Topics in interacting particle systems and anisotropic percolation models

• Autor(a) principal: Pablo Almeida Gomes
• Data de Publicação: 2020
• Tipo de documento: Tese
• Idioma: eng
• Título da fonte: Repositório Institucional da UFMG
• Texto Completo: http://hdl.handle.net/1843/38019; https://orcid.org/ 0000-0002-4472-3628
• topic: Contact process; Anisotropic percolation; Particle systems; Phase transition; Percolação (Física estatística) - Teses; Teoria do controle estocástico - Teses; Transformações de fase (Física estatística) - Teses

This thesis consists of the presentation of works on stochastic processes, more precisely, on interacting particle systems and anisotropic percolation models. The first topic studies the renewal contact process. We study an infection propagation on a finite population. We consider a finite and connected graph where an individual is attached to each vertex. The population starts with a single infected individual and the infection propagates through neighbors according to marks of independent Poisson processes. For each individual, recovery occurs according to marks of a renewal process with heavy-tailed stable law, associated to its vertex. We show a phase transition for infection survival, according to the size of the graph. In the second topic we study a half-line mechanical particle system. A constant force acts solely on the charged particle starting at origin, producing in it an accelerated motion to the right. All the other particles are force neutral and initially randomly placed in space. Collisions will thus take place in the system. We establish central limit theorems for the position and velocity of the charged particle. Finally, we study the phase diagram of the Bernoulli edge percolation model. A well-known phenomenon for independent percolation on the d-dimensional hipercubic lattice is that, in some sense, for d large, it resembles the model on the (d+1)-regular tree. We investigate this phenomenon for anisotropic percolation models. On the oriented case, we show that, if the sum of the local probabilities is strictly greater than one and each of them is not too large, then percolation occurs. We also show that, for d larger than 10, the crossover critical exponent is equal to one, the same value known for regular trees.