In this thesis we investigate the geometric and algorithmic aspects of the random-cluster model, a correlated bond percolation model of great importance in the field of mathematics and statistical mechanics. We focus on the computational and statistical efficiency of the single-bond or heat-bath Markov chain for the random-cluster model and develop algorithmic techniques that allow for an improvement from a previously known polynomial to a poly-logarithmic runtime scaling of updates for general graphs. The interplay between the (critical) cluster structure of the random-cluster model and algorithmic, as well as statistical, efficiencies is considered, leading to new exact identities. A complementary analysis of certain fragility properties of the Fortuin-Kasteleyn clusters provides new insights into fragmentation phenomena, culminating in a revised scaling relation for a related fragmentation power law exponent, previously only shown for the marginal bond percolation case. By utilising the established structural results, a dynamic fragmentation process is studied that allows for an extraction of characteristics of the equilibrium cluster structure by a careful analysis of the limiting fragments, as well as the entire evolution of the fragmentation process. Besides focussing on structural and computational aspects, in this dissertation we also analyse the efficiency of the coupling from the past perfect sampling algorithm for the random-cluster model via large-scale numerical simulations. Two key results are the particular, close to optimal, efficiency in the off-critical setting and the intriguing observation of its superiority compared to the alternative Chayes-Machta-Swendsen-Wang approach in three dimensions. Governed by a random runtime, the efficiency of the coupling from the past algorithm depends crucially on the fluctuations of the runtime. In this connection a compelling appearance of universal Gumbel fluctuations in the distribution of the runtime of the coupling from the past algorithm is established, both at and off criticality. Fluctuations at a tricritical point and at a discontinuous phase transition are shown to deviate from this Gumbel law. The above findings in two and three dimensions are supported by a rigorous analysis of certain aspects of the algorithm in one dimension, including a proof of the limiting Gumbel law.

Cette thèse s'intéresse au Continuum Random Cluster Model (CRCM), modèle gibbsien de boules aléatoires où la densité dépend du nombre de composantes connexes de la structure. Ce modèle est une version continue du Random Cluster Model introduit pour unifier l'étude des modèles d'Ising et de Potts. Le CRCM fut introduit pour sa relation avec le modèle de Widom-Rowlinson, fournissant une nouvelle preuve de la transition de phase pour ce modèle. Dans cette thèse nous étudions dans un premier temps l'existence du CRCM en volume infinie. Dans le cas extrême des rayons non-intégrables, nous démontrons un résultat de non-unicité du CRCM en petite activité. Nous conjecturons de plus que l'unicité serait obtenue en grande activité. Une version faible de cette conjecture est démontré en dimension 1. Dans un second temps nous étudions la percolation du CRCM, qui s'intéresse aux propriétés de connectivité et en particulier à l'existence d'une composante connexe infinie. La percolation est d'autant plus cohérente pour le CRCM dont l'interaction dépend directement de la connectivité de la structure. Nous montrons dans cette thèse l'absence de percolation en petite activité et la percolation en grande activité. Ce résultat permet de généraliser la transition de phase du modèle de Widom-Rowlinson à des rayons non bornés. / This thesis focuses on the Continuum Random Cluster Model (CRCM), defined as a Gibbs model of random balls where the density depends on the number of cluster in the structure. This model is a continuum version of the Random Cluster Model introduced to unify the study of the Ising and Potts model. The CRCM was introduced for its links with the Widom-Rowlinson model, which led to a new proof of the phase transition for this model. In this thesis we first study the existence of the model in the infinite volume regime. In the extreme setting of non integrable radii, we prove for small activities the non-uniqueness of a CRCM. We conjecture that the uniqueness would be revovered for large activities. A weak version of the conjecture is proved.We alson study the percolation of the CRCM, which is the existence of at least one unbounded connected component. Percolation is more relevant for the CRCM since the interaction depends on the connectivity of the structure. We prove the absence of percolation for small activities and percolation for large activities. This results leads to the phase transition of the Widom-Rowlinson model with unbounded radii.

