Large scale dynamics and fluctuations in non-equilibrium stochastic particle systems.

Statistical Mechanics (SM) is a branch of theoretical physics which inquires the properties of systems made up of a large number of interacting components. Starting from the details of the dynamics describing the interaction among the smaller components, the goal of statistical mechanics is to determine the emergent collective behaviour of the system resulting from the interaction of its constitutive parts. In this way, SM constitutes a bridge between the microscopic and macroscopic worlds. Starting from the beginning of the last century, SM has proven to be successful in describing many phenomena related to condensed matter physics like, just to cite a few, ferromagnetism, law of gases, glass states etc. Notwithstanding, the fields of application of this subject goes far beyond pure physics and they reach artificial intelligence, quantitative sociology, economics, biology etc. For all these reasons, SM is a vibrant and active discipline which keeps holding the attention of the scientific community. To describe the emergent collective behaviours, we need to de ne macroscopic observables which are linked to microscopic quantities. In the present thesis, we will mainly focus on the density of particles in a system and their current, that is the net flow of particles in a specific direction. In particular, we will be interested in the stationary regime of the macroscopic current which is given by a suitable average over the microscopic and quickly fluctuating particle flow. When the detailed balance condition is satisfied the system is said to be in equilibrium, and on average there is no net stationary current. In general, a system may be able to exhibit many equilibrium configurations which can be selected by choosing suitable values of a tuning parameter, such as temperature. This abrupt change as a parameter varies is often called a phase transition and determining the critical thresholds among phases is one of the fundamental questions which SM tries to answer. Instead, when the macroscopic stationary current has a value different from zero, the system is in a Non-Equilibrium Stationary State (NESS) and this will be of central interest for the present work.