyson. Estudamos a estabilidade da transição de fase no modelo de Ising ferromagnético de primeiros vizinhos quando adicionamos uma perturbação no campo externo crítico pela qual se torna mais fraca ao estar distante da raiz da árvore de Cayley. Estudamos a relação entre g-medidas e medidas de Gibbs, mostrando que a medida de Gibbs do modelo de Dyson a temperaturas suficientemente baixas não é uma g-medida. Também estudamos contagem de contornos em árvores, mostramos uma caracterização das árvores que possuem um número infinito de contornos de um tamanho fixo envolvendo um vértice, e comparamos entre diversas definições de contornos. Estudamos também as medidas de grafos aleatórios spatial Gibbs, e suas convergências locais.

For a general attractive Probabilistic Cellular Automata on S-Zd, we prove that the (time-) convergence towards equilibrium of this Markovian parallel dynamics, exponentially fast in the uniform norm, is equivalent to a condition (A). This condition means the exponential decay of the influence from the boundary for the invariant measures of the system restricted to finite boxes. For a class of reversible PCA dynamics on {1,+1}(Zd), wit a naturally associated Gibbsian potential rho, we prove that a (spatial-) weak mixing condition (WM) for rho implies the validity of the assumption (A); thus exponential (time-) ergodicity of these dynamics towards the unique Gibbs measure associated to rho hods. On some particular examples we state that exponential ergodicity holds as soon as there is no phase transition.

Kollektive Migration und Schwarmverhalten sind Beispiele für Selbstorganisation und können in verschiedenen biologischen Systemen beobachtet werden, beispielsweise in Vogel-und Fischschwärmen oder Bakterienpopulationen. Im Zentrum dieser Arbeit steht ein räumlich diskretes und zeitlich stetiges Model, welches das kollektive Migrieren von Individuen mittels eines stochastischen Vielteilchensystems (VTS) beschreibt und analysierbar macht. Das konstruierte Modell ist in keiner Klasse gut untersuchter Vielteilchensysteme enthalten, sodass der größte Teil der Arbeit der Entwicklung von Methoden zur Untersuchung des Langzeitverhaltens bestimmter VTS gewidmet ist. Eine entscheidende Rolle spielen hier Gibbs-Maße, die zu zeitlich invarianten Maßen in Beziehung gesetzt werden. Durch eine Simulationsstudie und die Analyse des Einflusses der Parameter Migrationsgeschwindigkeit, Sensitivität der Individuen und (räumliche) Dichte der Anfangsverteilung können Eigenschaften kollektiver Migration erklärt und Hypothesen für weitere Analysen aufgestellt werden. / Collective migration and swarming behavior are examples of self-organization and can be observed in various biological systems, such as in flocks of birds, schools of fish or populations of bacteria. In the center of this thesis lies a stochastic interacting particle system (IPS), which is a spatially discrete model with a continuous time scale that describes collective migration and which can be treated using analytical methods. The constructed model is not contained in any class of well-understood IPS’s. The largest part of this work is used to develop methods that can be used to study the long-term behavior of certain IPS’s. Thereby Gibbs-Measures play an important role and are related to temporally invariant measures. One can explain the properties of collective migration and propose a hypothesis for further analyses by a simulation study and by analysing the parameters migration velocity, sensitivity of individuals and (spatial) density of the initial distribution.

