Fluctuations are of great importance in systems of small length and energy scales. Measuring the pulling of single molecules or the stationary fiow of mesospheres dragged through a viscous media enables the direct analysis of work and entropy distributions. These probability distributions are the result of a large number of repetitions of the same experiment. Due to the small scale of these experiments, the outcome can vary significantly from one realization to the next. Strong theoretical predictions exist, collectively called Fluctuation Theorems, that restrict the shape of these distributions due to an underlying time reversal symmetry of the microscopic dynamics. Fluctuation Theorems are the strongest existing statements on the entropy production of systems that are out of equilibrium. Being the most important ingredient for the Fluctuation Theorems, the probability distribution of the entropy change is itself of great interest. Using numerically exact methods we characterize entropy distributions for various stochastic reaction-diffusion systems that present different properties in their underlying dynamics. We investigate these systems in their steady states and in cases where time dependent forces act on them. This study allows us to clarify the connection between the microscopic rules and the resulting entropy production. The present work also adds to the discussion of the steady state properties of stationary probabilities and discusses a non-equilibrium current amplitude that allows us to quantify the distance from equilibrium. The presented results are part of a greater endeavor to find common rules that will eventually lead to a general understanding of non-equilibrium systems. / Ph. D.

We study the large deviation behaviour of simple random walks in dimension three or more in this thesis. The first part of the thesis concerns the number of lattice sites visited by the random walk. We call this the range of the random walk. We derive a large deviation principle for the probability that the range of simple random walk deviates from its mean. Our result describes the behaviour for deviation below the typical value. This is a result analogous to that obtained by van den Berg, Bolthausen, and den Hollander for the volume of the Wiener sausage. In the second part of the thesis, we are interested in the number of lattice sites visited by two independent simple random walks starting at the origin. We call this the intersection of ranges. We derive a large deviation principle for the probability that the intersection of ranges by time n exceeds a multiple of n. This is also an analogous result of the intersection volume of two independent Wiener sausages.

Methods of efficient Monte-Carlo simulation when rare events are involved have been studied for several decades. Rare events are very important in the context of evaluating high quality computer/communication systems. Meanwhile, the efficient simulation of systems involving rare events poses great challenges. A simulation method is said to be efficient if the number of replicas required to get accurate estimates grows slowly, compared to the rate at which the probability of the rare event approaches zero. Despite the great success of the two mainstream methods, importance sampling (IS) and importance splitting, either of them can become inefficient under certain conditions, as reported in some recent studies. The purpose of this study is to look for possible enhancement of fast simulation methods. I focus on the ``level/phase process', a Markov process in which the level and the phase are two state variables. Furthermore, changes of level and phase are induced by events, which have rates that are independent of the level except at a boundary. For such a system, the event of reaching a high level occurs rarely, provided the system typically stays at lower levels. The states at those high levels constitute the rare event set. Though simple, this models a variety of applications involving rare events. In this setting, I have studied two efficient simulation methods, the rate tilting method and the adaptive splitting method, concerning their efficiencies. I have compared the efficiency of rate tilting with several previously used similar methods. The experiments are done by using queues in tandem, an often used test bench for the rare event simulation. The schema of adaptive splitting has not been described in literature. For this method, I have analyzed its efficiency to show its superiority over the (conventional) splitting method. The way that a system approaches a designated rare event set is called the system's large deviation behavior. Toward the end of gaining insight about the relation of system behavior and the efficiency of IS simulation, I quantify the large deviation behavior and its complexity. This work indicates that the system's large deviation behavior has a significant impact on the efficiency of a simulation method.

Methods of efficient Monte-Carlo simulation when rare events are involved have been studied for several decades. Rare events are very important in the context of evaluating high quality computer/communication systems. Meanwhile, the efficient simulation of systems involving rare events poses great challenges. A simulation method is said to be efficient if the number of replicas required to get accurate estimates grows slowly, compared to the rate at which the probability of the rare event approaches zero. Despite the great success of the two mainstream methods, importance sampling (IS) and importance splitting, either of them can become inefficient under certain conditions, as reported in some recent studies. The purpose of this study is to look for possible enhancement of fast simulation methods. I focus on the ``level/phase process', a Markov process in which the level and the phase are two state variables. Furthermore, changes of level and phase are induced by events, which have rates that are independent of the level except at a boundary. For such a system, the event of reaching a high level occurs rarely, provided the system typically stays at lower levels. The states at those high levels constitute the rare event set. Though simple, this models a variety of applications involving rare events. In this setting, I have studied two efficient simulation methods, the rate tilting method and the adaptive splitting method, concerning their efficiencies. I have compared the efficiency of rate tilting with several previously used similar methods. The experiments are done by using queues in tandem, an often used test bench for the rare event simulation. The schema of adaptive splitting has not been described in literature. For this method, I have analyzed its efficiency to show its superiority over the (conventional) splitting method. The way that a system approaches a designated rare event set is called the system's large deviation behavior. Toward the end of gaining insight about the relation of system behavior and the efficiency of IS simulation, I quantify the large deviation behavior and its complexity. This work indicates that the system's large deviation behavior has a significant impact on the efficiency of a simulation method.

