This work is a study of complex many-body systems with non-trivial interactions. Many such systems can be described with models that are much simpler than the real thing but which can still give good insight into the behavior of realistic systems. We take a look at two such systems. The first part looks at a model that elucidates the variety of magnetic phases observed in rare-earth heterostructures at low temperatures: the six-state clock model. We use an ANNNI-like model Hamiltonian that has a three-dimensional parameter space and yields two-dimensional multiphase regions in this space. A low-temperature expansion of the free energy reveals an example of Villain's `order from disorder' [81, 60] when an infinitesimal temperature breaks the ground-state degeneracy. The next part of our work describes biological systems. Using ECIS (Electric Cell-Substrate Impedance Sensing), we are able to extract complex impedance time series from a confluent layer of live cells. We use simple statistics to characterize the behavior of cells in these experiments. We compare experiment with models of fractional Brownian motion and random walks with persistence. We next detect differences in the behavior of single cell types in a toxic environment. Finally, we develop a very simple model of micromotion that helps explain the types of interactions responsible for the long-term and short-term correlations seen in the power spectra and autocorrelation curves extracted from the times series produced from the experiments.

I många nätverk antas det att någon form av fix infrastruktur existerar och att nätverkets olika noder kan använda denna för att kommunicera med varandra. I ett ad hoc-nätverk antar man att det inte finns någon fix infrastruktur och att noderna måste använda varandra för att kunna kommunicera. Ett exempel på ett ad hoc-nätverk kan vara bärbara datorer sammankopplade med infraröda länkar under ett möte. När ad hoc-nätverket är mobilt innebär det att noderna rör sig. I detta arbete har de tre protokollen Epidemic, GeoMean och GeoMove tillsammans med de två rörelsemodellerna Waypoint och den utökade slumpmässiga vandringen implementerats i en nyskriven simulator för denna typ av nätverk. De två Geo-protokollen är nyutvecklade och syftar till att använda geografisk information för att underlätta kommunikationen i denna kategori av nätverk tillsammans med den nya utvidgade slumpmässiga vandringsmodellen. / In many networks, some kind of fix infrastructure is assumed to exist and the nodes of the network can use this infrastructure to communicate with each other. In an ad hoc network one assumes that there don't exist any kind of fix infrastructure and that the nodes must use each other to be able to communicate. One example of an ad hoc network could be laptops connected together with infrared links during a meeting. When an ad hoc network is mobile it implies that the nodes are moving. In this work, the three protocols Epidemic, GeoMean and GeoMove together with the two mobility models Waypoint and Extended Random Walk, have been implemented in a newly written simulator for this kind of network. The two Geo-protocols are newly developed and aim to use geographical information to aid communication in this category of networks together with the new Extended Random Walk model.

Ce mémoire porte sur l’étude des maxima de champs gaussiens. Plus précisément, l’étude portera sur la convergence en loi, la convergence du premier ordre et la convergence du deuxième ordre du maximum d’une collection de variables aléatoires gaussiennes. Les modèles de champs gaussiens présentés sont le modèle i.i.d., le modèle hiérarchique et le champ libre gaussien. Ces champs gaussiens diffèrent par le degré de corrélation entre les variables aléatoires. Le résultat principal de ce mémoire sera que la convergence en probabilité du premier ordre du maximum est la même pour les trois modèles. Quelques résultats de simulations seront présentés afin de corroborer les résultats théoriques obtenus. / In this study, results about maxima of different Gaussian fields will be presented. More precisely, results for the convergence of the first order of the maximum of a set of Gaussian variables will be presented. Some results on the convergence of the second order, and of the law will also be explained. The models presented here are the Gaussian field of i.i.d. variables, the hierarchical model and the Gaussian free fields model. These fields differ from one another by their correlation structure. The main result of this study is that the first order convergence in probability of the maximum is the same for the three models. Finally, numerical simulations results will be presented to confirm theoretical results.

Ce mémoire étudie le comportement des particules dont la position est maximale au temps t dans la marche aléatoire branchante et le mouvement brownien branchant sur R, pour des valeurs de t grandes. Plus exactement, on regarde le comportement du maximum d’une marche aléatoire branchante dans un environnement inhomogène en temps, au sens où la loi des accroissements varie en fonction du temps. On compare avec des modèles connus ou simplifiés, en particulier le modèle i.i.d., où l’on observe des marches aléatoires indépendantes et le modèle de la marche aléatoire homogène. On s’intéresse par la suite aux corrélations entre les particules maximales d’un mouvement brownien branchant. Plus précisément, on étudie le temps de branchement entre deux particules maximales. Finalement, on applique les méthodes et les résultats des premiers chapitres afin d’étudier les corrélations dans un mouvement brownien branchant dans un environnement inhomogène. Le résultat principal du mémoire stipule qu’il y a existence de temps de branchement au centre de l’intervalle [0, t] dans le mouvement brownien branchant inhomogène, ce qui n’est pas le cas pour le mouvement brownien branchant standard. On présentera également certaines simulations numériques afin de corroborer les résultats numériques et pour établir des hypothèses pour une recherche future. / This thesis studies the behavior of particles that are maximal at time t in branching random walk and branching Brownian motion on R, for large values of t. Precisely, we look at the behavior of the maximum in a branching random walk in a time-inhomogeneous environment, where the law of the increments varies with respect to time. We compare with known or simplified models such as the model where random walks are taken to be i.i.d. and the branching random walk in a time-homogeneous environment model. We then take a look at the correlations between maximal particles in a branching brownian motion. Specifically, we look at the branching time between those maximal particles. Finally, we apply results and methods from the first chapters to study those same correlations in branching Brownian motion in a inhomogeneous environment. The thesis’ main result establishes existence of branching time at the center of the interval [0, t] for the branching Brownian motion in a inhomogeneous environment, which is not the case for standard branching brownian motion.We also present results of simulations that agree with theoretical results and help establishing new hypotheses for future research.

