This thesis is divided into two parts. In Part I we study metastability properties of queue-based random-access protocols for wireless networks. The network is modeledas a bipartite graph whose... Show moreThis thesis is divided into two parts. In Part I we study metastability properties of queue-based random-access protocols for wireless networks. The network is modeledas a bipartite graph whose edges represent interference constraints. In Part II we study spectra of inhomogeneous Erdős-Rényi random graphs. We focus in particularon the limiting spectral distribution of the adjacency and Laplacian matrices and on the largest eigenvalue of the adjacency matrix. Show less
This thesis deals with two different models in two different contexts. The first part deals with dynamical Gibbs-non-Gibbs transitions. Gibbs measures describe the equilibrium states of a system... Show moreThis thesis deals with two different models in two different contexts. The first part deals with dynamical Gibbs-non-Gibbs transitions. Gibbs measures describe the equilibrium states of a system consisting of a large number of components that interact with each other. Due to the large number of particles, it is natural to assume that the state of the system is random. Gibbs measures capture this randomness. This description involves some particular ``regularity'' conditions for the conditional probabilities. The question of interest is whether this condition remains valid after the system is subjected to a stochastic dynamic. Is it still possible to describe the evolved measure as a Gibbs measure? The second part deals with stochastic geometry. The relevant information about the particles is their position. Particles may be placed at random in any region of the space. Subsequently, each particle is displaced independently of each other according to a d-dimensional Brownian Motion during t time, and the trace produced by that motion is recorded. The question of interest is whether the final set obtained from all the traces has an infinite connected component or not. If so, then is it unique? Show less
This thesis is dedicated to the study of random walks in dynamic random environments. These are models for the motion of a tracer particle in a disordered medium, which is called a static random... Show moreThis thesis is dedicated to the study of random walks in dynamic random environments. These are models for the motion of a tracer particle in a disordered medium, which is called a static random environment if it stays constant in time, or dynamic otherwise. The evolution of the random walk is defined by assigning to it random jump rates which depend locally on the random environment. Such models belong to the greater area of \emph{disordered systems}, and have been studied extensively since the early seventies in the physics and mathematics literature. The goal is to understand the scaling properties, as time goes to infinity, of the path of the random walk. Several results are available in the literature for dynamic random environments which are uniformly elliptic and have uniform and fast enough mixing in space-time. However, very little is known when either of these conditions fail. In this thesis, we study examples of such situations, namely, non-elliptic cases in Chapter 2, a dynamic random environment with fast but non-uniform mixing in Chapter 4, and a dynamic random environment with both slow and non-uniform mixing in Chapters 3 and 5. Show less
In this thesis we use both the two-layer and the large-deviation approach to study the conservation and loss of the Gibbs property for both lattice and mean-field spin systems. Chapter 1 gives... Show moreIn this thesis we use both the two-layer and the large-deviation approach to study the conservation and loss of the Gibbs property for both lattice and mean-field spin systems. Chapter 1 gives general backgrounds on Gibbs and non-Gibbs measures and outlines the the two-layer and the large-deviation approach. Chapter 2 studies the transforms of one-dimensional lattice spin systems. We start from a Gibbs measure with infinite range interaction and consider both deterministic and stochastic transformations K. Using the two-layer approach we prove that the constrained system has a unique Gibbs measure for every choice of transformed configuration, as long as the range of K is finite. This implies that the associated transformed Gibbs measures are always Gibbs. Further, we prove that if the initial interaction is exponentially decaying, then the transformed interaction decays exponentially as well, while if the initial interaction is polynomially decaying (with an exponent large enough so that the system is in the uniqueness regime), then the transformed interaction decays polynomially as well (with a smaller power). The proofs of these results use the house-of-cards coupling argument. Chapters 3 and 4 provide new and explicitly computable examples of Gibbs-non-Gibbs transitions by using the large-deviation approach. These examples include independent Brownian motions, Ornstein-Uhlenbeck processes, and birth-death processes. Chapter 4 computes the Feng-Kurtz Hamiltonian and Lagrangian associated to the large deviations of the trajectory of the empirical distribution for independent Markov processes, and of the empirical measure for translation invariant interacting Markov processes. We treat both the case of jump processes (continuous-time Markov chains and interacting particle systems) and the case of diffusion processes. For di usion processes, the Lagrangian is a quadratic form of the deviation of the trajectory from the Kolmogorov forward equation. In all cases, the Lagrangian can be interpreted as a relative entropy (density) per unit time. Show less