International Telemetering Conference Proceedings / October 18-21, 2004 / Town & Country Resort, San Diego, California / Powered by single-segment switched interconnection, Ethernet can be used in time-critical data acquisition applications. Unlike synchronous time division multiple access, asynchronous packet streams result in congestions and uncertain multiplexing delays. With the delay analysis in the worst case and probabilistic guaranteeing conditions, we restrict the packet-sizes, intervals or traffic burstiness a priori to regulate delay deviations within acceptable scales. Some methods of combinatorics and stochastic theory, e.g. Cumulant Generating Function and the Large Deviation Principle, are used and verified by some simulation-based computations. The influence of time varying delay for telemetry applications is also discussed in some sense.

The study on the longest stretch of consecutive successes in \random" trials dates back to 1916 when the German philosopher Karl Marbe wrote a paper concerning the longest stretch of consecutive births of children of the same sex as appearing in the birth register of a Bavarian town. The result was actually used by parents to \predict" the sex of their children. The longest stretch of same-sex births during that time in 200 thousand birth registrations was actually 17 t log2(200 103): During the past century, the research of longest stretch of consecutive successes (longest runs) has found applications in various areas, especially in the theory of reliability. The aim of this thesis is to study large deviations on longest runs in the setting of Markov chains. More precisely, we establish a general large deviation principle for the longest success run in a two-state (success or failure) Markov chain. Our tool is based on a recent result regarding a general large deviation for the longest success run in Bernoulli trails. It turns out that the main ingredient in the proof is to implement several global and local estimates of the cumulative distribution function of the longest success run.

Nesta dissertação estudamos uma probabilidade obtida a partir de conceitos da Mecânica Estatística Quântica do ponto de vista da Teoria Ergódica. A probabilidade é obtida a partir de um estado KMS sobre um lattice unidimensional de spins quânticos. Mostramos que esta probabilidade é mixing para o shift. Além disso, mostramos que vale um princípio dos grandes desvios para uma certa classe de funções e exploramos algumas propriedades do Jacobiano. Iremos considerar o estado KMS associado a um certo Hamiltoniano específico agindo sobre o lattice de spins quânticos. Nas seções iniciais vamos apresentar alguns conceitos e prerequisitos básicos (como operadores densidade, produto tensorial, C*-algebras e estados KMS) para o entendimento do resultado principal / In this dissertation we study a probability derived from Quantum Statistical Mechanics through the viewpoint of Ergodic Theory. The probability is obtained from a KMS state acting on a one dimensional lattice of quantum spins. We show that this probability is mixing for the shift map. Moreover, we show that a large deviation principle is true for a certain class of functions and we explore some properties of the Jacobian. We will consider the KMS state associated to a certain specific Hamiltonian acting on the quantum spin lattice. In the initial sections we will present some concepts and prerequisites (such as density operators, tensor product, C*-algebras and KMS states) for the understanding of our main results.

This thesis studies some fundamental aspects of wireless systems with partial channel state information at the transmitter (CSIT), with a special emphasis on the high signal-to-noise ratio (SNR) regime. The first contribution is a study on multi-layer variable-rate communication systems with quantized feedback, where the expected rate is chosen as the performance measure. Iterative algorithms exploiting results in the literature of parallel broadcast channels are developed to design the system parameters. Necessary and sufficient conditions for single-layer coding to be optimal are derived. In contrast to the ergodic case, it is shown that a few bits of feedback information can improve the expected rate dramatically. The next part of the thesis is devoted to characterizing the tradeoff between diversity and multiplexing gains (D-M tradeoff) over slow fading channels with partial CSIT. In the multiple-input multiple-output (MIMO) case, we introduce the concept of minimum guaranteed multiplexing gain in the forward link and show that it influences the D-M tradeoff significantly. It is demonstrated that power control based on the feedback is instrumental in achieving the D-M tradeoff, and that rate adaptation is important in obtaining a high diversity gain even at high rates. Extending the D-M tradeoff analysis to decode-and-forward relay channels with quantized channel state feedback, we consider several different scenarios. In the relay-to-source feedback case, it is found that using just one bit of feedback to control the source transmit power is sufficient to achieve the multiantenna upper bound in a range of multiplexing gains. In the destination-to-source-and-relay feedback scenario, if the source-relay channel gain is unknown to the feedback quantizer at the destination, the diversity gain only grows linearly in the number of feedback levels, in sharp contrast to an exponential growth for MIMO channels. We also consider the achievable D-M tradeoff of a relay network with the compress-and-forward protocol when the relay is constrained to make use of standard source coding. Under a short-term power constraint at the relay, using source coding without side information results in a significant loss in terms of the D-M tradeoff. For a range of multiplexing gains, this loss can be fully compensated for by using power control at the relay. The final part of the thesis deals with the transmission of an analog Gaussian source over quasi-static fading channels with limited CSIT, taking the SNR exponent of the end-to-end average distortion as performance measure. Building upon results from the D-M tradeoff analysis, we develop novel upper bounds on the distortion exponents achieved with partial CSIT. We show that in order to achieve the optimal scaling, the CSIT feedback resolution must grow logarithmically with the bandwidth ratio for MIMO channels. The achievable distortion exponent of some hybrid schemes with heavily quantized feedback is also derived. As for the half-duplex fading relay channel, combining a simple feedback scheme with separate source and channel coding outperforms the best known no-feedback strategies even with only a few bits of feedback information. / QC 20100817

