<p>This dissertation includes my research works during Ph.D. career about three different kinds of random walks in (dynamical) random environments. It includes my two published papers “Functional weak limit of random walks in cooling random environments” which has been published in electronic communications in probability in 2020, and “Variable speed symmetric random walk driven by the simple symmetric exclusion process” which is the joint work with Peterson and Menezes and has been published in electronic journals of probability in 2021. This dissertation also includes my two other projects, one is the joint work with Janjigian and Emrah about moderate deviation and exit time estimates in integrable directed polymer models. The other one is the joint work with Peterson and Conrado that extends the weak limit of random walks in cooling randon environments with underlying environment is in the transient case.</p>

In this thesis several classes of directed paths are considered as models of linear polymers in a dilute solution. We obtain the generating functions for each model by considering factorization arguments. Information about the polymer behaviour can be extracted from the singularity structure of the associated generating functions. By using modified versions of these models we study the adsorption and localization of polymer molecules, the behaviour of polymers subject to a tensile force, the effects of stiffness, as well as the behaviour of polymers in confined geometries. In each of these situations the resulting generating functions contain at least two physical singularities. We identify the phase transitions in these systems by a changeover in the dominant singularity of the generating function. In the study of localization and polymers subject to a force, we utilize both homopolymer and random copolymer models. For copolymers, the physically relevant properties are obtained by considering a quenched average of the free energy over all possible monomer sequences. This procedure is intractable even for the simplest models. By considering the Morita approximation for several walk models we obtain results which give a bound on the corresponding features of the quenched system. We use a mapping between a simple model of duplex DNA and an adsorbing Motzkin path in order to study the mechanical unzipping of duplex DNA. From this model, we obtain force-temperature diagrams which show re-entrant behaviour of the force. We also develop a simple low temperature theory to describe the behaviour of the force close to T=0 and find that the shape of the force-temperature curve is associated with entropy in the ground state of the system. We consider the effect of stiffness on polymer adsorption and find that the phase transition is second order for all finite stiffness parameters. For systems of polymers in confined geometries, we find that the behaviour of the polymer depends on the distance between the confining surfaces and the associated interactions with each surface. In this problem, there exist regimes where the polymer exerts a force on the surfaces which can be attractive, repulsive or zero.