Kollektive Migration und Schwarmverhalten sind Beispiele für Selbstorganisation und können in verschiedenen biologischen Systemen beobachtet werden, beispielsweise in Vogel-und Fischschwärmen oder Bakterienpopulationen. Im Zentrum dieser Arbeit steht ein räumlich diskretes und zeitlich stetiges Model, welches das kollektive Migrieren von Individuen mittels eines stochastischen Vielteilchensystems (VTS) beschreibt und analysierbar macht. Das konstruierte Modell ist in keiner Klasse gut untersuchter Vielteilchensysteme enthalten, sodass der größte Teil der Arbeit der Entwicklung von Methoden zur Untersuchung des Langzeitverhaltens bestimmter VTS gewidmet ist. Eine entscheidende Rolle spielen hier Gibbs-Maße, die zu zeitlich invarianten Maßen in Beziehung gesetzt werden. Durch eine Simulationsstudie und die Analyse des Einflusses der Parameter Migrationsgeschwindigkeit, Sensitivität der Individuen und (räumliche) Dichte der Anfangsverteilung können Eigenschaften kollektiver Migration erklärt und Hypothesen für weitere Analysen aufgestellt werden. / Collective migration and swarming behavior are examples of self-organization and can be observed in various biological systems, such as in flocks of birds, schools of fish or populations of bacteria. In the center of this thesis lies a stochastic interacting particle system (IPS), which is a spatially discrete model with a continuous time scale that describes collective migration and which can be treated using analytical methods. The constructed model is not contained in any class of well-understood IPS’s. The largest part of this work is used to develop methods that can be used to study the long-term behavior of certain IPS’s. Thereby Gibbs-Measures play an important role and are related to temporally invariant measures. One can explain the properties of collective migration and propose a hypothesis for further analyses by a simulation study and by analysing the parameters migration velocity, sensitivity of individuals and (spatial) density of the initial distribution.

Dans cette thèse, nous étudions les sujets suivants :- la stabilité ergodique pour les systèmes conservatifs ;- la généricité de l'existence d'exposants positifs pour certains produits tordus avec fibres de dimension deux ;- rigidité des mesures $u$-Gibbs pour certains systèmes partiellement hyperboliques ;- la transitivité robuste.Nous donnons une preuve de la stabilité ergodique pour certains systèmes partiellement hyperboliques sans utiliser l'accessibilité. Ces systèmes ont été introduits par Pierre Berger et Pablo Carrasco, et ils ont les propriétés suivantes : ils possèdent une direction centrale bidimensionnelle ; ils sont non-uniformément hyperboliques avec un exposant positif et un exposant négatif le long de la direction centrale pour presque tout point, et la décomposition d'Oseledets n'est pas dominée.Dans un autre travail, nous donnons des critères de stabilité ergodique pour des systèmes ayant une décomposition dominée. En particulier, nous explorons la notion d'hyperbolicité par chaîne introduite par Sylvain Crovisier et Enrique Pujals. À l'aide de cette notion, nous donnons des critères explicites de stabilité ergodique et nous donnons quelques applications.Dans un travail commun avec Mauricio Poletti, nous prouvons que le produit aléatoire de difféomorphismes de surface conservatifs possède génériquement une région avec des exposants positifs. Nos résultats s'appliquent également aux produits tordus plus généraux.Nous étudions également les perturbations dissipatives de l'exemple de Berger-Carrasco. Nous classifions toutes les mesures $u$-Gibbs qui peuvent apparaître dans un voisinage de l'exemple. Dans ce voisinage, nous prouvons que toute mesure $u$-Gibbs est soit l'unique mesure SRB du système, soit la désintégration dans le feuilletage central est atomique. Dans un travail commun avec Pablo Carrasco, nous prouvons que cet exemple est robustement transitif (en fait robustement topologiquement mélangeant). / In this thesis we study the following topics:-stable ergodicity for conservative systems;-genericity of the existence of positive exponents for some skew products with two dimensional fibers;-rigidity of $u$-Gibbs measure for certain partially hyperbolic systems;-robust transitivity.We give a proof of stable ergodicity for a certain partially hyperbolic system without using accessibility. This system was introduced by Pierre Berger and Pablo Carrasco, and it has the following properties: it has a two dimensional center direction; it is non-uniformly hyperbolic having both a positive and a negative exponent along the center for almost every point, and the Oseledets decomposition is not dominated.In a different work, we find criteria of stable ergodicity for systems with a dominated splitting. In particular, we explore the notion of chain-hyperbolicity introduced by Sylvain Crovisier and Enrique Pujals. With this notion we give explicit criteria of stable ergodicity, and we give some applications.In a joint work with Mauricio Poletti, we prove that the random product of conservative surface diffeomorphisms generically has a region with positive exponents. Our results also hold for more general skew products.We also study dissipative perturbations of the Berger-Carrasco example. We classify all the $u$-Gibbs measures that may appear inside a neighborhood of the example. In this neighborhood, we prove that any $u$-Gibbs measure is either the unique SRB measure of the system or it has atomic disintegration along the center foliation. In a joint work with Pablo Carrasco, we prove that this example is robustly transitive (indeed robustly topologically mixing).