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Submitted by Erika Demachki (erikademachki@gmail.com) on 2014-11-13T16:45:09Z No. of bitstreams: 2 Dissertação - Daniel Antônio Mendonça da Silva - 2013.pdf: 1681960 bytes, checksum: 045f2940a2c8539d589a5c6227cf55e6 (MD5) license_rdf: 23148 bytes, checksum: 9da0b6dfac957114c6a7714714b86306 (MD5) / Approved for entry into archive by Erika Demachki (erikademachki@gmail.com) on 2014-11-13T16:45:43Z (GMT) No. of bitstreams: 2 Dissertação - Daniel Antônio Mendonça da Silva - 2013.pdf: 1681960 bytes, checksum: 045f2940a2c8539d589a5c6227cf55e6 (MD5) license_rdf: 23148 bytes, checksum: 9da0b6dfac957114c6a7714714b86306 (MD5) / Made available in DSpace on 2014-11-13T16:45:43Z (GMT). No. of bitstreams: 2 Dissertação - Daniel Antônio Mendonça da Silva - 2013.pdf: 1681960 bytes, checksum: 045f2940a2c8539d589a5c6227cf55e6 (MD5) license_rdf: 23148 bytes, checksum: 9da0b6dfac957114c6a7714714b86306 (MD5) Previous issue date: 2013-08-23 / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - CAPES / This work has the main objective to study a of the propagation of information sets N e Z:We consider three models of spreading information. The first model takes into account the power of each individual in transmitting information. The second model takes into account the power of every individual to receive information. In the third model this power is given by the amount of life and the probability that individuals have to jump right. / O trabalho tem como objetivo principal, o estudo da propagação de uma informação sobre os conjuntos N e Z: Consideramos três modelos de propagação da informação. O primeiro modelo leva em conta o poder de cada indivíduo em transmitir uma informação. O segundo modelo leva em conta o poder de cada indivíduo em receber informação. No terceiro modelo este poder é dado pela quantidade de vida e pelas probabilidades que os indivíduos têm de saltar para direita.

Estudamos um modelo de passeios aleatórios simples em grafos, conhecido como modelo dos sapos. Esse modelo pode ser descrito de maneira geral da seguinte forma: existem partículas ativas e partículas desativadas num grafo G. Cada partícula ativa desempenha um passeio aleatório simples a tempo discreto e a cada momento ela pode morrer com probabilidade 1-p. Quando uma partícula ativa entra em contato com uma partícula desativada, esta é ativada e também passa a realizar, de maneira independente, um passeio aleatório pelo grafo. Apresentamos limites superior e inferior para o parâmetro crítico de sobrevivência do modelo dos sapos na árvore, e demonstramos que este parâmetro crítico não é uma função monótona do grafo em que está definido. / We study a system of simple random walks on graphs, known as frog model. This model can be described generally speaking as follows: there are active and sleeping particles living on some graph G. Each particle performs a simple random walk with discrete time and at each moment it may disappear with probability 1 - p. When an active particle hits a sleeping particle, the latter becomes active and starts to perform, independently, a simple random walk on the graph. We present lower and upper bounds for the surviving critical parameter on the tree, and we show that this parameter is not a monotonic function of the graph it is defined on.

Wireless communication has grown very popular, and communication is the key to success in many situations. However, most of the common technologies today rely on infrastructure and in disaster situations infrastructure might be lost or get severely overloaded. This master thesis concerns intermittently connected mobile ad hoc networks. A network in which the devices may move freely in any direction and still be able to communicate. To be able to demonstrate a network protocol called random-walk gossip-based manycast (RWG) my assignment has been to implement this protocol using off-the-shelf hardware and software. RWG is a multi-hop and partition-tolerant mobile ad hoc manycast network protocol. Multi-hop refers to information being able to hop between more than two nodes in a network and partition-tolerant means that the protocol works even though a network is partitioned. Manycast means that the information should be successfully delivered to K of all the potential nodes in the area. The RWG protocol makes use of four different packet types, request to forward (REQF), ac- knowledgement (ACK), ok to forward (OKTF) and be silent (BS). The actual data being sent is carried by REQFs, and is referred to as messages. When a message is sent it takes what could be described as a random walk among the nodes in the network, hence the name. The implementation of the RWG protocol resides in user-space and depends on the IEEE 802.11b standard and the raw socket that is specified in the BSD socket API. It is written in C and was developed on a machine running Ubuntu. It runs on systems that use Linux 2.6 kernels and it supports cross-compiling for ARM based devices such as the Nokia N810 internet tablet and the Android dev phone 1. To be able to demonstrate the protocol I developed my own client application. Moreover, an already existing application for Android, Portable Open Search and Identification Tool (POSIT), was successfully extended to run on top of the RWG implementation. The extension was developed by people in the POSIT project and tested in a physical experiment covering five devices. The report covers the RWG protocol, the system choice, the implementation and the testing of the implementation.