O avanço das pesquisas em redes complexas proporcionou desenvolvimentos significativos para a compreensão de sistemas complexos. Uma rede complexa é modelada matematicamente por meio de um grafo, onde cada vértice representa uma unidade dinâmica e suas interações são simbolizadas por um conjunto de arestas. Para se determinar propriedades estruturais desse sistema, caminhadas aleatórias tem-se mostrado muito úteis pois dependem apenas de informações locais (vértices vizinhos). Entre elas, destaca-se o passeio auto-excludente (SAW) que possui a restrição de não visitar um vértice que já foi alcançado, ou seja, apresenta memória do caminho percorrido. Por este motivo o SAW tem apresentado melhores resultados do que caminhantes sem restrição, na exploração da rede. Entretanto, por não se tratar de um processo Markoviano ele apresenta grande complexidade analítica, tornando indispensável o uso de simulações computacionais para melhor compreensão de sua dinâmica em diferentes topologias. Mesmo com as dificuldades analíticas, o SAW se tornou uma ferramenta promissora na identificação de estruturas de comunidades. Apesar de sua importância, detecção de comunidades permanece um problema em aberto devido à alta complexidade computacional associada ao problema de optimização, além da falta de uma definição formal do significado de comunidade. Neste trabalho, propomos um método de detecção de comunidades baseado em SAW para extrair uma estrutura de comunidades da rede otimizando o parâmetro modularidade. Combinamos características extraídas desta dinâmica com a análise de componentes principais para posteriormente classificar os vértices em grupos por meio da clusterização hierárquica aglomerativa. Para avaliar a performance deste novo algoritmo, comparamos os resultados com outras quatro técnicas populares: Girvan-Newman, Fastgreedy, Walktrap e Infomap, aplicados em dois tipos de redes sintéticas e nove redes reais diversificadas e bem conhecidas. Para os benchmarks, esta nova técnica produziu resultados satisfatórios em diferentes combinações de parâmetros, como tamanho de rede, distribuição de grau e número de comunidades. Já para as redes reais, obtivemos valores de modularidade superior aos métodos tradicionais, indicando uma distribuição de grupos mais adequada à realidade. Feito isso, generalizamos o algoritmo para redes ponderadas e digrafos, além de incorporar metadados à estrutura topológica a fim de melhorar a classificação em grupos. / The progress in complex networks research has provided significant understanding of complex systems. A complex network is mathematically modeled by a graph, where each vertex represents a dynamic unit and its interactions are symbolized by groups of edges. To determine the system structural properties, random walks have shown to be a useful tool since they depend only on local information (neighboring vertices). Among them, the selfavoiding walk (SAW) stands out for not visiting vertices that have already been reached, meaning it can record the path that has been travelled. For this reason, SAW has shown better results when compared to non-restricted walkers network exploration methods. However, as SAW is not a Markovian process, it has a great analytical complexity and needs computational simulations to improve its dynamics in different topologies. Even with the analytical complexity, SAW has become a promising tool to identify the community structure. Despite its significance, detecting communities remains an unsolved problem due to its high computational complexity associated to optimization issues and the lack of a formal definition of communities. In this work, we propose a method to identify communities based on SAW to extract community structure of a network through optimization of the modularity score. Combining technical features of this dynamic with principal components analyses, we classify the vertices in groups by using hierarchical agglomerative clustering. To evaluate the performance of this new algorithm, we compare the results with four other popular techniques: Girvan-Newman, Fastgreedy, Walktrap and Infomap, applying the algorithm in two types of synthetic networks and nine different and well known real ones. For the benchmarks, this new technique shows satisfactory results for different combination of parameters as network size, degree distribution and number of communities. As for real networks, our data shows better modularity values when compared to traditional methods, indicating a group distribution most suitable to reality. Furthermore, the algorithm was adapted for general weighted networks and digraphs in addition to metadata incorporated to topological structure, in order to improve the results of groups classifications.

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.

The subject of this thesis is the asymptotic behaviour of generating functions of different combinatorial models of two-dimensional lattice walks and polygons, enumerated with respect to different parameters, such as perimeter, number of steps and area. These models occur in various applications in physics, computer science and biology. In particular, they can be seen as simple models of biological vesicles or polymers. Of particular interest is the singular behaviour of the generating functions around special, so-called multicritical points in their parameter space, which correspond physically to phase transitions. The singular behaviour around the multicritical point is described by a scaling function, alongside a small set of critical exponents. Apart from some non-rigorous heuristics, our asymptotic analysis mainly consists in applying the method of steepest descents to a suitable integral expression for the exact solution for the generating function of a given model. The similar mathematical structure of the exact solutions of the different models allows for a unified treatment. In the saddle point analysis, the multicritical points correspond to points in the parameter space at which several saddle points of the integral kernels coalesce. Generically, two saddle points coalesce, in which case the scaling function is expressible in terms of the Airy function. As we will see, this is the case for Dyck and Schröder paths, directed column-convex polygons and partially directed self-avoiding walks. The result for Dyck paths also allows for the scaling analysis of Bernoulli meanders (also known as ballot paths). We then construct the model of deformed Dyck paths, where three saddle points coalesce in the corresponding integral kernel, thereby leading to an asymptotic expression in terms of a bivariate, generalised Airy integral.

