Interacting Particle Systems: Percolation, Stochastic Domination and Randomly Evolving Environments

Abstract: In this thesis we first analyze the class of one-dependent trigonometric determinantal processes and show that they are all two-block-factors. We do this by constructing the two-block-factors explicitly. Second we investigate the dynamic stability of percolation for the stochastic Ising model and the contact process. This is a natural extension of what previously has been done for non-interacting particle systems. The main question we ask is: If we have percolation at a fixed time in a time-dependent but time-invariant system, do we have percolation at all times? A key tool in the analysis is the concept of $epsilon-$movability which we introduce here. We then proceed by developing and relating this concept to others previously studied. Finally, we introduce a new model which we refer to as the contact process in a randomly evolving environment. By using stochastic domination techniques we will investigate matters of extinction and that of weak and strong survival for this system. We do this by establishing stochastic relations between our new model and the ordinary contact process. In the process, we develop some sharp stochastic domination results for a hidden Markov chain and a continuous time analogue of this.