This thesis explores Breaking of Ensemble Equivalence (BEE) in random graph models by examining the spectral properties of adjacency matrices. The goal is to identify spectral characteristics that... Show moreThis thesis explores Breaking of Ensemble Equivalence (BEE) in random graph models by examining the spectral properties of adjacency matrices. The goal is to identify spectral characteristics that differentiate random graph ensembles, enhancing the understanding of complex network structures and behaviors. The research encompasses both theoretical analysis and practical applications, including a chapter on simulations and sampling methods.Chapter 1 introduces basic random graph theory and emphasizes the importance of maximum entropy models in real-world network modeling. It defines BEE and its characterization within statistical mechanics, highlighting the natural differences between canonical and microcanonical ensembles. It then introduces spectral theory of random graphs and why it is utilized to investigate BEE.Chapter 2 proposes a conjecture linking BEE to a gap between the expectations of the largest eigenvalue in the canonical and microcanonical ensembles, proving it for homogeneous graphs.Chapter 3 examines Chung-Lu random graphs, establishing central limit theorems for the largest eigenvalue and its eigenvector.Chapter 4 verifies the conjecture for inhomogeneous graphs, computing the expected largest eigenvalue of the configuration model.Chapter 5 offers numerical evidence through simulations, after a brief introduction to graph sampling. The thesis concludes with a summary of findings and suggestions for future research. Show less
This dissertation consists of two parts, each of which considers a different research area related to random interval maps. In the first part we are interested in random interval maps that are... Show moreThis dissertation consists of two parts, each of which considers a different research area related to random interval maps. In the first part we are interested in random interval maps that are critically intermittent. In Chapter 2 we consider a large class of such systems and demonstrate the existence of a phase transition, where the absolutely continuous invariant measure changes between finite and infinite. For a closely related class we derive in Chapter 3 statistical properties like decay of correlations and the Central Limit Theorem. In Chapter 4 we investigate whether a similar phase transition remains to exist when the critical behaviour is toned down. Random interval maps can also be used to generate number expansions, which will be the main object of study in the second part. In Chapter 5 we generalize Lochs’ Theorem, which compares the efficiency between representing real numbers in decimal expansions and regular continued fraction expansions, to a wide class of pairs of random interval maps that produce number expansions. Closely related to this result, we study in Chapter 6 the efficiency of beta-encoders as a potential source for pseudo-random number generation by comparing the output of a beta-encoder with its corresponding binary expansion. Show less
A Chung–Lu random graph is an inhomogeneous Erdős–Rényi random graph in which vertices are assigned average degrees, and pairs of vertices are connected by an edge with a probability that is... Show moreA Chung–Lu random graph is an inhomogeneous Erdős–Rényi random graph in which vertices are assigned average degrees, and pairs of vertices are connected by an edge with a probability that is proportional to the product of their average degrees, independently for different edges. We derive a central limit theorem for the principal eigenvalue and the components of the principal eigenvector of the adjacency matrix of a Chung–Lu random graph. Our derivation requires certain assumptions on the average degrees that guarantee connectivity, sparsity and bounded inhomogeneity of the graph. Show less
The subject of this thesis, ‘Approach to Markov Operators on Spaces of Measures by Means of Equicontinuity’, combines an analytical and probabilistic approach to Markov operators. We look at Markov... Show moreThe subject of this thesis, ‘Approach to Markov Operators on Spaces of Measures by Means of Equicontinuity’, combines an analytical and probabilistic approach to Markov operators. We look at Markov operators coming from deterministic dynamical systems and also stochastic processes which come from a probabilistic approach.In the study of Markov operators and Markov semigroups the central problems are to understand the behaviour of the processes and semigroups. Of particular interest is to identify the existence and uniqueness of invariant measures and long term behaviour of the process and dynamical system defined by the associated Markov operator or semigroup. Research on these questions dates back to the works of Andrey Markov, who described a Markov property for chains. A big part of theory for Markov chains can be found in the book by Meyn and Tweedie, who made a big contribution to the theory of Markov chains and gave a noteworthy description of e-chains, which was the motivation to working with equicontinuity properties for many authors. This theory is applicable when the underlying state space is locally compact. If it is not - in the generality of so-called Polish spaces - there is theory under development. Lasota and Szarek, and in recent years Worm generalized theory of Markov operators and families of Markov operators to this setting. In subsequent years, the theory was being developed starting with contractive Markov operators in the works of Lasota, through non-expansive Markov operators in Szarek’s,, and finally equicontinuous families of Markov operators in that of Szarek, Hille and Worm. We extend their results and give a new light to the existing ones by providing less restrictive conditions in cases. Show less
