A study of the long time behavior of three stochastic processes: a particle system with mechanical interactions, the REM under metropolis dynamics and the two-type contact process

Abstract

In this research project we are interested in the following stochastic processes: a particle system with mechanical interactions, the Random Energy Model (REM) and the two-type contact process.The first model that we are interested in was presented by Fontes, Neves and Sidoravicius in 2000. In this paper, was defined a model with infinite particles and there exists a leftmost particle at time 0 (and denoted by t.p). A constant force acts over the t.p particle and the rest of the particles are neutral to this force. At time zero, all the particles are at rest and each particle has mass 1. Also, each neutral particle is labeled as a particle of type 0 with probability p or as a particle of type 1 with probability 1-p. The particles of type 1 are perfectly inelastic with respect to collisions with the t.p particle and after the collision, they increase the mass of the t.p by 1. The particles of type 0 are perfectly elastic with respect to collisions with the t.p particle. About this model we are interested in studying convergence in distribution for the process seen from the t.p. For the Random Energy Model (REM) model, we are interested in the studying the case with the metropolis dynamics. For this model, we want to obtain a time scale for which the rescaled process converges to a non trivial process.In the case of the contact process, we would like to study the contact process with two types of particles and priorities in the environment. In this case, we are interested in completing the study of the metastability phenomenon in dimension 1, and proving that this phenomenon exists in the case of dimension 2. Also, we believe that we can explicitly provide the set of extremal invariant measures for the contact process in dimension 1 with two types of particles and a fixed priority in the environment. (AU)