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The theory of Gibbs measures belongs to the borderlandbetween statistical mechanics and probability theory. In thiscontext, the physical phenomenon of phase transitioncorresponds to the mathematical concept of non-uniqueness for acertain type of probability measures. The most studied model in statistical mechanics is thecelebrated Ising model. The Potts model is a natural extensionof the Ising model, and the beach model, which appears in adifferent mathematical context, is in certain respectsanalogous to the Ising model. The two main parts of this thesisdeal with the Potts model and the beach model,respectively. For theq-state Potts model on an infinite lattice, there areq+1 basic Gibbs measures: one wired-boundary measure foreach state and one free-boundary measure. For infinite trees,we construct "new" invariant Gibbs measures that are not convexcombinations of the basic measures above. To do this, we use anextended version of the random-cluster model together withcoupling techniques. Furthermore, we investigate the rootmagnetization as a function of the inverse temperature.Critical exponents to this function for different parametercombinations are computed. The beach model, which was introduced by Burton and Steif,has many features in common with the Ising model. We generalizesome results for the Ising model to the beach model, such asthe connection between phase transition and a certain agreementpercolation event. We go on to study aq-state variant of the beach model. Using randomclustermodel methods again we obtain some results on where in theparameter space this model exhibits phase transition. Finallywe study the beach model on regular infinite trees as well.Critical values are estimated with iterative numerical methods.In different parameter regions we see indications of both firstand second order phase transition. Keywords and phrases:Potts model, beach model,percolation, randomcluster model, Gibbs measure, coupling,Markov chains on infinite trees, critical exponent.

<p>The theory of Gibbs measures belongs to the borderlandbetween statistical mechanics and probability theory. In thiscontext, the physical phenomenon of phase transitioncorresponds to the mathematical concept of non-uniqueness for acertain type of probability measures.</p><p>The most studied model in statistical mechanics is thecelebrated Ising model. The Potts model is a natural extensionof the Ising model, and the beach model, which appears in adifferent mathematical context, is in certain respectsanalogous to the Ising model. The two main parts of this thesisdeal with the Potts model and the beach model,respectively.</p><p>For the<i>q</i>-state Potts model on an infinite lattice, there are<i>q</i>+1 basic Gibbs measures: one wired-boundary measure foreach state and one free-boundary measure. For infinite trees,we construct "new" invariant Gibbs measures that are not convexcombinations of the basic measures above. To do this, we use anextended version of the random-cluster model together withcoupling techniques. Furthermore, we investigate the rootmagnetization as a function of the inverse temperature.Critical exponents to this function for different parametercombinations are computed.</p><p>The beach model, which was introduced by Burton and Steif,has many features in common with the Ising model. We generalizesome results for the Ising model to the beach model, such asthe connection between phase transition and a certain agreementpercolation event. We go on to study a<i>q</i>-state variant of the beach model. Using randomclustermodel methods again we obtain some results on where in theparameter space this model exhibits phase transition. Finallywe study the beach model on regular infinite trees as well.Critical values are estimated with iterative numerical methods.In different parameter regions we see indications of both firstand second order phase transition.</p><p><b>Keywords and phrases:</b>Potts model, beach model,percolation, randomcluster model, Gibbs measure, coupling,Markov chains on infinite trees, critical exponent.</p>

