In this thesis we work on two different problems. First, we prove decoupling estimates for one-dimensional conservative particle systems. The second class of problems is related to noise sensitivity and sharp thresholds for percolation models. In our setting, a decoupling is a type of correlation estimate for monotone functions of the space-time configurations with far enough supports. We prove these estimates for two models: The exclusion process and the zero range process. These estimates are used to study processes evolving on top of these particle systems.
For the exclusion process, we consider a detection model: At time zero, place nodes on each site independently with probability p and let they evolve as a simple symmetric exclusion process. Also at time zero, a target is placed at the origin. The target moves only at integer times, and can move to any site that is within distance R from its current position. Assume also that the target can predict the future movement of all nodes. We prove that, for R large enough it is possible for the target to avoid detection forever with positive probability.
As for the decoupling of the zero range process, we use it to study the spread of an infection on top of this particle system. At time zero, the set of infected particles is composed by those which are in the negative axis, while particles at the right of the origin are considered healthy. A healthy particle immediately becomes infected if it shares a site with an infected particle. We prove that the front of the infection wave travels to the right with positive and finite velocities.
In the context of Boolean functions, noise sensitivity measures whether the outcome of such function can be predicted when one is given its value on a perturbation of the original configuration while threshold phenomena describes abrupt changes in the behavior of these functions.
We consider Poisson Voronoi percolation on R² and prove that box-crossing events in this model are noise sensitive and present a threshold phenomena with polynomial window.