Random walk on random walks: low densities

We consider a random walker in a dynamic random environment given by a system of independent simple symmetric random walks. We obtain ballisticity results under two types of perturbations: low particle density, and strong local drift on particles. Surprisingly, the random walker may behave very differently depending on whether the underlying environment particles perform lazy or non-lazy random walks, which is related to a notion of permeability of the system. We also provide a strong law of large numbers, a functional central limit theorem and large deviation bounds under an ellipticity condition.Comment: 28 page

Some case studies of random walks in dynamic random environments

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 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.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 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 5UBL - phd migration 201

Avaliação de sobrevivência e desempenho de "SEEDLINGS" de Umbuzeiro ( Spondias tuberosa ) após dois períodos de incubação de substrato contendo farelo de mamona.

O umbuzeiro (Spondias tuberosa Arruda Câmara) é uma frutífera nativa da região Nordeste do Brasil, representando fonte de emprego e renda no período da safra, para as populações na área de ocorrência natural das plantas. Estudos indicam que o uso racional de farelo de mamona poderá disponibilizar às plantas quantidades suficiente de nutrientes, dispensando o uso complementar de fertilizante mineral, em especial os nitrogenados

Random walk on random walks

In this paper we study a random walk in a one-dimensional dynamic random environment consisting of a collection of independent particles performing simple symmetric random walks in a Poisson equilibrium with density ¿¿(0,8). At each step the random walk performs a nearest-neighbour jump, moving to the right with probability p° when it is on a vacant site and probability p· when it is on an occupied site. Assuming that p°¿(0,1) and p·¿12, we show that the position of the random walk satisfies a strong law of large numbers, a functional central limit theorem and a large deviation bound, provided ¿ is large enough. The proof is based on the construction of a renewal structure together with a multiscale renormalisation argument

Random walk on random walks

In this paper we study a random walk in a one-dimensional dynamic random environment consisting of a collection of independent particles performing simple symmetric random walks in a Poisson equilibrium with density ¿¿(0,8). At each step the random walk performs a nearest-neighbour jump, moving to the right with probability p° when it is on a vacant site and probability p· when it is on an occupied site. Assuming that p°¿(0,1) and p·¿12, we show that the position of the random walk satisfies a strong law of large numbers, a functional central limit theorem and a large deviation bound, provided ¿ is large enough. The proof is based on the construction of a renewal structure together with a multiscale renormalisation argument

Tentativa de controle de Hymenolepis nana através de tratamentos clínicos repetidos, com praziquantel, em uma comunidade fechada

Praziquantel was used repeatedly in an orphanage to determine its effectiveness in controlling hymenolepis. At the same time, the possible mechanisms of the transmission of this parasitosis were studied. The first group examined consisted of 161 people of which 109 were the orphanage children, who varied in age from days or months to eight years, and 52 adults, employees of the institution. Stool tests of all were made about every two months, using the Hoffman, Pons and Janer method; cure control was carried out by the same method seven to fourteen days after treatment. Every two weeks, H. nana eggs were searched for under the finger nails of the children, in insects, in domestic wastes, on door knobs and refrigerator handles, and on toilet flush knobs and strings. Water collected from rinsed urinals and shower floors was also tested. Eggs and larva of helminths and protozoa cysts were found in domestic waste and cockroaches and on door knobs. H. nana eggs were found in puddles of water left on shower floors and the rinse water of urinals. Every patient whose feces had H. nana eggs was treated with a single oral 25mg/kg dose of praziquantel, taken after lunch. In the four groups treated (66 patients in all), no important side effects were observed, and follow-up indicated 100% cure. For the 5th and last treatment, a group of both positive and negative (for H. nana) people was divided into sub-groups and treated with one (25mg/kg) or two doses of the medicine for a period of four days (total: 50mg/kg). Follow-up examination two months after treatment showed that only six patients were still eliminating eggs of the parasite; all belonged to the sub-group treated with a single dose of the drug. In spite of the treatments given an of the high percentage of cure, control of hymenolepiasis was not achieved.Foi feita tentativa de controle do Hymenolepis nana em uma comunidade fechada utilizando-se o praziquantel em repetidos tratamentos. Concomitantemente, foram estudados os prováveis mecanismos de transmissão da parasitose. A comunidade trabalhada possuia inicialmente 161 pessoas, sendo 109 crianças internas, com idade variando de dias e/ou meses a 8 anos, e de 52 adultos, funcionários da instituição. O diagnóstico parasitológico foi realizado aproximadamente de 2 em 2 meses em toda a população, pelo método de Hoffman, Pons e Janer, e o controle de cura, pelo mesmo método, entre o 7.º e o 14.º dia. Quinzenalmente foram realizadas pesquisas para ovos de H. nana no leito ungueal das crianças, em insetos, no lixo doméstico, nas maçanetas das portas e geladeiras, nos botões e cordões de descarga. Posteriormente examinou-se água recolhida dos urinóis e do chão do "box" do chuveiro. Todos os pacientes eliminando ovos de H. nana nas fezes foram tratados com praziquantel, após exame clínico, na dose única oral de 25mg/kg, após o almoço. Em 4 tratamentos realizados (66 pacientes), não foram observadas reações colterais importantes, e o controle de cura foi sempre de 100%. No 5.º e último tratamento, grupos de pacientes positivos e negativos para H. nana foram divididos em subgrupos e tratados com uma dose da droga (25mg/kg) ou duas doses espaçadas de 4 dias (total: 50mg/kg). No levantamento realizado dois meses após o tratamento, foram encontrados apenas 6 indivíduos eliminando ovos do parasita. Estes pertenciam ao subgrupo de crianças com himenolepíase tratado com uma única dose da droga. Ovos e larvas de helmintos e cistos de protozoários foram encontrados no lixo doméstico, insetos (baratas) e maçanetas de portas, enquanto ovos de H. nana só foram achados em água aspirada do "box" do chuveiro e da lavagem dos urinóis. Apesar da elevada percentagem de cura e dos vários tratamentos realizados, não se conseguiu o controle da himenolepíase

Qualidade de seedlings de umbuzeiro (Spondias tuberosa) em função de farelo de mamona no substratato de propagação.

O umbuzeiro (Spondias tuberosa Arruda Câmara) é uma frutífera nativa da região Nordeste do Brasil, representando fonte de emprego e renda no período da safra, para as populações na área de ocorrência natural das plantas. Estudos indicam que o uso racional de farelo de mamona poderá disponibilizar às plantas quantidades suficiente de nutrientes, dispensando o uso complementar de fertilizante mineral, em especial os nitrogenados

Random walk on random walks: Higher dimensions

We study the evolution of a random walker on a conservative dynamic random environment composed of independent particles performing simple symmetric random walks, generalizing results of [16] to higher dimensions and more general transition kernels without the assumption of uniform ellipticity or nearest-neighbour jumps. Specifically, we obtain a strong law of large numbers, a functional central limit theorem and large deviation estimates for the position of the random walker under the annealed law in a high density regime. The main obstacle is the intrinsic lack of monotonicity in higher-dimensional, non-nearest neighbour settings. Here we develop more general renormalization and renewal schemes that allow us to overcome this issue. As a second application of our methods, we provide an alternative proof of the ballistic behaviour of the front of (the discrete-time version of) the infection model introduced in [23]