HTML clipboard Statistical physics seeks to explain macroscopic properties of matter in terms of microscopic interactions. Of particular interest is the phenomenon of phase transition: the sudden changes in macroscopic properties as external conditions are varied. Two models in particular are of great interest to mathematicians, namely the Ising model of a magnet and the percolation model of a porous solid. These models in turn are part of the unifying framework of the random-cluster representation, a model for random graphs which was first studied by Fortuin and Kasteleyn in the 1970’s. The random-cluster representation has proved extremely useful in proving important facts about the Ising model and similar models. In this work we study the corresponding graphical framework for two related models. The first model is the transverse field quantum Ising model, an extension of the original Ising model which was introduced by Lieb, Schultz and Mattis in the 1960’s. The second model is the space–time percolation process, which is closely related to the contact model for the spread of disease. In Chapter 2 we define the appropriate space–time random-cluster model and explore a range of useful probabilistic techniques for studying it. The space– time Potts model emerges as a natural generalization of the quantum Ising model. The basic properties of the phase transitions in these models are treated in this chapter, such as the fact that there is at most one unbounded fk-cluster, and the resulting lower bound on the critical value in <img src="http://upload.wikimedia.org/math/a/b/8/ab820da891078a8245d7f4f3252aee4f.png" />. In Chapter 3 we develop an alternative graphical representation of the quantum Ising model, called the random-parity representation. This representation is based on the random-current representation of the classical Ising model, and allows us to study in much greater detail the phase transition and critical behaviour. A major aim of this chapter is to prove sharpness of the phase transition in the quantum Ising model—a central issue in the theory— and to establish bounds on some critical exponents. We address these issues by using the random-parity representation to establish certain differential inequalities, integration of which gives the results. In Chapter 4 we explore some consequences and possible extensions of the results established in Chapters 2 and 3. For example, we determine the critical point for the quantum Ising model in <img src="http://upload.wikimedia.org/math/a/b/8/ab820da891078a8245d7f4f3252aee4f.png" /> and in ‘star-like’ geometries. / HTML clipboard Statistisk fysik syftar till att förklara ett materials makroskopiska egenskaper i termer av dess mikroskopiska struktur. En särskilt intressant egenskap är är fenomenet fasövergång, det vill säga en plötslig förändring i de makroskopiska egenskaperna när externa förutsättningar varieras. Två modeller är särskilt intressanta för en matematiker, nämligen Ising-modellen av en magnet och perkolationsmodellen av ett poröst material. Dessa två modeller sammanförs av den så-kallade fk-modellen, en slumpgrafsmodell som först studerades av Fortuin och Kasteleyn på 1970-talet. fk-modellen har sedermera visat sig vara extremt användbar för att bevisa viktiga resultat om Ising-modellen och liknande modeller. I den här avhandlingen studeras den motsvarande grafiska strukturen hos två näraliggande modeller. Den första av dessa är den kvantteoretiska Isingmodellen med transverst fält, vilken är en utveckling av den klassiska Isingmodellen och först studerades av Lieb, Schultz och Mattis på 1960-talet. Den andra modellen är rumtid-perkolation, som är nära besläktad med kontaktmodellen av infektionsspridning. I Kapitel 2 definieras rumtid-fk-modellen, och flera probabilistiska verktyg utforskas för att studera dess grundläggande egenskaper. Vi möter rumtid-Potts-modellen, som uppenbarar sig som en naturlig generalisering av den kvantteoretiska Ising-modellen. De viktigaste egenskaperna hos fasövergången i dessa modeller behandlas i detta kapitel, exempelvis det faktum att det i fk-modellen finns högst en obegränsad komponent, samt den undre gräns för det kritiska värdet som detta innebär. I Kapitel 3 utvecklas en alternativ grafisk framställning av den kvantteoretiska Ising-modellen, den så-kallade slumpparitetsframställningen. Denna är baserad på slumpflödesframställningen av den klassiska Ising-modellen, och är ett verktyg som låter oss studera fasövergången och gränsbeteendet mycket närmare. Huvudsyftet med detta kapitel är att bevisa att fasövergången är skarp—en central egenskap—samt att fastslå olikheter för vissa kritiska exponenter. Metoden består i att använda slumpparitetsframställningen för att härleda vissa differentialolikheter, vilka sedan kan integreras för att lägga fast att gränsen är skarp. I Kapitel 4 utforskas några konsekvenser, samt möjliga vidareutvecklingar, av resultaten i de tidigare kapitlen. Exempelvis bestäms det kritiska värdet hos den kvantteoretiska Ising-modellen på <img src="http://upload.wikimedia.org/math/a/b/8/ab820da891078a8245d7f4f3252aee4f.png" /> , samt i ‘stjärnliknankde’ geometrier. / QC 20100705

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.