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.

Master of Science / Department of Computing and Information Sciences / William H. Hsu / The project explores the possibility of increasing efficiency in the clusters formed out of massive data sets which are formed using threshold blocking algorithm. Clusters thus formed are denser and qualitative. Clusters that are formed out of individual clustering algorithms alone, do not necessarily eliminate outliers and the clusters generated can be complex, or improperly distributed over the data set. The threshold blocking algorithm, a current research paper from Michael Higgins of Statistics Department on other hand, in comparison with existing algorithms performs better in forming the dense and distinctive units with predefined threshold. Developing a hybridized algorithm by implementing the existing clustering algorithms to re-cluster these units thus formed is part of this project. Clustering on the seeds thus formed from threshold blocking Algorithm, eases the task of clustering to the existing algorithm by eliminating the overhead of worrying about the outliers. Also, the clusters thus generated are more representative of the whole. Also, since the threshold blocking algorithm is proven to be fast and efficient, we now can predict a lot more decisions from large data sets in less time. Predicting the similar songs from Million Song Data Set using such a hybridized algorithm is considered as the data set for the evaluation of this goal.

Master of Science / Department of Computing and Information Sciences / William H. Hsu / (Higgins, Savje, & Sekhon, 2016) Provides us with a sampling blocking algorithm that enables large and complex experiments to run in polynomial time without sacrificing the precision of estimates on a covariate dataset. The goal of this project is to run the different clustering algorithms on top of clusters formed from above mentioned blocking algorithm and analyze the performance and compatibility of the clustering algorithms. We first start with applying the blocking algorithm on a covariate dataset and once the clusters are formed, we then apply our clustering algorithm HAC (Hierarchical Agglomerative Clustering) or PAM (Partitioning Around Medoids) on the seeds of the clusters. This will help us to generate more similar clusters. We compare our performance and precision of our hybridized clustering techniques with the pure clustering techniques to identify a suitable hybridized blocking model.

In cellular networks, channels should be allocated efficiently to support communication betweenmobile hosts. In addition, in cellular networks, base stations may fail. Therefore, designing a faulttolerantchannel allocation algorithm is important. That is, the algorithm should tolerate failuresof base stations. Many existing algorithms are neither fault-tolerant nor efficient in allocatingchannels.We propose channel allocation algorithms which are both fault-tolerant and efficient. In theproposed algorithms, to borrow a channel, a base station (or a cell) does not need to get channelusage information from all its interference neighbors. This makes the algorithms fault-tolerant,i.e., the algorithms can tolerate base station failures, and perform well in the presence of thesefailures.Channel pre-allocation has effect on the performance of a channel allocation algorithm. Thiseffect has not been studied quantitatively. We propose an adaptive channel allocation algorithmto study this effect. The algorithm allows a subset of channels to be pre-allocated to cells. Performanceevaluation indicates that a channel allocation algorithm benefits from pre-allocating allchannels to cells.Channel selection strategy also inuences the performance of a channel allocation algorithm.Given a set of channels to borrow, how a cell chooses a channel to borrow is called the channelselection problem. When choosing a channel to borrow, many algorithms proposed in the literaturedo not take into account the interference caused by borrowing the channel to the cells which havethe channel allocated to them. However, such interference should be considered; reducing suchinterference helps increase the reuse of the same channel, and hence improving channel utilization.We propose a channel selection algorithm taking such interference into account.Most channel allocation algorithms proposed in the literature are for traditional cellular networkswith static base stations and the neighborhood relationship among the base stations is fixed.Such algorithms are not applicable for cellular networks with mobile base stations. We proposea channel allocation algorithm for cellular networks with mobile base stations. The proposedalgorithm is both fault-tolerant and reuses channels efficiently.KEYWORDS: distributed channel allocation, resource planning, fault-tolerance, cellular networks,3-cell cluster model.

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Yes / Multidimensional data streams are a major paradigm in data science. This work focuses on possibilistic clustering algorithms as means to perform clustering of multidimensional streaming data. The proposed approach exploits fuzzy outlier analysis to provide good learning and tracking abilities in both concept shift and concept drift.

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.

O modelo de cluster de partícula-alfa, o qual já foi bem-sucedido na descrição de dados espectroscópicos em núcleos leves próximos das duplas camadas fechadas no 16O e 40Ca, é aplicado ao núcleo 94Mo da região de massa dos meio-pesados. Para este propósito, vários estados deste núcleo são interpretados em termos de um sistema alfa + 90Zr onde um cluster-alfa interage com um caroço inerte de 90Zr através de um potencial fenomenológico local. São calculados os níveis de energia da banda do estado fundamental e as respectivas taxas de transição B(E2), havendo boa concordância com dados experimentais disponíveis. Os raios intercluster rms calculados para os níveis da banda do estado fundamental indicam que tal banda apresenta uma estrutura de cluster-alfa compacta. A comparação entre auto-funções de oscilador harmônico adaptadas ao sistema e as funções de onda radiais da banda do estado fundamental fornece a estimativa na qual há uma significativa contribuição do modelo de camadas na formação dos estados desta banda. O potencial de interação cluster-caroço faz prever a existência de uma banda de paridade negativa que se inicia a alguns MeV acima do limiar alfa + 90Zr, onde a aproximação de estado ligado é bastante apropriada. Da mesma forma, é previsto que uma banda de paridade positiva excitada se inicia logo abaixo da barreira coulombiana, mostrando uma característica típica de ressonância. / The alpha-particle cluster model, which has already been successful in describing the spectroscopic data in light nuclei near to the double shell closures at 16O and 40Ca, is applied to the 94Mo nucleus of the medium-heavy mass region. For this purpose, various states of this nucleus are interpreted in terms of an alpha + 90Zr system where an alpha-cluster interacts with an inert 90Zr core through a local phenomenological potential. The energy levels of the ground state band and the respective B(E2) transition rates are calculated, in good agreement with available experimental data. The intercluster rms radii calculated for the levels of the ground state band indicate that such band presents a compact alpha-cluster structure. The comparison between harmonic oscillator eigenfunctions adapted to the system and the radial wave functions of the ground state band provides the estimate in which there is a significant contribution of the shell model for the formation of the states of this band. The cluster-core potential predicts the existence of a negative parity band that starts at a few MeV above the alpha + 90Zr threshold, where the bound state approximation is very appropriate. In the same way, it is predicted that an excited positive parity band starts just below the Coulomb barrier, showing a typical feature of resonance.

Title: Cluster point processes in insurance mathematics Author: Veronika Veselá Department: Department of Probability and Mathematical Statistics Supervisor: RNDr. Zbyněk Pawlas, Ph.D. Abstract: In the present work we study point processes and their importance in insurance mathematics. With the help of cluster and marked point processes we can describe a model that considers times of claim occurence and times and hei- ghts of corresponding payments. We study two specific models which can be used to predict how much money is needed for claims which happened. The first model is chain ladder in the form of Mack's model. For this model we show chain ladder estimators of development factors, estimates of their variance and their proper- ties. We try to find one-step ahead prediction and multi-step ahead prediction, which we use for calculating prediction of reserves. We shortly review asymptotic properties of the estimators in Mack's model. The second model is the Poisson cluster model. Firstly we define this model and the variables entering the model. Then we devote attention to one-step ahead and multi-step ahead prediction. We also study prediction when some variables have specific distributions. Finally, we use both methods of prediction on simulated data and compare their average relative absolute errors....