Gegenstand der vorliegenden Dissertation ist das Verhalten sogenannter Selbstüberschneidungslokalzeiten $\\|\\ell_t\\|_p^p$ einer zeitstetigen Irrfahrt $(S_r)_r$ auf dem $d$-dimensionalen Gitter $\\Z^d$. Dabei ist für $p>1$ die Funktion $\\ell_t$ definiert durch $$ \\ell_t(z):=\\int_{0}^{t}\\1_{\\{S_r=z\\}}\\,\\d r\\nonumber $$ und bezeichnet die Aufenthaltsdauer der Irrfahrt bis zum Zeitpunkt $t\\in(0,\\infty)$ im Punkt $z\\in\\Z^d$. Ziel ist es, ein Prinzip großer Abweichungen zu entwickeln, d.h. das Hauptaugenmerk liegt auf dem asymptotischen Verhalten der Wahrscheinlichkeit, dass die Selbstüberschneidungslokalzeiten von ihrem Erwartungswert in erheblichem Maße nach oben abweichen. Mit anderen Worten; es soll das asymptotische Verhalten von $$ \\log\\P(\\|\\ell_t\\|_p^p\\geq r^p_t) $$ genau bestimmt werden, wobei $r_t^p\\in(0,\\infty)$ schneller als der Erwartungswert $\\E[\\|\\ell_t\\|_p^p]$ gegen unendlich streben soll. Dieses Verhalten kann dabei durch $t$, $r_t$ und eine gewisse Variationsformel beschrieben werden. Es wird sich herausstellen, dass es zwei Fälle zu betrachten gilt, in denen sich das probabilistisch beste Verhalten stark unterscheidet; die genaue Position des Phasenübergangs hängt dabei von den Parametern $p$ und $d$ ab. Im Vorgriff auf die Resultate kann man festhalten, dass die nötigen Selbstüberschneidungen in kleinen Dimensionen (im sogenannten subkritischen Fall) über einen großen Bereich erfolgen, aufgrund dessen bei der mathematischen Modellierung eine Reskalierung erforderlich ist. In hohen Dimensionen (dem sogenannten superkritischen Fall) ist dies nicht nötig, da die erforderlichen Selbstüberschneidungen innerhalb eines begrenzten Intervalles erfolgen. Das Interesse an der Untersuchung entstand unter anderem aus der Verbindung zu Modellen der statistischen Mechanik (parabolisches Anderson Modell) und zur Variationsanalysis. In der Vergangenheit wurde eine Vielzahl an Methoden benutzt, um dieses Problem zu lösen. In der vorliegenden Dissertation soll die sogenannte Momentenmethode bestmöglich ausgereizt werden und es wird gezeigt, welche Ergebnisse damit möglich sind.

Nesta dissertação estudamos uma probabilidade obtida a partir de conceitos da Mecânica Estatística Quântica do ponto de vista da Teoria Ergódica. A probabilidade é obtida a partir de um estado KMS sobre um lattice unidimensional de spins quânticos. Mostramos que esta probabilidade é mixing para o shift. Além disso, mostramos que vale um princípio dos grandes desvios para uma certa classe de funções e exploramos algumas propriedades do Jacobiano. Iremos considerar o estado KMS associado a um certo Hamiltoniano específico agindo sobre o lattice de spins quânticos. Nas seções iniciais vamos apresentar alguns conceitos e prerequisitos básicos (como operadores densidade, produto tensorial, C*-algebras e estados KMS) para o entendimento do resultado principal / In this dissertation we study a probability derived from Quantum Statistical Mechanics through the viewpoint of Ergodic Theory. The probability is obtained from a KMS state acting on a one dimensional lattice of quantum spins. We show that this probability is mixing for the shift map. Moreover, we show that a large deviation principle is true for a certain class of functions and we explore some properties of the Jacobian. We will consider the KMS state associated to a certain specific Hamiltonian acting on the quantum spin lattice. In the initial sections we will present some concepts and prerequisites (such as density operators, tensor product, C*-algebras and KMS states) for the understanding of our main results.