Walks organised at the community or 'grass roots' level both in Australia and internationally have been organised as a means to highlight environmental and social issues to the wider community. This thesis focuses on a coastal walk from Melbourne to Sydney during November 1993 to March 1994 called CoastWalk which was organised as part of the Australian Conservation Foundation's 'Coasts in Crisis' campaign. The Walk aimed to highlight environmental management problems specific to the coastal zone at both local and national levels using mass media, information evenings and targeting groups within local communities. Using a case study approach to the methodology, combined with principles from both social (interpretive) and empirical methods, this study involved determining the impact of CoastWalk, i.e. whether it changed individuals at the levels of awareness, understanding or action. The scope of the study does not include a detailed analysis of the communities themselves, nor does it explore the psychological aspects of individual and social change. It was found that the mass media aspect of the CoastWalk campaign created a short term awareness of the need to have concern for coastal management issues in those who did not participate in the Walk. For those who did participate, the impact was deeper and profound, changing individuals understanding and actions towards coastal management and translating into other areas of their lives. The impact of the Walk on local community groups who supported the Walk was negligible and it was determined that CoastWalk did not meet their needs. Community involvement in environmental management or campaigning equates to long term ownership and responsibility being taken for those issues. However, neither community nor government intervention in environmental management alone can resolve these issues successfully. This thesis argues that a combined approach from both community and government organisations is required - but as exemplified by CoastWalk, the success of this approach requires equality in communication and co-operation. As other environmental awareness walks have occurred, it is evident that they are perceived as worthy events by the community, and that there is potential for them to occur again in the future. It is therefore essential for an evaluation to occur of the techniques used to achieve their environmental education aims. Thus, the learning from previous Walks can be built into future Walks enhancing their success.

<p>Lévy flights and Lévy walks are two mathematical models used to describe anomalous diffusion(i.e. those having mean square displacements nonlinearly related to time (as opposed to Brownian motion)). Lévy flights follow probability distributions p(|<strong>r</strong>|) yielding infinite mean square displacements since some rare steps are very long. Lévy walks, however, have coupled space-time probability distributions penalising very long steps. Both Lévy flights and Lévy walks are dominated by a few long steps, but most steps are much, much smaller. The semi-experimental part ofthis work dealt with how fluorescent probes moved in systems of cationic starch and latex/solutions of dodecyl trimethyl ammonium bromide, respectively. Visually, no Lévy walks couldbe detected. However, mathematical regression suggested enhanced diffusion and subdiffusion. Moreover, time-dependent diffusion coefficients were calculated. Also examined was how Microsoft Excel could be used to generate normal diffusion as well as anomalous diffusion.</p> / <p>Lévyflygningar och Lévypromenader är matematiska modeller som används för att beskriva anomal diffusion (i.e. dessa då medelvärdet av kvadratförflyttningarna är icke-linjärt relaterat tilltiden (till skillnad från Brownsk rörelse)). Lévyflygningar följer sannolikhetsfördelningar p(|<strong>r</strong>|)som ger oändliga medelkvadratförflyttningar eftersom vissa steg är väldigt långa. Lévypromenader,å andra sidan, har kopplade rum-tid-sannolikhetsfördelningar som kraftigt reducerar demycket långa stegen. Både Lévyflygningar och -promenader domineras av ett fåtal långa steg ävenom de flesta steg är mycket, mycket mindre. Den semiexperimentella delen av detta arbetestuderade hur fluorescerande prober rör sig i katjonisk stärkelse respektive latex/lösningar avdodecyltrimetylammoniumbromid. Inga Lévypromenader kunde ses. Emellertid taladematematisk regression för att superdiffusion och subdiffusion förelåg. Tidsberoende diffusionskoefficienter beräknades också. I detta arbete undersöktes även hur Microsoft Excel kan användas för att generera både normal och anomal diffusion.</p>

Lévy flights and Lévy walks are two mathematical models used to describe anomalous diffusion(i.e. those having mean square displacements nonlinearly related to time (as opposed to Brownian motion)). Lévy flights follow probability distributions p(|r|) yielding infinite mean square displacements since some rare steps are very long. Lévy walks, however, have coupled space-time probability distributions penalising very long steps. Both Lévy flights and Lévy walks are dominated by a few long steps, but most steps are much, much smaller. The semi-experimental part ofthis work dealt with how fluorescent probes moved in systems of cationic starch and latex/solutions of dodecyl trimethyl ammonium bromide, respectively. Visually, no Lévy walks couldbe detected. However, mathematical regression suggested enhanced diffusion and subdiffusion. Moreover, time-dependent diffusion coefficients were calculated. Also examined was how Microsoft Excel could be used to generate normal diffusion as well as anomalous diffusion. / Lévyflygningar och Lévypromenader är matematiska modeller som används för att beskriva anomal diffusion (i.e. dessa då medelvärdet av kvadratförflyttningarna är icke-linjärt relaterat tilltiden (till skillnad från Brownsk rörelse)). Lévyflygningar följer sannolikhetsfördelningar p(|r|)som ger oändliga medelkvadratförflyttningar eftersom vissa steg är väldigt långa. Lévypromenader,å andra sidan, har kopplade rum-tid-sannolikhetsfördelningar som kraftigt reducerar demycket långa stegen. Både Lévyflygningar och -promenader domineras av ett fåtal långa steg ävenom de flesta steg är mycket, mycket mindre. Den semiexperimentella delen av detta arbetestuderade hur fluorescerande prober rör sig i katjonisk stärkelse respektive latex/lösningar avdodecyltrimetylammoniumbromid. Inga Lévypromenader kunde ses. Emellertid taladematematisk regression för att superdiffusion och subdiffusion förelåg. Tidsberoende diffusionskoefficienter beräknades också. I detta arbete undersöktes även hur Microsoft Excel kan användas för att generera både normal och anomal diffusion.

This paper investigates whether the inherent non-stationarity of macroeconomic time series is entirely due to a random walk or also to non-linear components. Applying the numerical tools of the analysis of dynamical systems to long time series for the US, we reject the hypothesis that these series are generated solely by a linear stochastic process. Contrary to the Real Business Cycle theory that attributes the irregular behavior of the system to exogenous random factors, we maintain that the fluctuations in the time series we examined cannot be explained only by means of external shocks plugged into linear autoregressive models. A dynamical and non-linear explanation may be useful for the double aim of describing and forecasting more accurately the evolution of the system. Linear growth models that find empirical verification on linear econometric analysis, are therefore seriously called in question. Conversely non-linear dynamical models may enable us to achieve a more complete information about economic phenomena from the same data sets used in the empirical analysis which are in support of Real Business Cycle Theory. We conclude that Real Business Cycle theory and more in general the unit root autoregressive models are an inadequate device for a satisfactory understanding of economic time series. A theoretical approach grounded on non-linear metric methods, may however allow to identify non-linear structures that endogenously generate fluctuations in macroeconomic time series. (authors' abstract) / Series: Working Papers Series "Growth and Employment in Europe: Sustainability and Competitiveness"

Existing methods to utilize capacity-heterogeneity in a P2P system either rely on constructing special overlays with capacity-proportional node degree or use topology adaptation to match a node's capacity with that of its neighbors. In existing P2P networks, which are often characterized by diverse node capacities and high churn, these methods may require large node degree or continuous topology adaptation, potentially making them infeasible due to their high overhead. In this thesis, we propose an unstructured P2P system that attempts to address these issues. We first prove that the overall throughput of search queries in a heterogeneous network is maximized if and only if traffic load through each node is proportional to its capacity. Our proposed system achieves this traffic distribution by biasing search walks using the Metropolis-Hastings algorithm, without requiring any special underlying topology. We then define two saturation metrics for measuring the performance of overlay networks: one for quantifying their ability to support random walks and the second for measuring their potential to handle the overhead caused by churn. Using simulations, we finally compare our proposed method with Gia, an existing system which uses topology adaptation, and find that the former performs better under all studied conditions, both saturation metrics, and such end-to-end parameters as query success rate, latency, and query-hits for various file replication schemes.